Graphs for the Analysis of Bipolar Fuzzy Information [1st ed.] 9789811587559, 9789811587566

Citation preview

Studies in Fuzziness and Soft Computing

Muhammad Akram Musavarah Sarwar Wieslaw A. Dudek

Graphs for the Analysis of Bipolar Fuzzy Information

Studies in Fuzziness and Soft Computing Volume 401

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by ISI, DBLP and Ulrichs, SCOPUS, Zentralblatt Math, GeoRef, Current Mathematical Publications, IngentaConnect, MetaPress and Springerlink. The books of the series are submitted for indexing to Web of Science.

More information about this series at http://www.springer.com/series/2941

Muhammad Akram Musavarah Sarwar Wieslaw A. Dudek •

•

Graphs for the Analysis of Bipolar Fuzzy Information

123

Muhammad Akram Department of Mathematics University of the Punjab Lahore, Pakistan

Musavarah Sarwar Department of Mathematics Government College Women University Sialkot, Pakistan

Wieslaw A. Dudek Faculty of Pure and Applied Mathematics Wrocław University of Technology Wrocław, Poland

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-981-15-8755-9 ISBN 978-981-15-8756-6 (eBook) https://doi.org/10.1007/978-981-15-8756-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

We dedicate this book to the memory of Professor Lotfi Zadeh!

Foreword

Graphs are considered as one of the fundamental structures studied in combinatorics. The subject started with the paper published by Euler on the seven bridges of Königsberg in 1736. Graph theory is highly applied in other areas of mathematics and other branches of science. However, “crisp graphs” are not appropriate for describing all of the existing relations between objects. Thus, Kauffman (1973) presented the first definition of fuzzy graphs based on Zadeh’s fuzzy relations. In addition, Rosenfeld (1975) introduced “fuzzy analogs” of several basic graph-theoretic concepts. In many real-world problems, “bipolarity” plays a role when it is important to distinguish between positive information representing satisfaction degree and negative information indicating dissatisfaction degree. Both types of information cannot be understood simultaneously by the fuzzy model, leading to the existence of bipolar information. This type of information cannot be well-represented by means of fuzzy graphs. In these situations, bipolar fuzzy set theory is applied to the graphs in order to describe the relationships. Muhammad Akram (2011) first introduced the notion of bipolar fuzzy graphs and published different papers with his colleagues and students on a wide variety of bipolar fuzzy graph-theoretic structures. This book presents some fundamental theories of such types of bipolar fuzzy graphs, complex bipolar fuzzy graphs, Cayley bipolar fuzzy graphs, bipolar fuzzy planar graphs, bipolar fuzzy matroids, double dominating energy of bipolar fuzzy graphs, bipolar single-valued neutrosophic competition graphs, and bipolar single-valued neutrosophic graph structures. The authors are well-known researchers in the field of fuzzy graphs. They have presented all concepts of bipolar fuzzy graph theory with applications in a variety of diverse fields and have paid careful attention to algorithmic issues. Therefore, the book can contribute to the development of fuzzy graph theory for students and researchers. Rajab Ali Borzooei Shahid Beheshti University Tehran, Iran

vii

Preface

The motivation of this monograph can be traced back to the work of late Lotfi A. Zadeh (1921–2017). Professor Zadeh introduced fuzzy set theory in 1965, as a model for the representation of uncertainty and vagueness in real-world systems. A mathematical framework—fuzzy logic—was initiated in order to give formal support to the notion of partial truth between “absolute true” and “absolute false.” This meant one of the most important, creative, and fruitful concepts introduced in science and technology during the second half of the twentieth century. One of its interactions with other branches of mathematics produced the concept of fuzzy graphs, which is based on Zadeh’s fuzzy relations, originated with Kafmann. In 1975, Rosenfeld laid the foundations for fuzzy graph theory. Mordeson and Nair also made early contributions to that field. Some aspects of data uncertainty are likely to appear in many contexts and scientific disciplines. Thus, different forms of vagueness have been recognized in applied fields. Some come from conflicting or incomplete information, as well as from multiple interpretations of some phenomena. In applications including expert systems, belief systems, and information fusion, it is not only convenient to use the one real value from the unit interval [0, 1] supported by the evidence, but also useful to consider the counter-membership against the evidence. Such ‘bipolarity’ is a core feature in many domains of information processing, both from the point of view of knowledge representation and from the perspective of processing and reasoning. Bipolarity appears whenever it is important to distinguish between (i) positive information that represents what is possible or preferred and (ii) negative information that represents what is impossible or forbidden. Both inputs cannot be simultaneously discerned by the fuzzy model. Motivated by this concern, Zhang introduced in 1994 a new component that expresses a negative degree of membership in addition to the structure of a fuzzy set. This idea launched the concept of bipolar fuzzy set that inspires the concept to which this monograph is devoted. With this work we aim at filling a gap, to wit, the lack of a mathematical approach towards bipolar information, that is, the coexistence of positive and negative information.

ix

x

Preface

This volume is based on a number of papers by the authors that have been published in various scientific journals. They are part of a long-lasting research program that concerns decision-making methods under bipolar fuzzy graphical models. Now they are wrapped together in a book that may be useful for researchers in mathematics, computer scientists, and social scientists alike. Let us describe its content. In Chap. 1, we review the notion of bipolar fuzzy sets. Then, we present the concept of bipolar fuzzy graphs and describe various methods of their construction. We discuss the concept of isomorphism of bipolar fuzzy graphs and investigate their fundamental properties. We present certain types of bipolar fuzzy graphs, complex bipolar fuzzy graphs, bipolar fuzzy walk, bipolar fuzzy bridge, strength of connectedness, and weak and strong bipolar fuzzy edges. We also endorse the importance of bipolar fuzzy digraphs with a number of real-world problems. In Chap. 2, we discuss certain properties of distance functions in bipolar fuzzy graphs. We establish the formulae of distance in complete bipolar fuzzy graphs, complete bipartite bipolar fuzzy graphs and products of bipolar fuzzy graphs. We present an algorithm for computing the distance matrix, eccentricity of the vertices, radius, and diameter in bipolar fuzzy graphs. We study the concepts of self-centered bipolar fuzzy graphs and antipodal bipolar fuzzy graphs. We describe applications of bipolar fuzzy graphs in product manufacturing. In Chap. 3, we present certain types of bipolar fuzzy graphs with various methods of construction. We discuss the concept of degree and isomorphism of irregular bipolar fuzzy graphs. We describe the concepts of bipolar fuzzy line graphs and the Cayley bipolar fuzzy graphs. In Chap. 4, we present different types of concepts concerning bipolar fuzzy competition graphs and their extensions. Various methods for the construction of bipolar fuzzy competition graphs of certain products of bipolar fuzzy digraphs are studied with theoretical proofs. We study certain types of competition graphs under complex bipolar fuzzy environment. We describe applications of bipolar fuzzy competition graphs in different domains. In Chap. 5, we apply the powerful technique of bipolar fuzzy sets to multigraphs and planar graphs. We present the notions of bipolar fuzzy multigraphs, bipolar fuzzy planar graphs, and bipolar fuzzy dual graphs. Chapter 6 considers various types of dominations in bipolar fuzzy graphs and their importance in decision-making models. In Chap. 7, we study the theory of matroids and circuits in bipolar fuzzy environment, soft environment, and bipolar fuzzy soft environment. We present certain applications of bipolar fuzzy matroids in decision support systems, secret sharing problems, and network analysis. Chapter 8 is concerned with various energy-like quantities of bipolar fuzzy graphs. In particular, we study Laplacian energy, signless Laplacian energy, dominating energy, and double dominating energy of bipolar fuzzy graphs. Two case studies illustrate these concepts.

Preface

xi

In Chap. 9, we present the concepts of bipolar single-valued neutrosophic competition graphs, bipolar neutrosophic single-valued p-competition graphs, and m-step bipolar single-valued neutrosophic competition graphs. We describe some applications of bipolar single-valued neutrosophic competition graphs and m-step bipolar single-valued neutrosophic competition graphs in economics, organizational designations, and business competition. In Chap. 10, we apply the concept of bipolar single-valued neutrosophic set to graph structures and study a framework to handle bipolar neutrosophic information by combining bipolar neutrosophic sets with graph structures. We study several operations on bipolar single-valued neutrosophic graph structures with interesting properties. We describe the isomorphism properties of various types of bipolar single-valued neutrosophic graph structures with u-complement. We explain the importance of bipolar single-valued neutrosophic graph structures with real-world applications in international relations, psychology, and global terrorism. Acknowledgements: The authors are grateful to the administration of University of Punjab, Lahore, for providing the facilities, which were required for successful completion of this monograph. Musavarah Sarwar is also thankful to the administration of Government College Women University, Sialkot, Pakistan, for providing the facilities which were required for writing this book. Lahore, Pakistan Sialkot, Pakistan Wrocław, Poland

Muhammad Akram Musavarah Sarwar Wieslaw A. Dudek

Contents

1

2

Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Bipolar Fuzzy Graphs . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Isomorphism of Bipolar Fuzzy Graphs . . . . . . . . . . . . 1.4 Complex Bipolar Fuzzy Graphs . . . . . . . . . . . . . . . . . 1.5 Bipolar Fuzzy Digraphs . . . . . . . . . . . . . . . . . . . . . . . 1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Bipolar Fuzzy Neurons in Medical Diagnosis . 1.6.2 Social Networking . . . . . . . . . . . . . . . . . . . . 1.6.3 Bipolar Fuzzy Organizational Model . . . . . . . 1.6.4 Bipolar Fuzzy Graphs in Marketability . . . . . 1.6.5 Vulnerability Assessment of Gas Pipeline Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.6 Bipolar Fuzzy Digraph in Travel Time . . . . . 1.6.7 Comparison Analysis . . . . . . . . . . . . . . . . . . 1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance Measures in Bipolar Fuzzy Graphs . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Distance in Bipolar Fuzzy Graphs . . . . . . . . . . 2.3 Metric in Bipolar Fuzzy Graphs . . . . . . . . . . . . 2.4 Self-centered Bipolar Fuzzy Graphs . . . . . . . . . 2.5 Antipodal Bipolar Fuzzy Graphs . . . . . . . . . . . 2.6 Applications of Bipolar Fuzzy Graphs . . . . . . . 2.6.1 Product Manufacturing . . . . . . . . . . . . 2.6.2 Safe Route and Shortest Path Problem .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

1 1 7 26 42 51

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

60 61 63 64 66

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

68 70 74 74 75 78

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

81 81 82 96 103 112 122 122 122

xiii

xiv

Contents

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Exercises 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3

Special Types of Bipolar Fuzzy Graphs . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Regular and Irregular Bipolar Fuzzy Graphs . 3.3 Bipolar Fuzzy Line Graphs . . . . . . . . . . . . . 3.4 Cayley Bipolar Fuzzy Graphs . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . Exercises 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

127 127 128 141 147 157 158 158

4

Bipolar Fuzzy Competition Graphs . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bipolar Fuzzy Competition Graphs . . . . . . . . . . . . . . . . . 4.3 Bipolar Fuzzy k-competition Graphs . . . . . . . . . . . . . . . . 4.4 Complex Bipolar Fuzzy Competition Graphs . . . . . . . . . . 4.5 Applications of Bipolar Fuzzy Competition Graphs . . . . . . 4.5.1 Variants of Bipolar Fuzzy Competition Graphs in Food Webs . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Competitive Market . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Political Competition . . . . . . . . . . . . . . . . . . . . . 4.5.4 Social Competition in Bipolar Fuzzy Environment 4.5.5 Interactions and Conflicts in Communication Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

161 161 163 187 191 195

. . . .

. . . .

. . . .

. . . .

196 205 206 210

. . . .

. . . .

. . . .

. . . .

213 219 220 221

Bipolar Fuzzy Planar Graphs . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Bipolar Fuzzy Planar Graphs . . . . . . . . . . . . . . . . . . . 5.3 Bipolar Fuzzy Bridges . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Bipolar Fuzzy Cut Vertices and Bipolar Fuzzy Blocks 5.5 Bipolar Fuzzy Cycles and Bipolar Fuzzy Trees . . . . . . 5.6 Applications of Bipolar Fuzzy Planar Graphs . . . . . . . 5.6.1 Road Network Model to Monitor Traffic . . . . 5.6.2 Modeling of Electrical Connections . . . . . . . . 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

223 223 225 234 238 241 246 247 248 249 250 250

5

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . . . . . .

. . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

Contents

xv

6

Domination in Bipolar Fuzzy Graphs . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Types of Domination in Bipolar Fuzzy Graphs . . . . . . . . . 6.3 Irredundance in Bipolar Fuzzy Graphs . . . . . . . . . . . . . . . 6.4 Applications of Domination in Decision-Making Problems 6.4.1 Facility Location Problem . . . . . . . . . . . . . . . . . . 6.4.2 Representatives in Youth Development Council . . 6.4.3 Transmission Tower Location Problem . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

253 253 254 272 274 274 275 277 278 279 279

7

Bipolar Fuzzy Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Bipolar Fuzzy Circuits . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Circuit Rectangles . . . . . . . . . . . . . . . . . . . 7.3 Bipolar Fuzzy Soft Circuits . . . . . . . . . . . . . . . . . . . 7.4 Decision Support Systems . . . . . . . . . . . . . . . . . . . . 7.4.1 Ordering of Machines Using Bipolar Fuzzy Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Network Analysis . . . . . . . . . . . . . . . . . . . . 7.4.3 Ordering of Tasks Using Bipolar Fuzzy Soft Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

281 281 282 294 295 301

8

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . . . . 301 . . . . . . . . 303 . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

304 305 305 306

Energy of Bipolar Fuzzy Graphs . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Energy of Bipolar Fuzzy Graphs . . . . . . . . . . . . . . . . . 8.3 Laplacian Energy of Bipolar Fuzzy Graphs . . . . . . . . . 8.4 Signless Laplacian Energy of Bipolar Fuzzy Graphs . . . 8.5 Dominating and Double Dominating Energy of Bipolar Fuzzy Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Applications of Energy to Decision-Making Problems . . 8.6.1 Smooth Communication Problem . . . . . . . . . . 8.6.2 Selection of a Business Partner . . . . . . . . . . . . 8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

309 309 311 317 324

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

328 335 335 338 343 343 347

. . . .

xvi

9

Contents

Bipolar Neutrosophic Competition Graphs . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Certain Types of Bipolar Neutrosophic Competition Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Bipolar Neutrosophic Neighborhood Graphs . . . . . . . 9.4 Applications of Competition Graphs in Bipolar Neutrosophic Environment . . . . . . . . . . . . . . . . . . . 9.4.1 Designation Competition in an Organization 9.4.2 Competition in Textile Market . . . . . . . . . . 9.4.3 Competition in Sports . . . . . . . . . . . . . . . . . 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . 349 . . . . . . . . 349 . . . . . . . . 350 . . . . . . . . 369 . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

377 379 382 385 389 390 390

10 Bipolar Neutrosophic Graph Structures . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Operations on Bipolar Neutrosophic Graph Structures . . 10.3 Isomorphism in Bipolar Neutrosophic Graph Structures 10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 International Relations . . . . . . . . . . . . . . . . . . 10.4.2 Psychological Improvement of Patients . . . . . . 10.4.3 Uncovering the Undercover Reasons of Global Terrorism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

393 393 394 416

. . . . . . .

. . . . . . 427 . . . . . . 427 . . . . . . 433 . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

438 444 444 446

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

About the Authors

Muhammad Akram is Professor at the Department of Mathematics, University of the Punjab, Lahore, Pakistan. He previously served at Punjab University College of Information Technology as Assistant Professor and Associate Professor. He earned his Ph.D. in Fuzzy Mathematics from Government College University, Lahore, Pakistan. His research interests include numerical algorithms, fuzzy graphs, fuzzy algebras, and fuzzy decision support systems. He has published 9 books and over 365 research articles in peer-reviewed international journals and supervised 12 Ph.D. students. Musavarah Sarwar is Assistant Professor at Government College Women University, Sialkot, Pakistan. A gold medalist in academic excellence, Dr. Sarwar received her Ph.D. in Mathematics from the University of the Punjab, Lahore, Pakistan. Her research interests include graph theory, fuzzy graphs and fuzzy decision support systems/decision-making models. She has published 18 research articles in international peer-reviewed journals. Wieslaw A. Dudek is Professor at the Department of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wrocław, Poland. He completed his Ph.D. in Mathematics at the Department of Higher Algebra, State University of Chisinau, Republic of Moldova, under the supervision of Prof. V. D. Belousov. He received his Degree of Habilitated Doctor from the Warsaw University of Technology. The main areas of his scientific interests include polyadic groups, quasigroups, fuzzy algebras, fuzzy graphs and their applications. Professor Dudek has authored 2 monographs, 4 textbooks, and published 200 research articles in international peer-reviewed journals. He is Editor-in-Chief of the journal Quasigroups and Related Systems and on the editorial board of several international journals. He has promoted 4 doctors of mathematics.

xvii

List of Figures

Fig. 1.1 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29

Order of magnitude ð  Þ of crisp and bipolar fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph . . . . . . . . . . . . . . . . . . . . . . Union of bipolar fuzzy graphs . . . . . . . . . . . . . . Bipolar fuzzy graph G1 . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G2 . . . . . . . . . . . . . . . . . . . . Union G1 [ G2 of bipolar fuzzy graphs . . . . . . . . Intersection of bipolar fuzzy graphs G1 and G2 . Join of bipolar fuzzy graphs G1 and G2 . . . . . . . Cartesian product G1 hG2 . . . . . . . . . . . . . . . . . . Direct product G1  G2 . . . . . . . . . . . . . . . . . . . . Strong product G1 hG2 . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy path P 5 . . . . . . . . . . . . . . . . . . . . . Strong bipolar fuzzy graph G . . . . . . . . . . . . . . . Complete bipolar fuzzy graph . . . . . . . . . . . . . . . Bipartite bipolar fuzzy graph. . . . . . . . . . . . . . . . Complete bipartite bipolar fuzzy graph . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . Complement of G . . . . . . . . . . . . . . . . . . . . . . . . Union of strong bipolar fuzzy graphs . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . 0 Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G3 . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G4 . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G1 . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G2 . . . . . . . . . . . . . . . . . . . . Strength of connectedness . . . . . . . . . . . . . . . . . . Strongest bipolar fuzzy path . . . . . . . . . . . . . . . . Phase term in bipolar fuzzy set . . . . . . . . . . . . . . Phase term in complex bipolar fuzzy set . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 9 10 11 11 1 13 14 17 18 19 21 22 22 23 24 24 25 26 28 28 34 34 36 36 40 40 43 44

xix

xx

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

List of Figures

1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26

Complex bipolar fuzzy graph . . . . . . . . . . . . . . . . . . . Complex bipolar fuzzy graph G . . . . . . . . . . . . . . . . . Complement of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ ¼ ðA; ~ Bipolar fuzzy digraph G BÞ . . . . . . . . . . . . . . . . Union of bipolar fuzzy digraphs . . . . . . . . . . . . . . . . . ~1 and G ~2 . . . . Intersection of bipolar fuzzy digraphs G ~ ~ Join of bipolar fuzzy digraphs G1 and G2 . . . . . . . . . . ~4 . . . . . . . . . . . . . . . . . . . . . . ~ 4 hP Cartesian product P ~ ~ Direct product G  P 3 . . . . . . . . . . . . . . . . . . . . . . . . ~4 . . . . . . . . . . . . . . . . . . . . . . . . ~ 4 P Strong product P ~3 . . . . . . . . . . . . . . . . . . ~3  P Lexicographic product P Bipolar fuzzy influence graph . . . . . . . . . . . . . . . . . . . Bipolar fuzzy organizational model . . . . . . . . . . . . . . . Bipolar fuzzy digraph of marketability . . . . . . . . . . . . Weighted digraph of a gas pipeline network . . . . . . . . Bipolar fuzzy graph of a road network . . . . . . . . . . . . Weighted digraph of bipolar fuzzy road network . . . . Cartesian product G1 hG2 . . . . . . . . . . . . . . . . . . . . . . Direct product G1  G2 . . . . . . . . . . . . . . . . . . . . . . . . Strong product G1 G2 . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy path P 5 . . . . . . . . . . . . . . . . . . . . . . . . . Cartesian product P 4 hP 4 . . . . . . . . . . . . . . . . . . . . . . Direct product P 3  P 3 . . . . . . . . . . . . . . . . . . . . . . . Direct product P 3  C3 . . . . . . . . . . . . . . . . . . . . . . . . Strong product P 3 P 3 . . . . . . . . . . . . . . . . . . . . . . . . Lexicographic product P 3  P 3 . . . . . . . . . . . . . . . . . . Radius and diameter of a bipolar fuzzy graph G. . . . . Eccentric vertices in bipolar fuzzy graph G . . . . . . . . Center of bipolar fuzzy graph G . . . . . . . . . . . . . . . . . Distance in bipolar fuzzy graph G . . . . . . . . . . . . . . . Self-centered bipolar fuzzy graph G . . . . . . . . . . . . . . Self-centered bipolar fuzzy graph . . . . . . . . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Antipodal bipolar fuzzy graph AðGÞ . . . . . . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Antipodal bipolar fuzzy graph AðGÞ . . . . . . . . . . . . . . Complement Gc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antipodal bipolar fuzzy graph AðGc Þ . . . . . . . . . . . . . ðAðGÞÞc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Antipodal bipolar fuzzy graph AðGÞ . . . . . . . . . . . . . . Complement Gc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antipodal bipolar fuzzy graph AðGc Þ . . . . . . . . . . . . .

....... ....... ....... ....... ....... ....... ....... ....... ....... ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

46 52 52 53 54 54 55 56 57 58 59 64 66 67 69 71 72 83 84 85 86 89 91 92 94 96 98 101 101 104 105 107 113 113 117 117 118 118 118 119 119 119 120

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

2.27 2.28 2.29 2.30 2.31 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

3.15 3.16 3.17 3.18 3.19 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Fig. Fig. Fig. Fig. Fig.

4.8 4.9 4.10 4.11 4.12

Fig. Fig. Fig. Fig. Fig.

4.13 4.14 4.15 4.16 4.17

xxi

ðAðGÞÞc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-median bipolar fuzzy graph . . . . . . . . . . . . . . . . . Bipolar fuzzy model of product manufacturing . . . . . . Bipolar fuzzy model of shortest path problem . . . . . . Bipolar fuzzy graphs . . . . . . . . . . . . . . . . . . . . . . . . . . Regular bipolar fuzzy graph . . . . . . . . . . . . . . . . . . . . Totally regular bipolar fuzzy graph . . . . . . . . . . . . . . . Regular and totally regular bipolar fuzzy graph G . . . Irregular bipolar fuzzy graph G. . . . . . . . . . . . . . . . . . Totally irregular bipolar fuzzy graph. . . . . . . . . . . . . . Complete bipolar fuzzy graph . . . . . . . . . . . . . . . . . . . Neighborly irregular bipolar fuzzy graph . . . . . . . . . . Neighborly totally irregular bipolar fuzzy graph . . . . . Highly irregular bipolar fuzzy graph . . . . . . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Neighborly irregular bipolar fuzzy graph . . . . . . . . . . Neighborly totally irregular bipolar fuzzy graph . . . . . Neighborly irregular bipolar fuzzy graph . . . . . . . . . . Bipolar fuzzy subgraph H which is not neighborly irregular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Totally irregular bipolar fuzzy graph G. . . . . . . . . . . . Bipolar fuzzy subgraph which is not totally irregular . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy line graph LðGÞ . . . . . . . . . . . . . . . . . . Cayley bipolar fuzzy graph . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy digraph . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy competition graph . . . . . . . . . . . . . . . . . Bipolar fuzzy digraph . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy competition graph . . . . . . . . . . . . . . . . . ~1 and G ~2 . . . . . . . . . . . . . . . Bipolar fuzzy digraphs G Bipolar fuzzy competition graphs . . . . . . . . . . . . . . . . Bipolar fuzzy graph GCðG~1 Þ hCð~G2 Þ [ Gh . . . . . . . . . . . ~ 1 hG ~2 . . . . . . . . . . . . . . . . . . . . . . Cartesian product G ~1 hG ~2 Þ . . . . . . . . Bipolar fuzzy competition graph CðG ~ ~ Bipolar fuzzy digraphs G5 and G6 . . . . . . . . . . . . . . . ~5 Þ and CðG ~6 Þ . . Bipolar fuzzy competition graphs CðG Bipolar fuzzy graph GCðG~5 Þ hCð~G6 Þ [ Gh . . . . . . . . . . . ~6 . . . . . . . . . . . . . . . . . . . . . . ~ 5 hG Cartesian product G ~5 hG ~6 Þ . . . . . . . . Bipolar fuzzy competition graph CðG ~3 and G ~4 . . . . . . . . . . . . . . . Bipolar fuzzy digraphs G Bipolar fuzzy competition graphs Cð~ G3 Þ and Cð~ G4 Þ . . ~3 Þ  CðG ~4 Þ [ G . . . . . . . . . Bipolar fuzzy graph ½CðG

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

120 121 123 124 125 129 130 130 132 132 132 134 135 135 136 137 138 140

140 141 141 142 143 149 164 164 165 166 . . . . . . . 170 . . . . . . . 171

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . 173 . . . . . . . 173 . . . . . . . 174 . . . . . . . 174 . . . . . . . 175 . . . . . . . 177 . . . . . . . 178 . . . . . . . 178 . . . . . . . 181 . . . . . . . 181 . . . . . . . 182

xxii

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 6.1 6.2

~3  G ~4 . . . . . . . . . . . . . . . . . . . . . . . . Direct product G ~3  G ~4 Þ . . . . . . . Bipolar fuzzy competition graph CðG ~ ~ Bipolar fuzzy digraphs G5 and G6 . . . . . . . . . . . . . . . Bipolar fuzzy competition graphs Cð~ G5 Þ and Cð~ G6 Þ . .  ~ ~ Bipolar fuzzy graph ½CðG5 Þ  CðG6 Þ [ G . . . . . . . . . ~5  G ~6 . . . . . . . . . . . . . . . . . . . . . Cartesian product G ~5  G ~6 Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CðG Bipolar fuzzy food web . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy competition graph of bipolar fuzzy food web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy common enemy graph . . . . . . . . . . . . . Bipolar fuzzy food web . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy competition common enemy graph . . . . Bipolar fuzzy niche graph . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy marketing digraph . . . . . . . . . . . . . . . . . Bipolar fuzzy competition graph of business network . Bipolar fuzzy digraph of political seats . . . . . . . . . . . . Bipolar fuzzy political competition graph . . . . . . . . . . Bipolar fuzzy social digraph . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy influence graph . . . . . . . . . . . . . . . . . . . Communication over a noisy channel . . . . . . . . . . . . . Bipolar fuzzy conflict graph . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy noisy channel . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy confusion graph . . . . . . . . . . . . . . . . . . Bipolar fuzzy graphs . . . . . . . . . . . . . . . . . . . . . . . . . . ~1 and G ~2 . . . . . . . . . . . . . . . Bipolar fuzzy digraphs G Bipolar fuzzy multigraph . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy complete multigraph . . . . . . . . . . . . . . . Bipolar fuzzy planar graph . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy faces in G . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy dual graph. . . . . . . . . . . . . . . . . . . . . . . Connected bipolar fuzzy graph . . . . . . . . . . . . . . . . . . Bipolar fuzzy bridges . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy cut vertices in G ¼ ðA; BÞ . . . . . . . . . . . Bipolar fuzzy block. . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy block which is not a firm . . . . . . . . . . . Bipolar fuzzy cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial bipolar fuzzy graph cycle . . . . . . . . . . . . . . . . . Bipolar fuzzy road model . . . . . . . . . . . . . . . . . . . . . . Electrical connections . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy planar graph . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Domination numbers of bipolar fuzzy graph . . . . . . . .

. . . . . . . 183 . . . . . . . 183 . . . . . . . 184 . . . . . . . 184 . . . . . . . 185 . . . . . . . 186 . . . . . . . 186 . . . . . . . 197 . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

198 200 202 203 204 206 207 209 210 212 214 215 216 218 219 220 221 226 227 228 231 233 235 236 238 240 241 242 242 247 248 249 255 256

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 7.1 7.2 7.3 7.4 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10

xxiii

EN degree of vertices in G . . . . . . . . . . . . . . . . . . . . . Equitable isolated vertex . . . . . . . . . . . . . . . . . . . . . . . Equitable dominating set of G . . . . . . . . . . . . . . . . . . Degree equitable bipolar fuzzy graph . . . . . . . . . . . . . EI set of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong equitable dominating set of G . . . . . . . . . . . . . Total ED set of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total 2-dominating set of G . . . . . . . . . . . . . . . . . . . . Restrained dominating set of G . . . . . . . . . . . . . . . . . . Global RD set of G . . . . . . . . . . . . . . . . . . . . . . . . . . Direct product of G1 and G2 . . . . . . . . . . . . . . . . . . . . Irredundant sets of G . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph of towns . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph of representatives . . . . . . . . . . . . Bipolar fuzzy graph of cities . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph . . . . . . . . . . . . . . . . . . . . . . . . . . 3polar fuzzy multigraph . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy soft graph G ¼ fHðe1 Þ; Hðe2 Þg: . . . . . . Wireless communication . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy digraph . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . ~... .... .... .... .... .... Bipolar fuzzy digraph G Bipolar fuzzy graph . . . . . . . . . . . . . . . . . . . . . . . . . . Signless Laplacian energy of bipolar fuzzy digraph . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy graph G . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy digraphs . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy digraphs . . . . . . . . . . . . . . . . . . . . . . . . ~ .......................... Cartesian product G ~... .... .... .... .... .... Bipolar fuzzy digraph G Bipolar fuzzy digraph G . . . . . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic graph G . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic digraph . . . . . . . . . . . . . . . . . . . ~ ................. Bipolar neutrosophic digraph G Bipolar neutrosophic competition graph . . . . . . . . . . . Bipolar neutrosophic digraph . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic 3-competition graph . . . . . . . . . . ~ ................. Bipolar neutrosophic digraph G ~ ................. Bipolar neutrosophic digraph G ~ ................. Bipolar neutrosophic digraph G 2-step bipolar single-valued neutrosophic competition graph . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

....... . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

....... ....... .......

260 260 261 261 262 263 265 266 267 267 269 273 275 276 277 284 289 297 304 312 317 318 323 325 327 329 334 337 339 343 344 344 353 354 355 356 357 357 359 360 360

. . . . . . . 361

xxiv

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig.

9.11 9.12 9.13 9.14 9.15 9.16

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21 10.22

Bipolar Bipolar Bipolar Bipolar Bipolar Bipolar

neutrosophic graph G . . . . . . . . . . . . . . . . . . . . . . . . neutrosophic open neighborhood graph N ðGÞ . . . . . neutrosophic closed neighborhood graph N ½G . . . . neutrosophic digraphs . . . . . . . . . . . . . . . . . . . . . . . . neutrosophic competition graphs . . . . . . . . . . . . . . . neutrosophic graph GCðG~1 Þ hCðG~2 Þ [ Gh . . . . . . . . . . ~ 1 hG ~2 . . . . . . . . . . . . . . . . . . . . . . . . . . . Cartesian product G ~ 1 hG ~2 Þ . . . . . . . Bipolar neutrosophic competition graph CðG Bipolar neutrosophic organization model . . . . . . . . . . . . . . . Bipolar neutrosophic competition graph . . . . . . . . . . . . . . . . Bipolar neutrosophic market model. . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic market competition graph . . . . . . . . . . Bipolar neutrosophic digraph of players . . . . . . . . . . . . . . . . 4-step bipolar single-valued neutrosophic competition graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic graph structure no. 1 . . . . . . . . . . . . . . Bipolar neutrosophic graph structure no. 2 . . . . . . . . . . . . . . Bipolar neutrosophic subgraph structure . . . . . . . . . . . . . . . . Bipolar neutrosophic induced subgraph structure . . . . . . . . . Bipolar neutrosophic spanning subgraph structure . . . . . . . . Strong bipolar single-valued neutrosophic graph structure . . Complete bipolar neutrosophic graph structure . . . . . . . . . . . Bipolar neutrosophic graph structures . . . . . . . . . . . . . . . . . .  b1  G  b2 . . . . . . . . . . . . . . . . . . . . . Lexicographic product G  b1 G  b2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong product G  b1 G  b2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong product G   Union Gb1 [ Gb2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  b1 þ G  b2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Join G  b1 . . . . Bipolar single-valued neutrosophic graph structure G  Bipolar neutrosophic graph structure Gb2 . . . . . . . . . . . . . . .  b1 . . . . . . . . . . . . . . . Bipolar neutrosophic graph structure G  b2 . . . . . . . . . . . . . . . Bipolar neutrosophic graph structure G wc   Gbn and u-complement Gbn . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic graph structure . . . . . . . . . . . . . . . . . . . International relations of countries . . . . . . . . . . . . . . . . . . . . Psychological behaviors of patients in a Mental Hospital . . . Highlighting undercover reasons of global terrorism . . . . . . .

. . . . . .

. . . . . .

.. . . . . . .

. . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

370 371 371 374 375 376 377 378 379 381 384 384 387 388 396 397 397 398 399 400 401 402 403 407 407 411 417 418 419 420 421 421 424 432 437 443

List of Tables

Table Table Table Table Table Table Table Table Table Table Table Table Table Table

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14

Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table

1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 3.1 3.2 3.3 3.4 3.5 3.6

Operation in bipolar space . . . . . . . . . . . . . . . . . . Operation in bipolar space . . . . . . . . . . . . . . . . . . Bipolar fuzzy set A on fx1 ; x2 ; x3 g . . . . . . . . . . . . . . Bipolar fuzzy relation B in fx1 ; x2 ; x3 g . . . . . . . . . . . Bipolar fuzzy set A on fx; y; zg . . . . . . . . . . . . . . . . Bipolar fuzzy relation B in fx; y; zg . . . . . . . . . . . . . Complex bipolar fuzzy set A . . . . . . . . . . . . . . . . . . Complex bipolar fuzzy relation D . . . . . . . . . . . . . . Bipolar fuzzy set A on fv1 ; v2 ; v3 ; v4 g. . . . . . . . . . . . Bipolar fuzzy relation ~ B in fv1 ; v2 ; v3 ; v4 g . . . . . . . . Bipolar fuzzy relation QðD ! SÞ . . . . . . . . . . . . . . . Bipolar fuzzy relation RðS ! PÞ . . . . . . . . . . . . . . . Composition T ¼ R  QðD ! PÞ . . . . . . . . . . . . . . . Names of employees in the organization with their designations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy set A of employees . . . . . . . . . . . . . . . Adjacency matrix corresponding to Fig. 1.42 . . . . . . Bipolar fuzzy set F . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy relation P . . . . . . . . . . . . . . . . . . . . . . Adjacency matrix of ranks of edges . . . . . . . . . . . . . Bipolar fuzzy relation on the set of crossings. . . . . . Weights of directed edges in Fig. 1.45 . . . . . . . . . . . Bipolar fuzzy relation QðD ! SÞ . . . . . . . . . . . . . . . Bipolar fuzzy relation RðS ! PÞ . . . . . . . . . . . . . . . Bipolar fuzzy set A on fa; b; c; dg . . . . . . . . . . . . . . Bipolar fuzzy relation B in fa; b; c; dg . . . . . . . . . . . Bipolar fuzzy set on fv1 ; v2 ; v3 g . . . . . . . . . . . . . . . . Bipolar fuzzy relation in fv1 ; v2 ; v3 g. . . . . . . . . . . . . Bipolar fuzzy set A . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy relation B . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

2 2 9 10 21 22 46 46 52 53 62 62 62

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

64 65 65 68 69 69 71 71 76 76 129 129 130 130 131 131

xxv

xxvi

List of Tables

Table Table Table Table Table Table Table Table Table Table Table

3.7 3.8 3.9 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Table Table Table Table Table Table Table Table Table Table Table Table Table

4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21

Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table

4.22 4.23 4.24 4.25 4.26 4.27 5.1 5.2 5.3 5.4 6.1 6.2 7.1 7.2 7.3 7.4 7.5 7.6 9.1

Bipolar fuzzy set A on X . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy relation B in X . . . . . . . . . . . . . . . . . . . . . . . Rða; bÞ for Cayley bipolar fuzzy graph. . . . . . . . . . . . . . . . Bipolar fuzzy out neighborhoods of vertices . . . . . . . . . . . Adjacency matrix of bipolar fuzzy relation ~ B. . . . . . . . . . . Bipolar fuzzy out neighborhoods of vertices . . . . . . . . . . . Bipolar fuzzy out neighborhoods . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy out neighborhoods . . . . . . . . . . . . . . . . . . . . Membership values of constructed edges . . . . . . . . . . . . . . ~ 1 hG ~2 . . . Bipolar fuzzy out neighborhoods of vertices in G Adjacent vertices of GCð~G5 Þ hCðG~6 Þ [ Gh . . . . . . . . . . . . . . . ~3 Þ  CðG ~4 Þ [ G . . . . . . . . . . . . Adjacent vertices of ½CðG Membership values of constructed edges . . . . . . . . . . . . . . Bipolar fuzzy in neighborhoods of species . . . . . . . . . . . . . Strength of competition between species . . . . . . . . . . . . . . Bipolar fuzzy out neighborhoods of species . . . . . . . . . . . . Strength of common enemies between species . . . . . . . . . . Bipolar fuzzy out and in neighborhoods of species . . . . . . Strength of competition between species . . . . . . . . . . . . . . Bipolar fuzzy out neighborhoods of companies . . . . . . . . . Strength of competition between companies . . . . . . . . . . . . Bipolar fuzzy out neighborhoods of candidates . . . . . . . . . Strength of competition of candidates for political seats . . Bipolar fuzzy in neighborhoods of families in social network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strength of competition in social network . . . . . . . . . . . . . Bipolar fuzzy out neighborhoods of noisy channel. . . . . . . Strength of conflict between wireless devices. . . . . . . . . . . Communication over a noisy channel . . . . . . . . . . . . . . . . . Bipolar fuzzy out neighborhoods of noisy channel. . . . . . . Strength of confusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy vertex set . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy multiedge set . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy vertex set A . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy multiedge set A . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy set A on set X . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy relation B in X . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy set A on fx1 ; x2 ; x3 g . . . . . . . . . . . . . . . . . . . Bipolar fuzzy relation B in fx1 ; x2 ; x3 g . . . . . . . . . . . . . . . . Bipolar fuzzy soft vertex set . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy soft edge set . . . . . . . . . . . . . . . . . . . . . . . . . Decision Support System . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy soft information . . . . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic set A . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . . .

. . . . .

142 142 148 164 165 165 170 171 172 173

.. . . 176 . . . . . . . . . . . .

. . . . . . . . . . . .

182 185 198 198 200 201 203 204 207 208 209 211

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

213 214 216 216 217 218 219 226 226 229 229 259 260 284 284 297 297 302 306 352

List of Tables

Table Table Table Table Table

9.2 9.3 9.4 9.5 9.6

Table 9.7 Table 9.8 Table 9.9 Table 9.10 Table 9.11 Table 9.12 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18

Table Table Table Table Table

10.19 10.20 10.21 10.22 10.23

Table 10.24 Table 10.25

xxvii

Bipolar neutrosophic relation A . . . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic out-neighborhoods of vertices . . . . . . ~1 . . . Bipolar neutrosophic out- and in-neighborhoods of G ~2 . . . Bipolar neutrosophic out- and in-neighborhoods of G Adjacent vertices of GCð~G1 Þ hCðG~2 Þ [ Gh . . . . . . . . . . . . . . . ~1 hG ~2 . . . . . . Bipolar neutrosophic out-neighborhoods of G Bipolar neutrosophic out-neighborhoods . . . . . . . . . . . . . . Strength of competition of applicants for particular designations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strength of competition of brands for particular quality . . . 4-step bipolar single-valued neutrosophic out-neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strength of competition of players for international games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic set B . . . . . . . . . . . . . . . . . . . . . . . . . Bipolar neutrosophic sets B1 , B2 , and B3 . . . . . . . . . . . . . . Bipolar neutrosophic information of countries . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Types of relations between countries . . . . . . . . . . . . . . . . . Bipolar neutrosophic information of patients . . . . . . . . . . . Psychological behavior of Albert with other patients . . . . . Psychological behavior of Charles with other patients . . . . Psychological behavior of Burton with other patients. . . . . Psychological behavior of Calvert with other patients . . . . Psychological behavior of Christopher with other patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Psychological behavior of David with other patients . . . . . Psychological behavior of Chapman with other patients . . Psychological behavior of Joseph with other patients . . . . . Various psychological behaviors of patients . . . . . . . . . . . . Bipolar neutrosophic information of terrorism in various countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasons of terrorism due to relationships of Pakistan with other countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasons of terrorism due to relationships of India with other countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 353 . . 355 . . 375 . . 375 . . 376 . . 378 . . 380 . . 381 . . 385 . . 387 . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

387 395 395 428 428 428 428 429 429 429 429 430 430 433 434 434 434 435

. . . . .

. . . . .

435 435 435 436 436

. . 439 . . 440 . . 440

xxviii

Table 10.26 Table 10.27 Table 10.28 Table 10.29 Table 10.30 Table 10.31 Table 10.32

List of Tables

Reasons of terrorism due to relationships of Afghanistan with other countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasons of terrorism due to relationships of America with other countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasons of terrorism due to relationship of Iraq with other countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasons of terrorism due to relationship of Israel with other countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasons of terrorism due to relationship of Palestine with other countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasons of terrorism due to relationship of Kashmir with other countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasons of terrorism due to relationship between countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 440 . . 440 . . 441 . . 441 . . 441 . . 441 . . 442

Chapter 1

Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

In this chapter, we first review the notion of bipolar fuzzy sets and present several basic concepts concerning bipolar fuzzy graphs and bipolar fuzzy digraphs. We discuss different methods of construction of bipolar fuzzy graphs and their isomorphism properties. We describe certain types of bipolar fuzzy graphs, bipolar fuzzy walk, bipolar fuzzy bridge, strength of connectedness, weak and strong bipolar fuzzy edges. We establish the relations on bipolar fuzzy graphs, complement of bipolar fuzzy graphs, and crisp graphs with different operations, α−cuts and (α, β)−cuts. We also study certain operations and properties of complex bipolar fuzzy graphs. Moreover, with the help of composition of bipolar fuzzy relations, connectivity, and weighted matrices, we study the importance of bipolar fuzzy digraphs with a number of realworld problems. This chapter is basically adapted from [1–3, 45, 46, 51].

1.1 Introduction Set theory and logic systems are strongly coupled in the development of modern logic. Classical logic corresponds to the crisp set theory, and fuzzy logic is associated with the fuzzy set theory, which owes its origin to the pioneering work of Zadeh [48] in 1965. The theory of fuzzy sets has become a vigorous area of research in different disciplines. A fuzzy set, as a superset of a crisp set, has been introduced to deal with the notion of partial truth between “absolute true” and “absolute false”. The notion of a fuzzy set is an essential mathematical structure, whose boundaries are vague. The membership degree range of a fuzzy set is [0, 1]. Considering fuzzy relations studied by Zadeh [49], Kaufmann [20] introduced the powerful notion of a fuzzy graph. The fuzzy relations constructed using fuzzy sets were additionally considered by © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_1

1

2

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Rosenfeld [41] for developing the structure of a fuzzy graph, proposing mainframe of several graph theoretical ideas. It comes to no surprise that bipolarity in data and information plays a vital role in various domains of science and technology. A wide variety of human decisionmaking, especially multiagent decision analysis, is based on bipolar judgmental thinking on a positive side and a negative side. Although 2-valued Boolean logic provides a theoretical basis for digital computer technology and fuzzy logic provides one way for dealing with vagueness and uncertainty both lack the representational and reasoning capabilities for directly modeling the coexistence and interaction of bipolar relationships. This is because the logical values in the two classical logical models lie in the positive interval [0, 1] and these are unipolar models in nature. Since positive and negative causal factors can be interactive but not counteractive at the same time, it is misleading to bury them into a summation; on other hand, it is neither cognitive nor convenient to represent bipolar perceptions with a unipolar structure. For instance, (large negative effect) ∨ (large positive effect) = (neutral effect). Therefore, a system based on unipolar variables can be rather misleading or less effective for modeling bipolar perceptions involved in the decision processes. In qualitative reasoning, decision has often been made by considering the bipolar space {0, 1} × {−1, 0} (cf. [57]) which makes up what is usually referred to as the Quantity Space or the Universe of Description. Two internal operations ⊕ and ⊗ on the space can be defined as in Tables 1.1 and 1.2. Similar to unipolar crisp values, the real-valued bipolar representation in the bipolar space {0, 1} × {−1, 0} suffers from the deficiency that it cannot be used to represent high-order fuzziness. To overcome this limitation, a negative-positive-neutral (NPN) fuzzy logic (cf. [56]) was defined in the space [−1, 1]. The two internal operators ⊕ and ⊗ are defined as Table 1.1 Operation ⊕ in bipolar space ⊕ (0, 0) (0, −1) (0, 0) (0, −1) (1, 0) (1, −1)

(0, 0) (0, −1) (1, 0) (1, −1)

(0, −1) (0, −1) (1, −1) (1, −1)

Table 1.2 Operation ⊗ in bipolar space ⊗ (0, 0) (0, −1) (0, 0) (0, −1) (1, 0) (1, −1)

(0, 0) (0, 0) (0, 0) (0, 0)

(0, 0) (1, 0) (0, −1) (1, −1)

(1, 0)

(1, −1)

(1, 0) (1, −1) (1, 0) (1, −1)

(1, −1) (1, −1) (1, −1) (1, −1)

(1, 0)

(1, −1)

(0, 0) (0, −1) (1, 0) (1, −1)

(0, 0) (1, −1) (1, −1) (1, −1)

1.1 Introduction

3

(x, y) ⊕ (u, v) = (max(x, u), min(y, v))

(1.1)

(x, y) ⊗ (u, v) = (max(x ⊗ u, x ⊗ v, y ⊗ u, y ⊗ v), min(x ⊗ u, x ⊗ v, y ⊗ u, y ⊗ v)),

(1.2)

where x, y, u, v ∈ [−1, 1] and x ≤ y, u ≤ v. This definition is not strictly bipolar because the allowance of (x, x) ∈ [−1, 1] violates the principle of polarity and fuzziness. Note that when real value pairs in the bipolar space [0, 1] × [−1, 0] are used, neutrality of (u, v) can be measured as u + v. Therefore, any bipolar logical value can be represented as a pair (x, y) ∈ [0, 1] × [−1, 0]. With bipolar fuzzy variables (x, y), (u, v) ∈ [0, 1] × [−1, 0], we have (y, x) ⊕ (v, u) = (min(y, v), max(x, u)) (y, x) ⊗ (v, u) = (max(x ⊗ u, y ⊗ v), min(x ⊗ v, y ⊗ u)) Complement : (¬a, b) = (1 − a, b), (a, ¬b) = (a, −1 − b), ¬(a, b) = (1 − a, −1 − b) Negation : ¬(x, y) = (−y, −x) or (0, −1) ⊗ (x, y) = (−y, −x).

(1.3) (1.4) (1.5) (1.6)

A = (⊕, ⊗, ¬), where A = {∀ (x, y) | (x, y) ∈ [0, 1] × [−1, 0]} defines a bipolar fuzzy logic in the strict sense and inevitable for dealing with both polarity and fuzziness principles (cf. [57]) as follows: 1. Representation principle: A bipolar fuzzy logic variable should have both a positive pole and a negative pole. It should be able to capture the nature of a positive side, a negative side, and the coexistence of both sides in different degrees of gray levels. 2. Computation principle: A bipolar fuzzy logic should be able to support bipolar reasoning by allowing computational interactions between the positive and negative sides of bipolar fuzzy variables. The max and min operators in (1.3) and (1.4) enable causal effects to propagate through a pair of strongest (maximal in absolute value) causal chains (a positive one and a negative one). This provides a basis for assessing both positive and negative factors. In bipolar fuzzy logic, ≤≤, ≥≥, ≤≥ or ≥≤ are used as bipolar comparison operators. The order of magnitude of crisp and bipolar fuzzy variables is depicted and compared in Fig. 1.1. The bipolar comparison is applied to the absolute values of the two poles, respectively. For instance, with bipolar comparison we have (0.4, −0.5) ≥≥ (0.3, −0.5) is true because 0.4 ≥ 0.3 and | − 0.5| ≥ | − 0.5|; (0.3, −0.8) ≤≥ (0.4, −0.5) is true because 0.3 ≤ 0.4 and | − 0.8| ≥ | − 0.5|; (0.4, −0.5) ≥≤ (0.3, −0.8) is true because 0.4 ≥ 0.3 and | − 0.5| ≤ | − 0.8|; and (0.3, −0.5) ≤≥ (0.4, −0.5) is true because 0.3 ≤ 0.4 and | − 0.5| ≤ | − 0.5|. Bipolarity plays a vital role in many research domains and gives more precision, flexibility, and comparability to the system as compared to the classical and fuzzy models. In 1994, Zhang [51] initiated the concept of bipolar fuzzy sets. Bipolar fuzzy sets are an extension of fuzzy sets, whose membership degree range is [−1, 1].

4

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs (1, −1) (1, −1)

(x, −1)

(0, −1)

(1, 0)

(0, 0)

(1, y)

(1, 0)

(x, y)

(0, −1)

(x, 0)

(0, y)

∀ (x, y) ∈ {0, 1} × {−1, 0}

(0, 0) ∀ (x, y) ∈ [0, 1] × [−1, 0]

Fig. 1.1 Order of magnitude (≥≥) of crisp and bipolar fuzzy logic

In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [−1, 0) of an element indicates that the element somewhat satisfies the implicit counter property. The idea which lies behind such description is connected with the existence of “bipolar information” (e.g., positive information and negative information) about a set under consideration. Positive information represent what is granted to be possible, while negative information represents what is considered to be impossible. Actually, a wide variety of human decision-making is based on double-sided or bipolar judgmental thinking on a positive side and a negative side. For instance, cooperation and competition, friendship and hostility, common interest and conflict of interest, effect and side effect, likelihood and unlikelihood, feedforward and feedback, and so forth are often the two sides in decision and coordination. In the traditional Chinese medicine, “yin” and “yang” are the two sides. Yin is the feminine or negative side of a system and yang is the masculine or positive side of a system. The coexistence, equilibrium, and harmony of the two sides are considered a key for the mental and physical health of a person as well as for the stability and prosperity of a social system. This idea motivates the necessity of bipolar fuzzy sets. Bipolar fuzzy sets indeed have potential impacts on many fields, including artificial intelligence, computer science, information science, cognitive science, decision science, management science, economics, neural science, quantum computing, medical science, and social sciences. In recent years, bipolar fuzzy sets seem to have been studied and applied a bit enthusiastically and a bit increasingly. Definition 1.1 A lattice L on a non-empty set X is a triple (X, ∨, ∧), where ∧ and ∨ are binary operations on X , both ∨ and ∧ are commutative, associative, idempotent, and satisfy absorption law, that is,

1.1 Introduction

1. 2. 3. 4.

5

Commutative Law: x ∨ y = y ∨ x, x ∧ y = y ∧ x, Associative Law: x ∨ (x ∨ z) = (x ∨ y) ∨ z, x ∧ (x ∧ z) = (x ∧ y) ∧ z, Absorption Law: x ∨ (x ∧ y) = x, x ∧ (x ∨ y) = x, Idempotent: x ∨ x = x, x ∧ x = x for all x, y, z ∈ X .

Example 1.1 1. L = [0, 1] (unit closed interval) is an example of a lattice with binary operations: x1 ∨ x2 = max(x1 , x2 ) and x1 ∧ x2 = min(x1 , x2 ). 2. L = [0, 1]2 (unit closed square) is a lattice with binary operation: (x1 , y1 ) ∨ (x2 , y2 ) = (max(x1 , x2 ), max(x1 , y2 )) and (x1 , y1 ) ∧ (x2 , y2 ) = (min(x1 , x2 ), min(x1 , y2 )). Definition 1.2 An L-subset (or an L-set) on the set X is a mapping μ : X → L, where L is a lattice. When L = [0, 1] (the closed unit interval with ordinary relation), then [0, 1]-set on X will be called a fuzzy set on X . When L = [0, 1]2 , then it is a [0, 1]2 -set. Definition 1.3 A bipolar fuzzy set A on a non-empty set X is an object of the form p p A = {(x, μ A (x), μnA (x)) | x ∈ X }, where μ A : X → [0, 1] and μnA : X → [−1, 0] are mappings. p The positive membership degree μ A (x) denotes the truth or satisfaction degree of an element x to a certain property corresponding to bipolar fuzzy set A and μnA (x) represents the satisfaction degree of an element x to some counter property of bipolar p fuzzy set A. If μnA (x) = 0 and μ A (x) = 0, it is the situation that x is not satisfying the p property of A but satisfying the counter property to A. If μ A (x) = 0 and μnA (x) = 0, it is the case when x has only positive satisfaction for A. It is possible for x to be p such that μ A (x) = 0 and μnA (x) = 0 when x satisfies the property of A as well as its counter property in some part of X . Question: Is bipolar fuzzy set equivalent to intuitionistic fuzzy set? The answer to Question 1 is negative. The intuitionistic fuzzy set theory is an extension of fuzzy set theory initiated by Atanassov [10] in 1986. An intuitionistic fuzzy set on the universe of discourse X is an object of the form A = {(x, μ A (x), ν A (x)) | x ∈ X }, where the functions μ A : X → [0, 1] and ν A : X → [0, 1] define the degrees of membership and non-membership, respectively, of the element x ∈ X to the set A, and for every x ∈ X , 0 ≤ μ A (x) + ν A (x) ≤ 1. When we p compare a bipolar fuzzy set A = {(x, μ A (x), μnA (x)) | x ∈ X } with an intuitionistic p fuzzy set A = {(x, μ A (x), ν A (x)) | x ∈ X } under the conditions μ A (x) = μ A (x) n and μ A (x) = −ν A (x), both the sets look similar to each other. However, they are different from each other in terms of modeling of problems: In bipolar fuzzy set A, p the positive membership degree μ A (x) characterizes the extent that the element x satisfies the property A, and the negative membership degree μnA (x) characterizes the extent that the element x satisfies the implicit counter property of A. On the other hand, in intuitionistic fuzzy set A, the membership degree μ A (x) denotes the extent to which the element x satisfies property A, and the membership degree ν A (x) denotes the extent to which the element x does not satisfy property A, that is, not-property of

6

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

A. Since a counter property is not usually equivalent to not-property, both bipolar fuzzy set and intuitionistic fuzzy set are different extensions of fuzzy sets. This difference can also be manifested in the interpretation of an element x to be indifferent (i.e., neutral). In the perspective of bipolar fuzzy set A, it is interpreted that the element x does not satisfy property A and its implicit counter property and so its membership degree is (0, 0). In case of intuitionistic fuzzy set, the element x is neutral if it does not satisfy property A, that is, it completely satisfies not-property of A, and so the degree of membership is (0, 1). The intuitionistic fuzzy set representation is useful when there are some uncertainties in assigning membership degrees as illustrated in the following practical examples. 1. The bipolar fuzzy set representation is useful when irrelevant elements and contrary elements are needed to be discriminated. For example, when we want to express effect and side effect of a drug, we can use bipolar fuzzy valuations. Because side effect is a negative effect. In an intuitionistic fuzzy set, we can not model negative effect. The non-membership degree of an element doesn’t correspond to negative effect. 2. In modeling the profit and loss of a product, we can use bipolar fuzzy set because loss is implicit counter property to profit and is treated as negative effect of product. In an intuitionistic fuzzy set, this scenario can be treated as “profit” and “no profit”. Clearly, the property “no-profit” is not equivalent to “loss”, that is, the intuitionistic fuzzy set cannot be used to model negative effect (loss). Definition 1.4 Let A and B be two bipolar fuzzy sets on X and Y , respectively. A bipolar fuzzy relation R from R : A → B defined as R =  A to B is a mapping   p (x, y), μ R (x, y), μnR (x, y) | (x, y) ∈ X × Y such that p

p

p

μ R (x, y) ≤ μ A (x) ∧ μ B (y) and μnR (x, y) ≥ μnA (x) ∨ μnB (y). R is also a bipolar fuzzy relation in X × Y defined by the mapping R : X × Y : [0, 1] × [−1, 0]. A bipolar fuzzy relation R in X is defined by a mapping R : X × X : [0, 1] × [−1, 0]. Definition 1.5 Let R and S be two bipolar fuzzyrelations in X × Y . The union of  p R and S is defined as a bipolar fuzzy set R ∪ S = (x, y), μ R∪S (x, y), μnR∪S (x, y) | (x, y) ∈ X × Y } such that p

p

p

μ R∪S (x, y) = μ R (x, y) ∨ μ S (x, y) and μnR∪S (x, y) = μnR (x, y) ∧ μnS (x, y). The intersection of R and S is denoted by R ∩ S such that for all (x, y) ∈ X × Y , p

p

p

μ R∩S (x, y) = μ R (x, y) ∧ μ S (x, y) and μnR∩S (x, y) = μnR (x, y) ∨ μnS (x, y). Definition 1.6 Let R be a bipolar fuzzy relation in X . Then R is called a bipolar fuzzy equivalence relation on X if it satisfies the following conditions. 1. R is bipolar fuzzy reflexive, i.e., R(x, x) = (1, −1), for each x ∈ X .

1.1 Introduction

7

2. R is bipolar fuzzy symmetric, i.e., R(x, y) =R(y, x), for any x, y ∈ X . 3. R is bipolar fuzzy transitive, i.e., R(x, z) ≥ y (R(x, y) ∧ R(y, z)). Definition 1.7 Let R be a bipolar fuzzy relation in X . Then R is called a bipolar fuzzy partial order relation on X if it satisfies the following conditions. 1. R is bipolar fuzzy reflexive, i.e., R(x, x) = (1, −1), for each x ∈ X . 2. R is bipolar fuzzy antisymmetric, i.e., R(x, y)  = R(y, x), for any x, y ∈ X . 3. R is bipolar fuzzy transitive, i.e., R(x, z) ≥ y (R(x, y) ∧ R(y, z)). Definition 1.8 Let R be a bipolar fuzzy relation in X . Then R is called a bipolar fuzzy linear order relation on X if it satisfies the following conditions. 1. R is bipolar fuzzy partial order relation. 2. R(x, y) = (0, 0) or R(y, x) = (0, 0), for all x, y ∈ X . For further terminologies and studies on fuzzy sets, bipolar fuzzy graphs and complex bipolar fuzzy graphs, readers are referred to [5–9, 11–19, 21–36, 38–40, 42–44, 47, 50, 52–55, 58].

1.2 Bipolar Fuzzy Graphs Graph theory was first initiated by Euler during the course of finding solution to the problem of “Königsberg bridge” in 1736. Graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network. It is used to study the mathematical structures of pairwise relations among objects. Graphs are used to model various practical processes and relations in social, physical, information, and biological systems. A graph is a figure consisting of points, called vertices, which are connected by lines, called edges. Mathematically, a pair G ∗ = (X, E) is a crisp graph, where E ⊆ X × X is a collection of 2−element subsets of a non-empty universe X . However, in some cases, the crisp graphs do not study the uncertain behavior and degree of dependence among the objects. In such cases, it is important to deal with uncertainty using the methods of fuzzy sets. The approach of fuzzification to crisp graph is called a fuzzy graph. Graphical models can be studied more precisely when two-sided behavior of objects are to be dealt with, emphasizing the need for a mathematical approach toward graphs under bipolar fuzzy environment. Considering bipolar fuzzy relations studied by Zhang [51], bipolar fuzzy graphs were introduced by Akram [1], in 2011, to overcome the limitations entailed in crisp graph theory and fuzzy graph theory. Akram and Dudek [4] considered regular bipolar fuzzy graphs. The concept of a bipolar fuzzy graph is obtained as a generalization of a fuzzy graph which is an extension of the term crisp graph. Its construction depends on the mathematical structure of a bipolar fuzzy set which is more a convenient, intuitive, and computationally tractable notation.

8

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Definition 1.9 Let A be a bipolar fuzzy set on a non-empty set X . A bipolar fuzzy relation B on A is a mapping B : A → A such that p

p

p

μ B (x y) ≤ μ A (x) ∧ μ A (y) and μnB (x y) ≥ μnA (x) ∨ μnA (y),

for all x, y ∈ X.

B is also a bipolar fuzzy relation in X defined by the mapping B : X × X → [0, 1] × [−1, 0]. p

Definition 1.10 The support of a bipolar fuzzy set A = (μ A , μnA ), denoted by supp(A), is defined as supp(A) = supp p (A) ∪ supp n (A), where supp p (A) = p {x | μ A (x) > 0}, supp n (A) = {x | μnA (x) < 0}. We call supp p (A) as positive support and supp n (A) as negative support. p

Definition 1.11 Let A = (μ A , μnA ) be a bipolar fuzzy set on X and α ∈ [0, 1]. The p α-cut of bipolar fuzzy set A is denoted by Aα and defined as Aα = Aα ∪ Anα , where p p p p Aα = {x | μα (x) ≥ α}, Aα = {x | μnα (x) ≤ −α}. We call Aα as positive α-cut and p Anα as negative α-cut. The height of a bipolar fuzzy set A = (μ A , μnA ) is defined p p as h(A) = max{μ A (x)|x ∈ X }. The depth of a bipolar fuzzy set A = (μ A , μnA ) is n defined as d(A) = min{μ A (x)|x ∈ X }. We shall say that bipolar fuzzy set A is norp mal, if there is at least one x ∈ X such that μ A (x) =1 or μnA (x) =−1. Definition 1.12 A bipolar polar fuzzy graph G on a non-empty set X is a pair G = (A, B) , where A : X → [0, 1] × [−1, 0] is a bipolar fuzzy set on the set X and B : X × X → [0, 1] × [−1, 0] is a bipolar polar fuzzy relation in X such that p

p

p

μ B (x y) ≤ μ A (x) ∧ μ A (y) and μnB (x y) ≥ μnA (x) ∨ μnA (y),

for all x, y ∈ X.

Remark 1.1 1. Clearly, B(x y) = (0, 0) for all x y ∈ X × X − E , where E ⊆ X × X is the set of edges. A is called a bipolar fuzzy vertex set of G and B is a bipolar fuzzy edge set of G. 2. In Definition 1.12, B is called a bipolar fuzzy relation on A. A bipolar fuzzy relation B on A is symmetric if B(x y) = B(yx) for all x, y ∈ X . 3. In a bipolar fuzzy graph G, the support of a bipolar fuzzy relation B is the set of edges in G. We will use the notation E to write the support of bipolar fuzzy relation B, that is, supp(B) = E. The following example illustrates the above definition. Example 1.2 Consider a bipolar fuzzy set A on the vertex set X = {x1 , x2 , x3 } as given in Table 1.3 and a bipolar fuzzy relation B in X as defined in Table 1.4. The bipolar fuzzy graph G = (A, B) on set X is demonstrated in Fig. 1.2. Definition 1.13 Let G = (A, B) be a bipolar fuzzy graph on X and (α, β) ∈ (0, 1] × p ∈ X | μ A (x) ≥ [−1, 0). Define (α, β)-cut of a bipolar fuzzy set A as, A(α,β) = {x  p α, μnA (x) ≤ β}. If B(α,β) = x y ∈ E | μ B (x y) ≥ α, μnB (x y) ≤ β is (α, β)-cut of bipolar fuzzy relation B, then the (α, β)-level graph of G is a crisp graph denoted by G (α,β) and defined as a pair G (α,β) = (A(α,β) , B(α,β) ).

1.2 Bipolar Fuzzy Graphs

9

Fig. 1.2 Bipolar fuzzy graph

0.4) .5, − x 1(0

(0.2, −0.3)

x2 (0.6, −0.5)

(0.2 , −0 .3)

.4) , −0 (0.6

x3 (0.7, −0.7)

Definition 1.14 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on non-empty sets X 1 and X 2 , respectively. The union of G 1 and G 2 is defined as a pair G 1 ∪ G 2 = (A1 ∪ A2 , B1 ∪ B2 ) such that for all x, y ∈ X 1 ∪ X 2 , 1. If x ∈ X 1 , x ∈ / X 2 , then (A1 ∪ A2 )(x) = A1 (x), 2. If x ∈ / X 1 , x ∈ X 2 , then (A1 ∪ A2 )(x) = A2 (x), p p 3. If x ∈ X 1 ∩ X 2 , then (A1 ∪ A2 )(x) = (μ A1 (x) ∨ μ A2 (x), μnA1 (x) ∧ μnA2 (x)). If supp(B1 ) = E 1 and supp(B2 ) = E 2 are the sets of edges in G 1 and G 2 , then B1 ∪ B2 can be defined as 1. If x y ∈ E 1 , x y ∈ / E 2 , then (B1 ∪ B2 )(x y) = B1 (x y), / E 2 , then (B1 ∪ B2 )(x y) = B2 (x y), 2. If x y ∈ E 1 , x y ∈ p p 3. If x y ∈ E 1 ∩ E 2 , then (B1 ∪ B2 )(x y) = (μ B1 (x y) ∨ μ B2 (x y), μnB1 (x y) ∧ n μ B2 (x y)). Example 1.3 Consider two bipolar fuzzy graphs G 1 and G 2 as shown in Fig. 1.3. The union G 1 ∪ G 2 is also given in Fig. 1.3. Example 1.4 Consider bipolar fuzzy graphs G 1 and G 2 as shown in Fig. 1.4 and 1.5, respectively. G 1 ∪ G 2 is a bipolar graph with set of vertices X 1 ∪ X 2 = {a, b, c, d, e, f }. The membership values of the vertices in G 1 ∪ G 2 can be computed as: p

p

1. (A1 ∪ A2 )(a) = (μ A1 (a) ∨ μ A2 (a), μnA1 (a) ∧ μnA2 (a)) = (0.2 ∨ 0.1, −0.4 ∧ −0.4) = (0.2, −0.4), p p 2. (A1 ∪ A2 )(d) = (μ A1 (d) ∨ μ A2 (d), μnA1 (d) ∧ μnA2 (d)) = (0.3 ∨ 0.2, −0.7 ∧ −0.6) = (0.3, −0.7) / X 2. 3. (A1 ∪ A2 )( f ) = A2 ( f ) as f ∈ Table 1.3 Bipolar fuzzy set A on {x1 , x2 , x3 } A x1 p

μ A (x) μnA (x)

0.5 −0.4

x2

x3

0.6 −0.5

0.7 −0.7

10

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Table 1.4 Bipolar fuzzy relation B in {x1 , x2 , x3 } B x1 x2 x1 x2 x3

x3

(0.2, −0.3) (0, 0) (0.6, −0.4)

(0, 0) (0.2, −0.3) (0.2, −0.3)

(0.2, −0.3) (0.6, −0.4) (0, 0)

G1 y(0.6, −0.2)

x(0.5, −0.3) (0.5, −0.2) G2

y(0.7, −0, 2)

w(0.5, −0.3)

(0.5, −0.2) G1 ∪ G2 x(0.5, −0.3)

w(0.5, −0.3)

y(0.7, −0, 2) (0.5, −0.2)

(0.5, −0.2)

Fig. 1.3 Union of bipolar fuzzy graphs

The membership values of the other vertices can be computed similarly. The membership values of the edges in G 1 ∪ G 2 can be computed as: p

p

1. (B1 ∪ B2 )(bc) = (μ B1 (bc) ∨ μ B2 (bc), μnB1 (bc) ∧ μnB2 (bc)) = (0.2 ∨ 0.1, −0.4 ∧ −0.4) = (0.2, −0.4), 2. (B1 ∪ B2 )(bd) = B2 (bd) as bd is not an edge in G 1 , 3. (B1 ∪ B2 )(ad) = B1 (ad) as ad is not an edge in G 2 . The union G 1 ∪ G 2 is shown in Fig. 1.6. Proposition 1.1 If G 1 and G 2 are bipolar fuzzy graphs, then G 1 ∪ G 2 is a bipolar fuzzy graph. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Also let E 1 and E 2 be the set of edges in G 1 and G 2 , respectively. If x y ∈ E 1 ∩ E 2 , then p

p

p

μ B1 ∪B2 (x y) = μ B1 (x y) ∨ μ B2 (x y)  p   p  p p ≤ μ A1 (x) ∧ μ A1 (y) ∨ μ A2 (x) ∧ μ A2 (y)  p   p  p p = μ A1 (x) ∨ μ A2 (x) ∧ μ A1 (y) ∨ μ A2 (y)

1.2 Bipolar Fuzzy Graphs

11

a(0.2, −0.4)

c(0.3, −0.6)

b(0.3, −0.5) (0.2, −0.4)

(0.1, −0.3)

(0.3, −0.4)

(0.1, −0.2)

5) 0. − , .3 (0

(0.2, −0.4) d(0.3, −0.7)

e(0.4, −0.6)

Fig. 1.4 Bipolar fuzzy graph G 1 a(0.1, −0.4)

c(0.3, −0.6)

b(0.2, −0.5)

(0.1, −0.4)

(0.1, −0.3)

(0.3, −0.5)

(0 .1 ,− 0. 4)

.3) −0 , .1 (0

f (0.4, −0.6)

d(0.2, −0.6)

Fig. 1.5 Bipolar fuzzy graph G 2 a(0.2, −0.4)

c(0.3, −0.6)

b(0.3, −0.5) (0.2, −0.4)

(0.1, −0.3)

(0.2, −0.4) d(0.3, −0.7)

e(0.4, −0.6)

(0 .3 ,− 0. 5)

(0.3, −0.5)

(0.3, −0.4)

(0.1, −0.2)

.3) −0 , .1 (0

4) 0. − , .1 (0

f (0.4, −0.6)

Fig. 1.6 Union G 1 ∪ G 2 of bipolar fuzzy graphs

p

p

= μ A1 ∪A2 (x) ∧ μ A1 ∪A2 (y), μnB1 ∪B2 (x y) = μnB1 (x y) ∧ μnB2 (x y)     ≥ μnA1 (x) ∨ μnA1 (y) ∧ μnA2 (x) ∨ μnA2 (y)     = μnA1 (x) ∧ μnA2 (x) ∨ μnA1 (y) ∧ μnA2 (y) = μnA1 ∪A2 (x) ∨ μnA1 ∪A2 (y). If x y ∈ E 1 ∩ E 2 , then

12

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs p

p

μ B1 ∪B2 (x y) = μ B1 (x y) p

p

≤ μ A1 (x) ∧ μ A1 (y)  p   p  p p ≤ μ A1 (x) ∨ μ A2 (x) ∧ μ A1 (y) ∨ μ A2 (y) p

p

= μ A1 ∪A2 (x) ∧ μ A1 ∪A2 (y), μnB1 ∪B2 (x y) = μnB1 (x y) ≥ μnA1 (x) ∨ μnA1 (y)     ≥ μnA1 (x) ∧ μnA2 (x) ∨ μnA1 (y) ∧ μnA2 (y) = μnA1 ∪A2 (x) ∨ μnA1 ∪A2 (y). Similarly, it can be proved that if x y ∈ E 2 ∩ E 1 , then p

p

p

μ B1 ∪B2 (x y) ≤ μ A1 ∪A2 (x) ∧ μ A1 ∪A2 (y), μnB1 ∪B2 (x y) ≥ μnA1 ∪A2 (x) ∨ μnA1 ∪A2 (y). 

This completes the proof.

Definition 1.15 The intersection of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is a pair G 1 ∩ G 2 = (A1 ∩ A2 , B1 ∩ B2 ), where A1 ∩ A2 and B1 ∩ B2 are defined as  p  p 1. (A1 ∩ A2 )(x) = μ A1 (x) ∧ μ A2 (x), μnA1 (x) ∨ μnA2 (x) , for all x ∈ X 1 ∩ X 2 ,  p  p 2. (B1 ∩ B2 )(x y) = μ B1 (x y) ∧ μ B2 (x y), μnB1 (x y) ∨ μnB2 (x y) , for all x y ∈ E 1 ∩ E2 . Example 1.5 Consider bipolar fuzzy graphs G 1 and G 2 as shown in Fig. 1.4 and 1.5, respectively. G 1 ∩ G 2 is a bipolar graph with set of vertices X 1 ∩ X 2 = {a, b, c, d}. The membership values of the vertices in G 1 ∩ G 2 can be computed as: p

p

1. (A1 ∩ A2 )(a) = (μ A1 (a) ∧ μ A2 (a), μnA1 (a) ∨ μnA2 (a)) = (0.2 ∧ 0.1, −0.4 ∨ −0.4) = (0.1, −0.4), p p 2. (A1 ∩ A2 )(d) = (μ A1 (d) ∧ μ A2 (d), μnA1 (d) ∨ μnA2 (d)) = (0.3 ∧ 0.2, −0.7 ∨ −0.6) = (0.2, −0.6) / X 1 ∩ X 2. 3. (A1 ∩ A2 )( f ) = (0, 0) as f ∈ The membership values of the other vertices can be computed similarly. The membership values of the edges in G 1 ∩ G 2 can be computed as: p

p

1. (B1 ∩ B2 )(bc) = (μ B1 (bc) ∧ μ B2 (bc), μnB1 (bc) ∨ μnB2 (bc)) = (0.2 ∧ 0.1, −0.4 ∨ −0.4) = (0.1, −0.4), / E1 ∩ E2 , 2. (B1 ∩ B2 )(bd) = (0, 0) as bd ∈ / E1 ∩ E2 . 3. (B1 ∩ B2 )(ad) = (0, 0) as ad ∈ The intersection G 1 ∩ G 2 is shown in Fig. 1.7.

1.2 Bipolar Fuzzy Graphs

13

a(0.1, −0.4)

c(0.3, −0.6)

b(0.2, −0.5) (0.1, −0.4)

(0.1, −0.3)

d(0.2, −0.6)

Fig. 1.7 Intersection of bipolar fuzzy graphs G 1 and G 2

Proposition 1.2 Let {G i | i ∈ } be a family of bipolar fuzzy graphs on non-empty set X . Then i∈ G i is a bipolar fuzzy graph. Proof Let G i = (Ai , Bi ), i ∈ , be bipolar fuzzy graphs on X . For any x, y ∈ X ,  p  p p p μ∩i∈ Bi (x y) = inf μ Bi (x y) ≤ inf μ Ai (x) ∧ μ Ai (y) i∈

i∈

p

p

= inf μ Ai (x) ∧ inf μ Ai (y) = μn∩i∈ Bi (x y)

= = =

Hence

 i∈

i∈ i∈ p p μ∩i∈ Ai (x) ∧ μ∩i∈ Ai (y),  sup μnBi (x y) ≥ sup μnAi (x) i∈ i∈ sup μnAi (x) ∨ sup μnAi (y) i∈ i∈ n μ∩i∈ Ai (x) ∨ ∩μn∩i∈ Ai (y).

G i is a bipolar fuzzy graph.

∨ μnAi (y)





Definition 1.16 The join of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is defined as a pair G 1 + G 2 = (A1 + A2 , B1 + B2 ) such that A1 + A2 = A1 ∪ A2 , for all x ∈ X 1 ∪ X 2 , and the membership values of the edges in G 1 + G 2 are defined as 1. B1 + B2 = B1 ∪ B2 , for all x y ∈ E 1 ∪ E 2 . 2. Let E  be the set of all edges joining the vertices of G 1 and G 2 , then for all x y ∈ p p E  , where x ∈ X 1 and y ∈ X 2 , (B1 + B2 )(x y) = μ A1 (x) ∧ μ A2 (y), μnA1 (x)∨  μnA2 (y) . Example 1.6 The join of two bipolar fuzzy graphs G 1 and G 2 is shown in Fig. 1.8. Proposition 1.3 If G 1 and G 2 are the bipolar fuzzy graphs, then G 1 + G 2 is a bipolar fuzzy graph.

14

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs x(0.5, −0.3)

x(0.5, −0.3)

z(0.7, −0.2)

z(0.7, −0.2)

(0.5, −0.2)

(0.5, −0.2)

3) 0. ,− .5 (0

(0 .6 ,− 0. 2)

(0.5, −0.2)

(0.5, −0.2)

(0.5, −0.2) y(0.6, −0.2)

G1

(0.5, −0.2)

y(0.6, −0.2)

w(0.5, −0.3) G2

w(0.5, −0.3)

G1 + G2

Fig. 1.8 Join of bipolar fuzzy graphs G 1 and G 2

Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. If x y ∈ E  , then p

p

p

μ B1 +B2 (x y) = μ A1 (x) ∧ μ A2 (y) p

p

≤ μ A1 ∪A2 (x) ∧ μ A1 ∪A2 (y) = μnB1 +B2 (x y)

= ≥ =

∵ By Proposition 1.1.

p p μ A1 +A2 (x) ∧ μ A1 +A2 (y), μnA1 (x) ∨ μnA2 (y) μnA1 ∪A2 (x) ∨ μnA1 ∪A2 (y) μnA1 +A2 (x) ∨ μnA1 +A2 (y).

If x y ∈ E 1 ∪ E 2 , then the result follows from Proposition 1.1. It completes the proof.  Proposition 1.4 Let G ∗1 = (X 1 , E 1 ) and G ∗2 = (X 2 , E 2 ) be crisp graphs such that X 1 ∩ X 2 = ∅. Let A1 and A2 be bipolar fuzzy sets on X 1 and X 2 and B1 , B2 be bipolar fuzzy relations in X 1 and X 2 , respectively. Then G 1 ∪ G 2 = (A1 ∪ A2 , B1 ∪ B2 ) is a bipolar fuzzy graph on X 1 ∪ X 2 with underlying graph G ∗1 ∪ G ∗2 if and only if G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) are bipolar fuzzy graphs with underlying graphs G ∗1 and G ∗2 , respectively. Proof Suppose that G 1 ∪ G 2 is a bipolar fuzzy graph. Let x y ∈ E 1 , then x y ∈ / E2 and x, y ∈ X 1 − X 2 , p

p

p

p

p

p

μ B1 (x y) = μ B1 ∪B2 (x y) ≤ μ A1 ∪A2 (x) ∧ μ A1 ∪A2 (y) = μ A1 (x) ∧ μ A1 (y), μnB1 (x y) = μnB1 ∩B2 (x y) ≥ μnA1 ∪A2 (x) ∨ μnA1 ∪A2 (y) = μnA1 (x) ∨ μnA1 (y).

1.2 Bipolar Fuzzy Graphs

15

It shows that G 1 = (A1 , B1 ) is a bipolar fuzzy graph. Similarly, we can show that G 2 = (A2 , B2 ) is a bipolar fuzzy graph. The converse part of the proposition is given by Proposition 1.1.  Proposition 1.5 Let G ∗1 = (X 1 , E 1 ) and G ∗2 = (X 2 , E 2 ) be crisp graphs such that X 1 ∩ X 2 = ∅. Let A1 and A2 be bipolar fuzzy sets on X 1 and X 2 and B1 , B2 be bipolar fuzzy relations in X 1 and X 2 , respectively. Then G 1 + G 2 = (A1 + A2 , B1 + B2 ) is a bipolar fuzzy graph on X 1 ∪ X 2 with underlying graph G ∗1 + G ∗2 if and only if G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) are bipolar fuzzy graphs with underlying graphs G ∗1 and G ∗2 , respectively. Definition 1.17 The composition of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is denoted by the pair G 1 ◦ G 2 = (A1 ◦ A2 , B1 ◦ B2 ) (or G 1 [G 2 ] = p p (A1 [A2 ], B1 [B2 ])), where A1 ◦ A2 = (μ A1 ◦A2 , μnA1 ◦A2 ) and B1 ◦ B2 = (μ B1 ◦B2 , n μ B1 ◦B2 ) are defined as  p  p 1. (A1 ◦ A2 )(x1 , x2 ) = μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , for all (x1 , x2 ) ∈ X 1 × X 2,  p  p 2. (B1 ◦ B2 )((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 ), μnA1 (x) ∨ μnB2 (x2 y2 ) , for all x ∈ X 1 , x2 y2 ∈ E 2 ,  p  p 3. (B1 ◦ B2 )((x1 , z)(y1 , z)) = μ B1 (x1 y1 ) ∧ μ A2 (z), μnB1 (x1 y1 ) ∨ μnA2 (z) , for all z ∈ X 2 , x1 y1 ∈ E 1 ,  p p p 4. (B1 ◦ B2 )((x1 , x2 )(y1 , y2 )) = μ A2 (x2 ) ∧ μ A2 (y2 ) ∧ μ B1 (x1 y1 ), μnA2 (x2 )  n n ∨μ A2 (y2 ) ∨ μ B1 (x1 y1 ) , for all x2 , y2 ∈ X 2 , x1 y1 ∈ E 1 . Proposition 1.6 If G 1 and G 2 are bipolar fuzzy graphs, then G 1 ◦ G 2 is a bipolar fuzzy graph. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let x ∈ X 1 and x2 y2 ∈ E 2 , then p

p

p

μ B1 ◦B2 ((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 )  p  p p ≤ μ A1 (x) ∧ μ A2 (x2 ) ∧ μ A2 (y2 )  p   p  p p = μ A1 (x) ∧ μ A2 (x2 ) ∧ μ A2 (x) ∧ μ A2 (y2 ) p

p

= μ A1 ◦A2 (x, x2 ) ∧ μ A1 ◦A2 (x, y2 ), μnB1 ◦B2 ((x, x2 )(x, y2 )) = μnA1 (x) ∨ μnB2 (x2 y2 )   ≥ μnA1 (x) ∨ μnA2 (x2 ) ∨ μnA2 (y2 )     = μnA1 (x) ∨ μnA2 (x2 ) , μnA1 (x) ∨ μnA2 (y2 ) = μnA1 ◦A2 (x, x2 ) ∨ μnA1 ◦A2 (x, y2 ). Let z ∈ X 2 and x1 y1 ∈ E 1 , then p

p

p

μ B1 ◦B2 ((x1 , z)(y1 , z)) = μ B1 (x1 y1 ) ∧ μ A2 (z)

16

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

 p  p p ≤ μ A1 (x1 ) ∧ μ A1 (y1 ) ∧ μ A2 (z)  p    p p p = μ A1 (x) ∧ μ A2 (z) ∧ μ A1 (y1 ) ∧ μ A2 (z) p

p

= μ A1 ◦A2 (x1 , z) ∧ μ A1 ◦A2 (y1 , z), μnB1 ◦B2 ((x1 , z)(y1 , z)) = μnB1 (x1 y1 ) ∨ μnA2 (z)   ≥ μnA1 (x1 ) ∨ μnA1 (y1 ) ∨ μnA2 (z)     = μ A1 (x) ∨ μnA2 (z) ∨ μnA1 (y1 ) ∨ μnA2 (z) = μnA1 ◦A2 (x1 , z) ∨ μnA1 ◦A2 (y1 , z). Let x1 y1 ∈ E 1 , x2 , y2 ∈ X 2 , x2 = y2 , then p

p

p

p

μ B1 ◦B2 ((x1 , x2 )(y1 , y2 )) = μ A2 (x2 ) ∧ μ A2 (y2 ) ∧ μ B1 (x1 y1 )  p   p  p p ≤ μ A2 (x2 ) ∧ μ A2 (y2 ) ∧ μ A1 (x1 ) ∧ μ A1 (y1 )  p   p  p p = μ A1 (x1 ) ∧ μ A2 (x2 ) ∧ μ A1 (y1 ) ∧ μ A2 (y2 ) p

p

= μ A1 ◦A2 (x1 , x2 ) ∧ μ A1 ◦A2 (y1 , y2 ), μnB1 ◦B2 ((x1 , x2 )(y1 , y2 )) = μnA2 (x2 ) ∨ μnA2 (y2 ) ∨ μnB1 (x1 y1 ))     ≥ μnA2 (x2 ) ∨ μnA2 (y2 ) ∨ μnA1 (x1 ) ∨ μnA1 (y1 )     = μnA1 (x1 ) ∨ μnA2 (x2 ) , μnA1 (y1 ) ∨ μnA2 (y2 ) = μnA1 ◦A2 (x1 , x2 ) ∨ μnA1 ◦A2 (y1 , y2 ). This completes the proof.



Definition 1.18 The Cartesian product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is denoted by the pair G 1 G 2 = (A1 A2 , B1 B2 ), where p p A1 A2 = (μ A1 A2 , μnA1 A2 ) and B1 B2 = (μ B1 B2 , μnB1 B2 ) are defined as  p  p 1. (A1 A2 )(x1 , x2 ) = μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , for all (x1 , x2 ) ∈ X 1 × X 2,  p  p 2. (B1 B2 )((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 ), μnA1 (x) ∨ μnB2 (x2 y2 ) , for all x ∈ X 1 , x2 y2 ∈ E 2 ,  p  p 3. (B1 B2 )((x1 , z)(y1 , z)) = μ B1 (x1 y1 ) ∧ μ A2 (z), μnB1 (x1 y1 ) ∨ μnA2 (z) , for all z ∈ X 2 , x1 y1 ∈ E 1 . Example 1.7 The Cartesian product of two bipolar fuzzy graphs G 1 and G 2 is shown in Fig. 1.9. Proposition 1.7 The Cartesian product of two bipolar fuzzy graphs is a bipolar fuzzy graph. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let x ∈ X 1 , x2 y2 ∈ E 2 , then

1.2 Bipolar Fuzzy Graphs

17

x1 (0.5, −0.3)

y1 (0.6, −0.2) G1

(0.5, −0.2) (y1 , x2 )(0.6, −0.2)

(x1 , x2 )(0.5, −0.2)

x2 (0.7, −0.2)

G1 G2

(0.5, −0.2)

(0.5, −0.2)

(0.5, −0.2)

(0.5, −0.2)

G2

(0.5, −0.2) (x1 , y2 )(0.5, −0.3)

(y1 , y2 )(0.5, −0.2)

y2 (0.5, −0.3)

Fig. 1.9 Cartesian product G 1 G 2 p

p

p

μ B1 B2 ((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 )  p  p p ≤ μ A1 (x) ∧ μ A2 (x2 ) ∧ μ A2 (y2 )  p   p  p p = μ A1 (x) ∧ μ A2 (x2 ) ∧ μ A2 (x) ∧ μ A2 (y2 ) p

p

= μ A1 A2 (x, x2 ) ∧ μ A1 A2 (x, y2 ), μnB1 B2 ((x, x2 )(x, y2 )) = μnA1 (x) ∨ μnB2 (x2 y2 )   ≥ μnA1 (x) ∨ μnA2 (x2 ) ∨ μnA2 (y2 )     = μnA1 (x) ∨ μnA2 (x2 ) ∨ μnA1 (x) ∨ μnA2 (y2 ) = μnA1 A2 (x, x2 ) ∨ μnA1 A2 (x, y2 ). Let z ∈ X 2 , x1 y1 ∈ E 1 , then p

p

p

μ B1 B2 ((x1 , z)(y1 , z)) = μ B1 (x1 y1 ) ∧ μ A2 (z)  p  p p ≤ μ A1 (x1 ) ∧ μ A1 (y1 ) ∧ μ A2 (z)  p    p p p = μ A1 (x1 ) ∧ μ A2 (z) ∧ μ A1 (y1 ) ∧ μ A2 (z) p

p

= μ A1 A2 (x1 , z) ∧ μ A1 A2 (y1 , z), μnB1 B2 ((x1 , z)(y1 , z)) = μnB1 (x1 y1 ) ∨ μnA2 (z)   ≥ μnA1 (x1 ) ∨ μnA1 (y1 ) ∨ μnA2 (z)     = μnA1 (x1 ) ∨ μnA2 (z) ∨ μnA1 (y1 ) ∨ μnA2 (z) = μnA1 A2 (x1 , z) ∨ μnA1 A2 (y1 , z).

18

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs x1 (0.5, −0.3)

y1 (0.6, −0.2) G1

(0.5, −0.2) (y1 , x2 )(0.6, −0.2)

(x1 , x2 )(0.5, −0.2)

(0.5, −0.2)

2) 0. − , .5 (0

(0 .5 ,− 0. 2)

(x1 , y2 )(0.5,

(y1 , y2 )(0.5,

0.3)

x2 (0.7, −0.2)

0.2)

y2 (0.5,

G2

0.3)

Fig. 1.10 Direct product G 1 × G 2



This completes the proof.

Definition 1.19 The direct product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is a pair G 1 × G 2 = (A1 × A2 , B1 × B2 ), where A1 × A2 = p p (μ A1 ×A2 , μnA1 ×A2 ) and B1 × B2 = (μ B1 ×B2 , μnB1 ×B2 ) are defined as  p  p 1. (A1 × A2 )(x1 , x2 ) = μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , for all (x1 , x2 ) ∈ X 1 × X 2,  p  p 2. (B1 × B2 )((x1 , x2 )(x1 , y2 ))= μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) , for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Example 1.8 The direct product of two bipolar fuzzy graphs G 1 and G 2 is shown in Fig. 1.10. Proposition 1.8 The direct product of two bipolar fuzzy graphs is a bipolar fuzzy graph. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let x1 y1 ∈ E 1 and x2 y2 ∈ E 2 , then p

p

p

μ B1 ×B2 ((x1 , x2 )(y1 , y2 )) = μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 )  p   p  p p ≤ μ A1 (x1 ) ∧ μ A1 (y1 ) ∧ μ A2 (x2 ) ∧ μ A2 (y2 )  p   p  p p = μ A1 (x1 ) ∧ μ A2 (x2 ) ∧ μ A1 (y1 ), μ A2 (y2 ) p

p

= μ A1 ×A2 (x1 , x2 ) ∧ μ A1 ×A2 (y1 , y2 ),

1.2 Bipolar Fuzzy Graphs

19

x1 (0.5, −0.3)

y1 (0.6, −0.2) (0.5, −0.2) (y1 , x2 )(0.6, −0.2)

(x1 , x2 )(0.5, −0.2)

G1

x2 (0.7, −0.2)

(0 .5 ,− 0. 2)

(0.5, −0.2)

(0.5, −0.2)

2) 0. − , .5 (0

(0.5, −0.2)

(0.5, −0.2)

G2

(0.5, −0.2) (x1 , y2 )(0.5, −0.3)

(y1 , y2 )(0.5, −0.2)

y2 (0.5,

0.3)

Fig. 1.11 Strong product G 1  G 2

μnB1 ×B2 ((x1 , x2 )(y1 , y2 )) = μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 )     ≥ μnA1 (x1 ) ∨ μnA1 (y1 ) ∨ μnA2 (x2 ) ∨ μnA2 (y2 )     = μnA1 (x1 ) ∨ μnA2 (x2 ) ∨ μnA1 (y1 ) ∨ μnA2 (y2 ) = μnA1 ×A2 (x1 , x2 ) ∨ μnA1 ×A2 (y1 , y2 ). The proof is complete.



Definition 1.20 The strong product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is denoted by the pair G 1  G 2 = (A1  A2 , B1  B2 ) and defined as  p  p 1. (A1  A2 )(x1 , x2 )= μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , for all (x1 , x2 ) ∈ X 1 × X 2,  p  p 2. (B1  B2 )((x, x2 )(x, y2 ))= μ A1 (x) ∧ μ B2 (x2 y2 ), μnA1 (x) ∨ μnB2 (x2 y2 ) , for all x ∈ X 1 , x2 y2 ∈ E 2 ,  p  p 3. (B1  B2 )((x1 , z)(y1 , z)) = μ B1 (x1 y1 ) ∧ μ A2 (z), μnB1 (x1 y1 ) ∨ μnA2 (z) , for all z ∈ X 2 , x1 y1 ∈ E 1 ,  p  p 4. (B1  B2 )((x1 , x2 )(x1 , y2 ))= μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) , for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Example 1.9 The strong product of two bipolar fuzzy graphs G 1 and G 2 is shown in Fig. 1.11. Proposition 1.9 The strong product of two bipolar fuzzy graphs is a bipolar fuzzy graph.

20

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let x ∈ X 1 and x2 y2 ∈ E 2 , then p

p

p

μ B1 B2 ((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 )  p  p p ≤ μ A1 (x) ∧ μ A2 (x2 ) ∧ μ A2 (y2 )  p   p  p p = μ A1 (x) ∧ μ A2 (x2 ) ∧ μ A2 (x) ∧ μ A2 (y2 ) p

p

= μ A1 A2 (x, x2 ) ∧ μ A1 A2 (x, y2 ), μnB1 B2 ((x, x2 )(x, y2 )) = μnA1 (x) ∨ μnB2 (x2 y2 )   ≥ μnA1 (x) ∨ μnA2 (x2 ) ∨ μnA2 (y2 )     = μnA1 (x) ∨ μnA2 (x2 ) ∨ μnA1 (x) ∨ μnA2 (y2 ) = μnA1 A2 (x, x2 ) ∨ μnA1 A2 (x, y2 ). Let z ∈ X 2 and x1 y1 ∈ E 1 , then p

p

p

μ B1 B2 ((x1 , z)(y1 , z)) = μ B1 (x1 y1 ) ∧ μ A2 (z)  p  p p ≤ μ A1 (x1 ) ∧ μ A1 (y1 ) ∧ μ A2 (z)  p    p p p = μ A1 (x1 ) ∧ μ A2 (z) ∧ μ A1 (y1 ) ∧ μ A2 (z) p

p

= μ A1 A2 (x1 , z) ∧ μ A1 A2 (y1 , z), μnB1 B2 ((x1 , z)(y1 , z)) = μnB1 (x1 y1 ) ∨ μnA2 (z)   ≥ μnA1 (x1 ) ∨ μnA1 (y1 ) ∨ μnA2 (z)     = μnA1 (x1 ) ∨ μnA2 (z) ∨ μnA1 (y1 ) ∨ μnA2 (z) = μnA1 A2 (x1 , z) ∨ μnA1 A2 (y1 , z). Let x1 y1 ∈ E 1 and x2 y2 ∈ E 2 , then p

p

p

μ B1 B2 ((x1 , x2 )(y1 , y2 )) = μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 )  p   p  p p ≤ μ A1 (x1 ) ∧ μ A1 (y1 ) ∧ μ A2 (x2 ) ∧ μ A2 (y2 )  p   p  p p = μ A1 (x1 ) ∧ μ A2 (x2 ) ∧ μ A1 (y1 ), μ A2 (y2 ) p

p

= μ A1 A2 (x1 , x2 ) ∧ μ A1 A2 (y1 , y2 ), μnB1 B2 ((x1 , x2 )(y1 , y2 )) = μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 )     ≥ μnA1 (x1 ) ∨ μnA1 (y1 ) ∨ μnA2 (x2 ) ∨ μnA2 (y2 )     = μnA1 (x1 ) ∨ μnA2 (x2 ) ∨ μnA1 (y1 ) ∨ μnA2 (y2 ) = μnA1 A2 (x1 , x2 ) ∨ μnA1 A2 (y1 , y2 ). This completes the proof.



1.2 Bipolar Fuzzy Graphs

21

Definition 1.21 The lexicographic product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ), denoted by G 1 • G 2 , is defined as a pair (A1 • A2 , B1 • B2 ) such that  p  p 1. (A1 • A2 )(x1 , x2 ) = μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , for all (x1 , x2 ) ∈ X 1 × X 2,  p  p 2. (B1 • B2 )((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 ), μnA1 (x) ∨ μnB2 (x2 y2 ) , for all x ∈ X 1 , x2 y2 ∈ E 2 ,  p  p 3. (B1 • B2 )((x1 , x2 )(x1 , y2 )) = μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) , for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Definition 1.22 Let G = (A, B) be a bipolar fuzzy graph on X . A bipolar fuzzy walk in a bipolar fuzzy graph G is an alternating sequence of vertices and edges yo , e1 , y1 , e2 , . . ., en−1 , yn−1 such that B(ek ) = (0, 0), for all 1 ≤ k ≤ n − 1. Definition 1.23 A bipolar fuzzy path in a bipolar fuzzy graph G = (A, B) is p a sequence of distinct vertices y1 , y2 , . . ., yn such that either μ B (yk yk+1 ) > 0 n or μ B (yk yk+1 ) < 0, for all 1 ≤ k ≤ n − 1. It is denoted by Pn . If y1 = yn and A(y1 ) = A(yn ), the bipolar fuzzy path is known as a bipolar fuzzy cycle denoted by Cn . A bipolar fuzzy graph G is connected if there exists a bipolar fuzzy path between each pair of distinct vertices. An example of a bipolar fuzzy path P5 is shown in Fig. 1.12. Definition 1.24 A bipolar fuzzy graph G = (A, B) on a non-empty set X is known as a strong bipolar fuzzy graph if  p   p  p B(x y) = μ B (x y), μnB (x y) = μ A (x) ∧ μ A (y), μnA (x) ∨ μnA (y) , for all x y ∈ E.

Example 1.10 Let A be a bipolar fuzzy set on X = {x, y, z} given in Table 1.5 and B be a bipolar fuzzy relation in X defined in Table 1.6. By routine computations, it is easy to see that the bipolar fuzzy graph G = (A, B), shown in Fig. 1.13, is a strong bipolar fuzzy graph.

a1 (0.2, −0.5)

a2 (0.3, −0.4)

a3 (0.2, −0.5)

(0.2, −0.4)

(0.1, −0.3)

a4 (0.5, −0.1)

(0.2, −0.1)

(0.1, −0.1)

Fig. 1.12 Bipolar fuzzy path P5 Table 1.5 Bipolar fuzzy set A on {x, y, z} A x p μA μnA

0.2 −0.4

y

z

0.3 −0.5

0.4 −0.5

a5 (0.4, −0.3)

22

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Table 1.6 Bipolar fuzzy relation B in {x, y, z} B x x y z

(0, 0) (0.2, −0.4) (0.2, −0.4)

Fig. 1.13 Strong bipolar fuzzy graph G

y

z

(0.2, −0.4) (0, 0) (0.3, −0.5)

(0.2, −0.4) (0.3, −0.5) (0, 0) z(0.4, −0.5)

y(0.3, −0.5) (0.3, −0.5)

(0.2, −0.4)

(0 .2 ,− 0. 4)

x(0.2, −0.4)

a2 (0.3, −0.4)

a1 (0.2, −0.5) (0.2, −0.4)

1) 0. − , .3 (0

(0 .2 ,− 0. 5)

(0.2, −0.4)

(0.2, −0.1)

Fig. 1.14 Complete bipolar fuzzy graph

(0.2, −0.1) a4 (0.5, −0.1)

a3 (0.2, −0.5)

Definition 1.25 A bipolar fuzzy graph G = (A, B) on a non-empty set X is known as a complete bipolar fuzzy graph if   p   p p B(x y) = μ B (x y), μnB (x y) = μ A (x) ∧ μ A (y), μnA (x) ∨ μnA (y) , for all x, y ∈ X. Example 1.11 Consider a bipolar fuzzy graph G on X = {a1 , a2 , a3 , a4 } as shown in Fig. 1.14. Routine calculations show that G is a complete bipolar fuzzy graph. Definition 1.26 A bipolar fuzzy graph G = (A, B) on a non-empty set X is known as a bipartite bipolar fuzzy graph if the set of vertices X can be written as the union of two disjoint sets X 1 and X 2 such that for some k and j,

1.2 Bipolar Fuzzy Graphs

23

(0.2, −0.1)

a1 (0.2, −0.5)

1) 0. − , .3 (0

a3 (0.4, −0.3)

a2 (0.3, −0.4)

(0 .2 ,− 0. 5)

b1 (0.5, −0.1)

2) 0. − , .2 (0

(0 .2 ,− 0. 3)

b2 (0.2, −0.5)

b3 (0.2,

0.6)

Fig. 1.15 Bipartite bipolar fuzzy graph

1. B(xk x j ) = (0, 0), if xk , x j ∈ X 1 or xk , x j ∈ X 2 , 2. B(xk x j ) = (0, 0), if xk ∈ X 1 and x j ∈ X 2 or xk ∈ X 2 and x j ∈ X 1 . Example 1.12 Let G be a bipolar fuzzy graph on X = {a1 , a2 , a3 } ∪ {b1 , b2 , b3 } as shown in Fig. 1.15. By Definition 1.26, G is a bipartite bipolar fuzzy graph. Definition 1.27 A bipolar fuzzy graph G = (A, B) is said to be n-partite (multipartite) bipolar fuzzy graph if the vertex set X can be partitioned into n disjoint subsets X 1 , X 2 , . . . X n such that 1. B(x y) = (0, 0), for every x, y ∈ X k , 1 ≤ k ≤ n. 2. B(x y) = (0, 0), if x ∈ X i , y ∈ X j , X i = X j , 1 ≤ i, j ≤ n. Definition 1.28 A bipolar fuzzy graph G = (A, B) on a non-empty set X is called a complete bipartite bipolar fuzzy graph if the set of vertices X can be written as the union of two disjoint sets X 1 and X 2 such that, for all x, y ∈ X , 1. B(x y) = (0, 0), if x, y ∈ X 1 or x, y ∈ X 2 ,  p p 2. B(x y) = μ A (x) ∧ μ A (y), μnA (x) ∨ μnA (y) , if x ∈ X 1 and y ∈ X 2 or x ∈ X 2 and y ∈ X 1 . Example 1.13 A complete bipartite bipolar fuzzy graph on X = {a1 , a2 , a3 } ∪ {b1 , b2 , b3 } is shown in Fig. 1.16. 





Definition 1.29 A bipolar fuzzy graph G = (A , B ) on a non-empty set X is known as a bipolar fuzzy subgraph of bipolar fuzzy graph G = (A, B) if  p p 1. A ⊆ A, that is, μ A (x) ≤ μ A (x) and μnA (x) ≥ μnA (x), for all x ∈ X .  p p 2. B ⊆ B, that is, μ B  (x y) ≤ μ B (x y) and μnB  (x y) ≥ μnB (x y), for all x y ∈ E. A spanning subgraph of a bipolar fuzzy graph G = (A, B) is a bipolar fuzzy graph H = (C, D) such that C(x) = A(x), for all x ∈ X .

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

(0. 2, − (0 0.5 .2 ) ,− 0. 5)

4, (0.

1) 0. − , .3 (0

b1 (0.5, −0.1)

.1) −0

3) 0. − , .2 (0

(0 .2 ,− 0. 4)

(0.2, −0.4)

(0.2, −0.1)

a3 (0.4, −0.3)

a2 (0.3, −0.4)

a1 (0.2, −0.5)

(0.2, −0.3)

24

b3 (0.2, −0.6)

b2 (0.2, −0.5)

Fig. 1.16 Complete bipartite bipolar fuzzy graph

Definition 1.30 The complement of a bipolar fuzzy G = (A, B) is a pair G c = p (Ac , B c ) or G = (A, B), where Ac = A and B c = (μ B c , μnB c ) is defined as p p p p n n μ B c (x y) = μ A (x) ∧ μ A (y) − μ B (x y) and μ B c (x y) = μ A (x) ∨ μnA (y) − μnB (x y). Definition 1.31 The complement of a strong bipolar fuzzy graph G = (A, B), on a non-empty set X , is a strong bipolar fuzzy graph G c = (Ac , B c ) on X such that p Ac = A and B c = (μ B c , μnB c ) is defined as  p μ B c (x y)

= 

μnB c (x y)

=

p

0 if μ B (x y) > 0, p p p μ A (x) ∧ μ A (y) if μ B (x y) = 0, 0 if μnB (x y) < 0, μnA (x) ∨ μnA (y) if μnB (x y) = 0.

Example 1.14 Consider a bipolar fuzzy graph G = (A, B) on X = {a, b, c} as shown in Fig. 1.17. The complement of G is given in Fig. 1.18.

Fig. 1.17 Bipolar fuzzy graph G

b(0.1, −0.4)

c(0.2, −0.3) (0.1, −0.3)

(0.2, −0.3)

a(0.3, −0.4)

1.2 Bipolar Fuzzy Graphs

25 b(0.1, −0.4)

Fig. 1.18 Complement of G

c(0.2, −0.3)

(0.1, −0.4)

a(0.3, −0.4)

Proposition 1.10 If G 1 and G 2 are strong bipolar fuzzy graphs, then G 1 G 2 , G 1 ◦ G 2 , G 1 × G 2 , G 1  G 2 and G 1 • G 2 are strong bipolar fuzzy graphs. Remark 1.2 The union of two strong bipolar fuzzy graphs is not necessarily a strong bipolar fuzzy graph as it can be seen in Example 1.15. Example 1.15 Let G 1 and G 2 be two strong bipolar fuzzy graphs on X = {a, b, c}. The union G 1 ∪ G 2 shown in Fig. 1.19 is not a strong bipolar fuzzy graph. Proposition 1.11 If G 1 G 2 is a strong bipolar fuzzy graph, then at least one of G 1 or G 2 must be strong. Proof Suppose that G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) are not strong bipolar fuzzy graphs. Then there exist x1 y1 ∈ E 1 and x2 y2 ∈ E 2 such that p

p

p

p

p

p

μ B1 (x1 y1 ) < μ A1 (x1 ) ∧ μ A1 (y1 ), μ B2 (x1 y1 ) < μ A2 (x2 ) ∧ μ A2 (y2 ), μnB1 (x1 y1 ) > μnA1 (x1 ) ∨ μnA1 (y1 ), μnB2 (x1 y1 ) > μnA2 (x2 ) ∨ μnA2 (y2 ). p

p

p

p

p

Assume that μ B2 (x2 y2 ) ≤ μ B1 (x1 y1 ) < μ A1 (x1 ) ∧ μ A1 (y1 ) ≤ μ A1 (x1 ) and E = {(x, x2 )(x, y2 )|x ∈ X 1 , x2 y2 ∈ E 2 } ∪ {(x1 , z)(y1 , z)|z ∈ X 2 , x1 y1 ∈ E 1 }. Consider (x, x2 )(x, y2 ) ∈ X 1 × X 2 such that x ∈ X 1 and x2 y2 ∈ X 2 , then p

p

p

p

p

p

μ B1 B2 ((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 ) < μ A1 (x) ∧ μ A2 (x2 ) ∧ μ A2 (y2 ), p

p

p

p

p

p

μ A1 A2 (x, x2 ) = μ A1 (x) ∧ μ A2 (x2 ), μ A1 A2 (x, y2 ) = μ A1 (x) ∧ μ A2 (y2 ), p

p

p

p

p

⇒ μ A1 A2 (x, x2 ) ∧ μ A1 A2 (x, y2 ) = μ A1 (x) ∧ μ A2 (x2 ) ∧ μ A2 (y2 ), p

p

p

⇒ μ B1 B2 ((x, x2 )(x, y2 )) < μ A1 A2 (x, x2 ) ∧ μ A1 A2 (x, y2 ). Similarly, it can be easily shown that μnB1 B2 ((x, x2 )(x, y2 )) > max(μnA1 A2 (x, x2 ), μnA1 A2 (x, y2 )). That is, G 1 G 2 is not a strong bipolar fuzzy graph, a contradiction.

26

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

a(0.6, −0.5)

b(0.4, −0.7)

b(0.3, −0.4)

c(0.7, −0.6) (0.3, −0.4)

(0.4, −0.5)

G2 is strong.

(0.2, −0.5)

(0.4, −0.6)

G1 is strong.

a(0.2, −0.5)

c(0.5, −0.6)

a(0.6, −0.5)

b(0.4, −0.7) (0.4, −0.5)

(0 .2 ,− 0. 5)

(0.4, −0.6)

G1 ∪ G2 is not strong.

c(0.7, −0.6)

Fig. 1.19 Union of strong bipolar fuzzy graphs

Hence if G 1 G 2 is strong bipolar fuzzy graph, then at least one of G 1 or G 2 must be strong.  Proposition 1.12 If G 1 ◦ G 2 is a strong bipolar fuzzy graph, then at least one of G 1 or G 2 must be strong.

1.3 Isomorphism of Bipolar Fuzzy Graphs In this section, the concept of isomorphism in bipolar fuzzy graphs is studied with certain properties which remain invariant between isomorphic bipolar fuzzy graphs. Definition 1.32 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. A homomorphism of bipolar fuzzy graphs G 1 and G 2 , written as G 1 is homomorphic to G 2 , is a mapping f : X 1 → X 2 that satisfies

1.3 Isomorphism of Bipolar Fuzzy Graphs p

27

p

1. μ A1 (x1 ) ≤ μ A2 ( f (x1 )), μnA1 (x1 ) ≥ μnA2 ( f (x1 )), for all x1 ∈ X 1 p p 2. μ B1 (x1 y1 ) ≤ μ B2 ( f (x1 ) f (y1 )), μnB1 (x1 y1 ) ≥ μnB2 ( f (x1 ) f (y1 )), for all x1 , y1 ∈ X 1. Definition 1.33 A weak isomorphism of bipolar fuzzy graphs G 1 and G 2 is a bijective homomorphism f : X 1 → X 2 that satisfies p

p

μ A1 (x1 ) = μ A2 ( f (x1 )), μnA1 (x1 ) = μnA2 ( f (x1 )), for all x1 ∈ X 1 . Definition 1.34 A bijective homomorphism f : X 1 → X 2 is called a co-weak isomorphism from a bipolar fuzzy graph G 1 to a bipolar fuzzy graph G 2 if it satisfies p

p

μ B1 (x1 y1 ) = μ B2 ( f (x1 ) f (y1 )), μnB1 (x1 y1 ) = μnB2 ( f (x1 ) f (y1 )), for all x1 , y1 ∈ X 1 . Definition 1.35 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. An isomorphism of bipolar fuzzy graphs G 1 and G 2 , written as G 1 ∼ = G 2 , is a bijective mapping f : X 1 → X 2 that satisfies p

p

1. μ A1 (x1 ) = μ A2 ( f (x1 )), μnA1 (x1 ) = μnA2 ( f (x1 )), for all x1 ∈ X 1 p p 2. μ B1 (x1 y1 ) = μ B2 ( f (x1 ) f (y1 )), μnB1 (x1 y1 ) = μnB2 ( f (x1 ) f (y1 )), for all x1 , y1 ∈ X 1. Remark 1.3 1. If G 1 = G 2 = G, then the homomorphism f over itself is called an endomorphism. An isomorphism f over G is called an automorphism. p 2. Let A = (μ A , μnA ) be a bipolar fuzzy graph with an underlying set X . Let Aut(G) be the set of all automorphisms f : X → X of G. Let e : X → X be a map defined by e(x) = x, for all x ∈ X . Clearly, e ∈ Aut (G). 3. If G 1 = G 2 , then the weak and co-weak isomorphisms actually become isomorphisms. 4. If f : X 1 → X 2 is a bijective map then f −1 : X 2 → X 1 is also bijective map. 





Example 1.16 Consider bipolar fuzzy graphs G = (A, B) and G = (A , B ) on X = {a, b, c, d, e, f } and X  = {a  , b , c , d  , e , f  } as shown in Figs. 1.20 and    1.21, respectively. Define a mapping f : X → X by f (a) = c , f (c) = a , f (b) =       d , f (d) = b , f (e) = e and f ( f ) = f . Clearly, A(a) = (0.2, −0.3) = A (c ) =       A ( f (a)), A(b) = (0.5, −0.4) = A (d ) = A ( f (b)) A(c) = (0.5, −0.6) = A (a ) =      A ( f (c)), B(ab) = (0.2, −0.3) = B (c d ) = B ( f (a) f (b)). Similarly, A(x) =    A ( f (x)) and B(x y) = B ( f (x) f (y)), for all x, y ∈ X . Hence G and G are isomorphic bipolar fuzzy graphs. Definition 1.36 A bipolar fuzzy graph G is called self-complementary if G c ∼ = G. Theorem 1.1 Let G = (A, B) be a self-complementary bipolar fuzzy graph on X , then

28

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs a(0.2, −0.3)

b(0.5, −0.4) (0.2, −0.3)

(0.1, −0.2)

(0.1, −0.1)

2) 0. − , .5 (0

(0.1, −0.1) e(0.5, −0.5)

Fig. 1.21 Bipolar fuzzy  graph G

d(0.8, −0.2) b (0.8, −0.2)

a (0.5, −0.6) −0 .3)

(0 .1 ,− 0. 2)

(0 .2,



(0 .1 ,− 0. 1)

e (0.5, −0.5)

.3) −0

x= y

1 p p μ (x) ∧ μ A (y), 2 x= y A

.1) −0

2) 0. ,− .1 (0

.2, (0

d (0.5, −0.4)

p

μ B (x y) =

.1, (0

(0.5, −0.2)

c (0.2, −0.3)



(0.2, −0.3)

(0.1, −0.2)

f (0.1, −0.5)

c(0 .5, − 0.6)

Fig. 1.20 Bipolar fuzzy graph G

f (0.1, −0.5)

μnB (x y) =

x= y

1 n μ (x) ∨ μnA (y). 2 x= y A

Proof Since G = (A, B) is a self-complementary bipolar fuzzy graph, there exists an isomorphism f : X → X from G to G c = (Ac , B c ) such that p

p

1. μ A (x) = μ Ac ( f (x)), μnA (x) = μnAc ( f (x)), for all x ∈ X p p 2. μ B (x y) = μ B c ( f (x) f (y)), μnB (x y) = μnB c ( f (x) f (y)), for all x, y ∈ X . Now by Definition 1.30, p

p

p

p

μ B c ( f (x) f (y)) = μ Ac ( f (x)) ∧ μ Ac ( f (y)) − μ B ( f (x) f (y)) p

⇒

x = y

p

μ B (x y) +

x = y

p

p

p

⇒ μ B (x y) = μ A (x) ∧ μ A (y) − μ B ( f (x) f (y)) p p p μ B ( f (x) f (y)) = μ A (x) ∧ μ A (y)

x = y p p p ⇒ 2 μ B (x y) = μ A (x) ∧ μ A (y) x = y x = y



1.3 Isomorphism of Bipolar Fuzzy Graphs ⇒

x = y

p

μ B (x y) =

29 1 p p μ A (x) ∧ μ A (y). 2 x = y

Similarly, μnB c ( f (x) f (y)) = μnAc ( f (x)) ∨ μnAc ( f (y)) − μnB ( f (x) f (y))

⇒

x = y

μnB (x y) +

x = y

⇒ μnB (x y) = μnA (x) ∨ μnA (y) − μnB ( f (x) f (y)) μnB ( f (x) f (y)) = μnA (x) ∨ μnA (y)

⇒ 2

x = y

⇒



x = y

x = y

μnB (x y) =



x = y

1 μnB (x y) = 2

μnA (x) ∨ μnA (y)



x = y

μnA (x) ∨ μnA (y).



The proof is complete. Theorem 1.2 Let G = (A, B) be a bipolar graph such that for all x, y ∈ X , p

μ B (x y) =

1 p 1 p (μ (x) ∧ μ A (y)) and μnB (x y) = (μnA (x) ∨ μnA (y)), 2 A 2

then G is a self-complementary bipolar fuzzy graph. Proof Let G c = (Ac , B c ) be the complement of a bipolar fuzzy graph G = (A, B), then by Definition 1.30, p

p

p

p

μ B c (x y) = μ A (x) ∧ μ A (y) − μ B (x y),

μnB c (x y) = μnA (x) ∨ μnA (y) − μnB (x y),

1 p p p p p μ B c (x y) = μ A (x) ∧ μ A (y) − (μ A (x) ∧ μ A (y)), 2 1 p p p μ B c (x y) = (μ A (x) ∧ μ A (y)), 2 p p μ B c (x y) = μ B (x y).

μnB c (x y) = μnA (x) ∨ μnA (y) −

1 n (μ (x) ∨ μnA (y)), 2 A

1 n (μ (x) ∨ μnA (y)), 2 A μnB c (x y) = μnB (x y). μnB c (x y) =

It follows that B = B c , that is, G is a self-complementary bipolar fuzzy graph.  Proposition 1.13 Let G 1 and G 2 be strong bipolar fuzzy graphs, then G 1 ∼ = G 2 if and only if G c1 ∼ = G c2 . Proof Assume that G 1 and G 2 are isomorphic to each other, then there exists a bijective map f : X 1 → X 2 satisfying p

p

μ A1 (x) = μ A2 ( f (x)), μnA1 (x) = μnA2 ( f (x)), for all x ∈ X 1 , p

p

μ B1 (x y) = μ B2 ( f (x) f (y)), μnB1 (x y) = μnB2 ( f (x) f (y)), for all x y ∈ E 1 .

30

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Case 1. If for x, y ∈ X 1 , B1 (x y) = (0, 0), then B2 ( f (x) f (y)) = (0, 0). By Definition 1.30, p

p

p

p

p

p

μ B c (x y) = μ A1 (x) ∧ μ A1 (y) = μ A2 ( f (x)) ∧ μ A2 ( f (y)) = μ B c ( f (x) f (y)), 1

2

μnB1c (x y) = μnA1 (x) ∨ μnA1 (y) = μnA2 ( f (x)) ∨ μnA2 ( f (y)) = μnB2c ( f (x) f (y)), forall x y ∈ E 1 . Case 2. If for x, y ∈ X 1 , B1 (x y) ∈ (0, 1] × [−1, 0), then B2 ( f (x) f (y)) ∈ (0, 1] × [−1, 0). By Definition 1.30, B1c (x y) = (0, 0) = B2c ( f (x) f (y)). Hence G c1 ∼ = G c2 . c ∼ c Conversely, let G 1 = G 2 , then there exists a bijective mapping g : X 1 → X 2 which satisfies p

p

μ Ac (x) = μ Ac (g(x)), μnAc1 (x) = μnAc2 (g(x)) , for all x ∈ X 1 , 1

p

2

p

μ B c (x y) = μ B c ( f (x) f (y)) , μnB1c (x y) = μnB2c ( f (x) f (y)) , for all x y ∈ E 1 . 1

2

Case 1. If for x, y ∈ X 1 , B1 (x y) = (0, 0), then B1c (x y) ∈ (0, 1] × [−1, 0). It implies that B2c ( f (x) f (y)) ∈ (0, 1] × [−1, 0). By Definition 1.30, B2 ( f (x) f (y)) = (0, 0). Case 2. If B1 (x y) ∈ (0, 1] × [−1, 0), then by Definition 1.30, B1c (x y) = (0, 0) = B2c ( f (x) f (y)) = (0, 0). Thus we have p

p

p

p

p

μ B2 ( f (x) f (y)) = μ A2 ( f (x)) ∧ μ A2 ( f (y)) = μ A2 ( f (x)) ∧ μ A2 ( f (y)) p

p

= μ Ac ( f (x)) ∧ μ Ac ( f (y)) 2

p

2

p

= μ Ac (x) ∧ μ Ac (y) 1

p

1

p

= μ A1 (x) ∧ μ A1 (y) p

= μ B1 (x y), μnB2 ( f (x) f (y)) = μnA2 ( f (x)) ∨ μnA2 ( f (y)) = μnA2 ( f (x)) ∨ μnA2 ( f (y)) = μnAc2 ( f (x)) ∨ μnAc2 ( f (y)) = μnAc1 (x) ∨ μnAc1 (y) = μnA1 (x) ∨ μnA1 (y) = μnB1 (x y). It follows that G 1 ∼ = G2.



1.3 Isomorphism of Bipolar Fuzzy Graphs

31

Proposition 1.14 Let G 1 and G 2 be strong bipolar fuzzy graphs. If there is a weak isomorphism between G 1 and G 2 , then there is a weak isomorphism between G c1 and G c2 . Proof Let f be a weak isomorphism between G 1 and G 2 then f : X 1 → X 2 defined by f (x1 ) = x2 , for all x1 ∈ X 1 , is a bijective map that satisfies p

p

μ A1 (x1 ) = μ A2 ( f (x1 )), μnA1 (x1 ) = μnA2 ( f (x1 )), for all x1 ∈ X 1 , p

p

μ B1 (x1 y1 ) ≤ μ B2 ( f (x1 ) f (y1 )), μnB1 (x1 y1 ) ≥ μnB2 ( f (x1 ) f (y1 )), for all x1 y1 ∈ E 1 . Since f : X 1 → X 2 is a bijective map, f −1 : X 2 → X 1 is also bijective map such p p that f −1 (x2 ) = x1 , for all x2 ∈ X 2 . Thus μ A1 ( f −1 (x2 )) = μ A2 (x2 ), μnA1 ( f −1 (x2 )) = n μ A2 (x2 ), for all x2 ∈ X 2 . Case 1. If for x, y ∈ X 1 , B1 (x y) = (0, 0), then by Definition 1.30 μ B c ( f −1 (x2 ) 1 p p p p p p f −1 (y2 )) = μ B c (x1 y1 ) = μ A1 (x1 ) ∧ μ A1 (y1 ) = μ A2 ( f (x1 )) ∧ μ A2 ( f (y1 )) ≥ μ B c 1 2 p ( f (x1 ) f (y1 )) = μ B c (x2 y2 ) because, 2 p p p p 1. If μ B2 ( f (x1 ) f (y1 )) = 0, then μ B c ( f (x1 ) f (y1 )) = μ A2 ( f (x1 )) ∧ μ A2 ( f (y1 )). 2 p p p p 2. If μ B2 ( f (x1 ) f (y1 )) > 0, then μ B c ( f (x1 ) f (y1 )) = 0 < μ A2 ( f (x1 )) ∧ μ A2 2 ( f (y1 )). Analogously, it can be proved that μnB c ( f −1 (x2 ) f −1 (y2 )) ≤ μnB c (x1 y1 ). 1 2 Case 2. If for x, y ∈ X 1 , B1 (x y) = (0,0), then on the same lines as case 1, p p it can be proved that μ B c (x2 y2 ) ≤ μ B c f −1 (x2 ) f −1 (y2 ) and μnB c (x2 y2 ) ≥ μnB c 2 1 2 1  −1  f (x2 ) f −1 (y2 ) . Thus, f −1 : X 2 → X 1 is a bijective map which is a weak iso morphism between G c1 and G c2 . The proof of converse part is straightforward. p

Proposition 1.15 Let G 1 and G 2 be strong bipolar fuzzy graphs. If there is a coweak isomorphism between G 1 and G 2 , then there is a homomorphism between G c1 and G c2 . Theorem 1.3 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively, such that X 1 ∩ X 2 = ∅, then 1. (G 1 + G 2 )c ∼ = G c1 ∪ G c2 , 2. (G 1 ∪ G 2 )c ∼ = G c1 + G c2 . Proof 1. To prove that (G 1 + G 2 )c and G c1 ∪ G c2 are isomorphic to each other, it is to be shown that the identity mapping I : X 1 ∪ X 2 → X 1 ∪ X 2 is the required isomorphism, that is, for all x, y ∈ X 1 ∪ X 2 , (A1 + A2 )c (x) = Ac1 (x) ∪ Ac2 (x) and (B1 + B2 )c (x y) = B1c (x y) ∪ B2c (x y). (A1 + A2 )c (x) = (A1 + A2 )(x) = (A1 ∪ A2 )(x) p p = (μ A1 (x) ∨ μ A2 (x), μnA1 (x) ∧ μnA2 (x)) p

p

= (μ Ac (x) ∨ μ Ac (x), μnAc1 (x) ∧ μnAc2 (x)) 1

2

32

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

= (Ac1 ∪ Ac2 )(x), p

p

p

p

μ(B1 +B2 )c (x y) = μ A1 +A2 (x) ∧ μ A1 +A2 (y) − μ B1 +B2 (x y)  p p p μ A1 ∪A2 (x) ∧ μ A1 ∪A2 (y) − μ B1 ∪B2 (x y), if x y ∈ E 1 ∪ E 2 =  p p p p μ A1 ∪A2 (x) ∧ μ A1 ∪A2 (y) − μ A1 (x) ∧ μ A2 (y), if x y ∈ E and x ∈ X 1 , y ∈ X 2 ⎧ p p p ⎪ ⎪ ⎨μ A1 (x) ∧ μ A1 (y) − μ B1 (x y), if x y ∈ E 1 =

=

p

p

p

μ A2 (x) ∧ μ A2 (y) − μ B2 (x y), if x y ∈ E 2 ⎪ ⎪ ⎩μ p (x) ∧ μ p (y) − μ p (x) ∧ μ p (y), if x y ∈ E  and x ∈ X , y ∈ X 1 2 A1 A2 A1 A2 ⎧ p ⎪ ⎨ μ B c (x y), if x y ∈ E 1 p

⎪ ⎩

1

μ B c (x y), if x y ∈ E 2 2



if x y ∈ E and x ∈ X 1 , y ∈ X 2

0,

p

= μ B c ∪B c (x y), 1

2

μn(B +B )c (x y) = μnA +A (x) ∨ μnA +A (y) − μnB +B (x y) 1 2 1 2 1 2 1 2  n μ A ∪A (x) ∨ μnA ∪A (y) − μnB ∪B (x y), if x y ∈ E 1 ∪ E 2 1 2 1 2 1 2 =  μnA ∪A (x) ∨ μnA ∪A (y) − μnA (x) ∨ μnA (y), if x y ∈ E and x ∈ X 1 , y ∈ X 2 1 2 1 2 1 2 ⎧ n n n ⎪ ⎪ ⎨μ A1 (x) ∨ μ A1 (y) − μ B1 (x y), if x y ∈ E 1 n n = μ A (x) ∨ μ A (y) − μnB (x y), if x y ∈ E 2 2 2 2 ⎪ ⎪ ⎩μn (x) ∨ μn (y) − μn (x) ∨ μn (y), if x y ∈ E  and x ∈ X , y ∈ X 1 2 A1 A2 A1 A2 ⎧ n (x y), if x y ∈ E μ ⎪ c 1 ⎪ ⎨ B1 = μnB c (x y), if x y ∈ E 2 ⎪ 2 ⎪  ⎩ 0, if x y ∈ E and x ∈ X 1 , y ∈ X 2 = μnB c ∪B c (x y). 1

2

2. To prove that (G 1 ∪ G 2 )c and G c1 + G c2 are isomorphic to each other, it is to be shown that (A1 ∪ A2 )c (x) = Ac1 (x) + Ac2 (x) and (B1 ∪ B2 )c (x y) = B1c (x y) + B2c (x y). (A1 ∪ A2 )c (x) = = (A1 ∪ A2 )(x) p p = (μ A1 (x) ∨ μ A2 (x), μnA1 (x) ∧ μnA2 (x)) p

p

= (μ Ac (x) ∨ μ Ac (x), μnAc1 (x) ∧ μnAc2 (x)) 1

2

= (Ac1 ∪ Ac2 )(x) = (Ac1 + Ac2 )(x) p

p

p

p

μ(B1 ∪B2 )c (x y) = μ A1 ∪A2 (x) ∧ μ A1 ∪A2 (y) − μ B1 ∪B2 (x y)

1.3 Isomorphism of Bipolar Fuzzy Graphs

33

⎧ p p p ⎪ ⎨μ A1 (x) ∧ μ A1 (y) − μ B1 (x y), if x y ∈ E 1 = μ Ap 2 (x) ∧ μ Ap 2 (y) − μ Bp 2 (x y), if x y ∈ E 2 ⎪ ⎩ p p μ A1 (x) ∧ μ A2 (y) − 0, if x ∈ X 1 , y ∈ X 2 ⎧ p ⎪ ⎨μ B1c (x y), if x y ∈ E 1 p = μ B c (x y), if x y ∈ E 2 ⎪ ⎩ p2 p μ Ac (x) ∧ μ Ac (y), if x ∈ X 1 , y ∈ X 2 1

p

2

p

μ(B1 ∪B2 )c (x y) = μ B c +B c (x y), 1

2

μn(B1 ∪B2 )c (x y) = μnA1 ∪A2 (x) ∨ μnA1 ∪A2 (y) − μnB1 ∪B2 (x y) ⎧ n n n ⎪ ⎨μ A1 (x) ∨ μ A1 (y) − μ B1 (x y), if x y ∈ E 1 = μnA2 (x) ∨ μnA2 (y) − μnB2 (x y), if x y ∈ E 2 ⎪ ⎩ n μ A1 (x) ∨ μnA2 (y) − 0, if x ∈ X 1 , y ∈ X 2 ⎧ n ⎪ ⎨μ B1c (x y), if x y ∈ E 1 = μnB c (x y), if x y ∈ E 2 ⎪ ⎩ n2 μ Ac (x) ∨ μnAc (y), if x ∈ X 1 , y ∈ X 2 1

2

μn(B1 ∪B2 )c (x y) = μnB1c +B2c (x y). 

The proof is complete.

Definition 1.37 If G = (A, B) is a bipolar fuzzy graph on a non-empty set X , then the order of G is denoted by O(G) and defined as O(G) =

 p  μ A (x), μnA (x) . x∈X

The size of a bipolar fuzzy graph G is denoted by S(G) and defined as S(G) =

 p  μ B (x y), μnB (x y) . x y∈E

Theorem 1.4 The order and size of isomorphic bipolar fuzzy graphs are the same. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let f : G 1 → G 2 be an isomorphism from G 1 to G 2 , then O(G 1 ) =

 p  p   μ A1 (x1 ), μnA1 (x1 ) = μ A2 ( f (x1 )), μnA2 ( f (x1 )) x1 ∈X 1

x1 ∈X 1

=

 p  μ A2 (x2 ), μnA2 (x2 ) = O(G 2 ).

x2 ∈X 2

34

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs e1 (0.1, −0.3) (0.1, −0.1) (0.1, −0.1)

(0.2, −0.2)

a1 (0.2, −0.5)

(0.2, −0.2) b1 (0.5, −0.4)

(0.1, 0) d1 (0.1, −0.1)

c1 (0.9, −0.2)

Fig. 1.22 Bipolar fuzzy graph G 3

(0.2, −0.2)

(0 .1,

−0 .1)

(0 .1 ,− 0. 1)

a2 (0.2, −0.5)

c2 (0.9, −0.2)

e2 (0.1, −0.3)

Fig. 1.23 Bipolar fuzzy graph G 4

) 0.2 2, − (0.

(0. 1,

b2 (0.5, −0.4)

S(G 1 ) =

0)

d2 (0.1, −0.1)

 p  μ B1 (x1 y1 ), μnB1 (x1 y1 ) x1 y1 ∈E 1

=

 p  μ B1 ( f (x1 ) f (y1 )), μnB1 ( f (x1 ) f (y1 ))

x1 y1 ∈E 1

=

 p  μ B2 (x2 y2 ), μnB2 (x2 y2 ) = S(G 2 ).

x2 y2 ∈E 2

The proof is complete.



Remark 1.4 The converse of Theorem 1.4 is not true in general as it can be seen in Example 1.17. Example 1.17 Consider bipolar fuzzy graphs G 3 = (A3 , B3 ) and G 4 = (A4 , B4 ) as shown in Figs. 1.22 and 1.23. Clearly O(G 3 ) = O(G 4 ) = (1.8, −1.5) and S(G 1 ) = S(G 4 ) = (0.7, −0.6). The order and size of G 3 and G 4 are the same but these are not isomorphic to each other. Remark 1.5 1. If bipolar fuzzy graphs are weak isomorphic to each other, then orders are the same but the converse of this statement is not true in general.

1.3 Isomorphism of Bipolar Fuzzy Graphs

35

2. If bipolar fuzzy graphs are co-weak isomorphic to each other, then sizes are the same but the converse of this statement is not true in general. Definition 1.38 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X then the degree vertex x ∈ X is a pair = (deg p (x), degn (x)), where  of any  deg(x) p p n n deg (x) = u∈X μ B (xu) and deg (x) = u∈X μ B (xu). Theorem 1.5 If G 1 and G 2 are isomorphic bipolar fuzzy graphs, then the degrees of corresponding vertices are preserved. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let f : X 1 → X 2 be an isomorphism of bipolar fuzzy graphs, then by Definition 1.38  p  μ B1 (x1 y1 ), μn1 B(x1 y1 )

deg(x1 ) =

y1 ∈X 1

 p  μ B1 ( f (x1 ) f (y1 )), μn1 B( f (x1 ) f (y1 ))

=

y1 ∈X 1

 p  μ B2 (x2 y2 ), μn2 B(x2 y2 ) = deg(x2 ).

=

y2 ∈X 2



This completes the proof.

Remark 1.6 If the degrees of vertices of bipolar fuzzy graphs are preserved, then they may not be isomorphic to each other as it can be seen in Example 1.18. Example 1.18 Consider the example of bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) on X 1 = {a1 , b1 , c1 , d1 , e1 } and X 2 = {a2 , b2 , c2 , d2 , e2 } as shown in Figs. 1.24 and 1.25, respectively. Define a mapping f : X 1 → X 2 by f (a1 ) = a2 , f (b1 ) = c2 , f (c1 ) = b2 , f (d1 ) = d2 and f (e1 ) = e2 . By routine calculations, deg(b1 ) = (0.3, −0.3) = deg(c2 ) = deg(a1 ) = (0.2, −0.2) = deg( f (a1 )), deg( f (b1 )). Similarly, the degrees of all the vertices are preserved. But B1 (a1 b1 ) = B2 ( f (a1 )(b1 )), i.e., the edges are not preserved which shows that f is not an isomorphism. p

Definition 1.39 A bipolar fuzzy set A = (μ A , μnA ) on a semigroup S is called a bipolar fuzzy subsemigroup of S if it satisfies p

p

p

μ A (x y) ≥ μ A (x) ∧ μ A (y), μnA (x y) ≤ μnA (x) ∨ μnA (y), for all x, y ∈ S. p

A bipolar fuzzy set A = (μ A , μnA ) on a group G is called a bipolar fuzzy subgroup of a group G if it is a bipolar fuzzy subsemigroup of G and satisfies μ A (x −1 ) = μ A (x), μnA (x −1 ) = μnA (x), for all x ∈ G. p

p

36

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs a1 (0.2, −0.1) b1 (0.3, −0.6) .2) (0.2, −0

Fig. 1.24 Bipolar fuzzy graph G 1

(0. 1,

(0 .1 ,− 0. 4)

−0 .1)

e1 (0.1, −0.5)

c1 (0.1, −0.1)

d1 (0.5, −0.4)

b2 (0.1, −0.1) e

a2 (0.2, −0.1)

2 (0.1, −0.5)

.1, (0 .1) −0

(0.1, −0.4 )

(0.1, −0.1)

(0.1 , −0 .1)

Fig. 1.25 Bipolar fuzzy graph G 2

(0.1, −0.1)

4) ) 0. − 0.1 , − .5 1, (0 (0. d2

c2 (0.3, −0.6)

Consider the case to associate a bipolar fuzzy group with a bipolar fuzzy graph in a natural way. Proposition 1.16 Let G = (A, B) be a bipolar fuzzy graph and Aut (G) be the set of all automorphisms of G, then (Aut (G), ◦) forms a bipolar fuzzy group. Proof Let φ, ψ ∈ Aut (G) and x, y ∈ X , then p

p

μ B ((φ ◦ ψ)(x)(φ ◦ ψ)(y)) = μ B ((φ(ψ(x))(φ(ψ))(y)) p p ≥ μ B (ψ(x)ψ(y)) ≥ μ B (x y), μnB ((φ ◦ ψ)(x)(φ ◦ ψ)(y)) = μnB ((φ(ψ(x))(φ(ψ))(y)) ≤ μnB (ψ(x)ψ(y)) ≤ μnB (x y). p

p

p

p

μ B ((φ ◦ ψ)(x)) = μ B ((φ(ψ(x))) ≥ μ B (ψ(x)) ≥ μ B (x), p μ N ((φ ◦ ψ)(x)) = μnB ((φ(ψ(x))) ≤ μnB (ψ(x)) ≤ μnB (x). Thus φ ◦ ψ ∈ Aut (G), that is, Aut (G) satisfies associativity under the operation p p ◦, φ ◦ e = φ = e ◦ φ, μ A (φ −1 ) = μ A (φ), μnA (φ −1 ) = μnA (φ), for all φ ∈ Aut (G). Hence (Aut (G), ◦) forms a bipolar fuzzy group. The proof is complete. 

1.3 Isomorphism of Bipolar Fuzzy Graphs

37

Proposition 1.17 Let G = (A, B) be a bipolar fuzzy graph and Aut (G) be the set of all automorphisms of G. Let g = (μgp , μng ) be a bipolar fuzzy set on Aut (G) defined by p

μgp (φ) = sup{μ B (φ(x)φ(y)) | x, y ∈ X } and μng (φ) = inf{μnB (φ(x)φ(y)) | x, y ∈ X }, for all φ ∈ Aut (G). Then g = (μgp , μng ) is a bipolar fuzzy group on Aut (G). Proposition 1.18 Every bipolar fuzzy group has an embedding into the bipolar fuzzy group of the group of automorphisms of some bipolar fuzzy graph. Proposition 1.19 The relation of isomorphism between bipolar fuzzy graphs is an equivalence relation. Proof Let G 1 = (A1 , B1 ), G 2 = (A2 , B2 ) and G 3 = (A3 , B3 ) be bipolar fuzzy graphs on non-empty sets X 1 , X 2 and X 3 , respectively. 1. Reflexivity: Define I : X 1 → X 1 by I (x1 ) = x1 , for all x1 ∈ X 1 . Then I is a bijective homomorphism and A1 (x1 ) = A1 (I (x1 )) and B1 (x1 y1 ) = B1 (I (x1 )I (y1 )), for all x1 , y1 ∈ X 1 . I is an isomorphism of a bipolar fuzzy graph to itself. 2. Symmetry: Let f : X 1 → X 2 be an isomorphism defined by f (x1 ) = x2 . Since f is a bijective mapping, f −1 : X 2 → X 1 exists and f −1 (x2 ) = x1 , for all x2 ∈ X 2 . Then, A2 (x2 ) = A2 ( f (x1 )) = A1 (x1 ) = A1 ( f −1 (x2 )). B2 (x2 y2 ) = B2 ( f (x1 ) f (y1 )) = B1 (x1 y1 ) = B1 ( f −1 (x2 ) f −1 (y2 )), x1 , y1 ∈ X 1 , x2 , y2 ∈ X 2 . Hence f −1 is an isomorphism. 3. Transitivity: Let f : X 1 → X 2 and g : X 2 → X 3 be the isomorphisms from G 1 into G 2 and G 2 into G 3 defined by f (x1 ) = x2 and g(x2 ) = x3 , respectively. By Definition 1.35, A1 (x1 ) = A2 ( f (x1 )) = A2 (x2 ) = A3 (g(x2 )) = A3 (g( f (x1 ))) = A3 (g ◦ f (x1 )), B1 (x1 y1 ) = B2 (x2 y2 ) = B3 (g(x2 )g(y2 )) = B3 (g( f (x1 ))g( f (y1 ))) = B3 (g ◦ f (x1 )g ◦ f (y1 )), where x1 , y1 ∈ X 1 , x2 , y2 ∈ X 2 and x3 , y3 ∈ X 3 . Clearly, g ◦ f is an isomorphism from G 1 into G 3 . Hence isomorphism of bipolar fuzzy graphs is an equivalence relation.



Proposition 1.20 Let G 1 , G 2 and G 3 be bipolar fuzzy graphs, then the weak isomorphism between these bipolar fuzzy graphs is a partial order relation.

38

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Proof The reflexivity is obvious. To prove the antisymmetry, let f : X 1 → X 2 be a weak isomorphism from G 1 into G 2 . Then f is a bijective map defined by f (x1 ) = x2 , for all x1 ∈ X 1 , satisfying, p

p

μ A1 (x1 ) = μ A2 ( f (x1 )), μnA1 (x1 ) = μnA2 ( f (x1 )), for all x1 ∈ X 1 , p

p

μ B1 (x1 y1 ) ≤ μ B2 ( f (x1 ) f (y1 )), μnB1 (x1 y1 ) ≥ μnB2 ( f (x1 ) f (y1 )), for all x1 y1 ∈ E 1 . (1.7) Let g : X 2 → X 1 be a weak isomorphism from G 2 into G 1 , then g is a bijective map defined by g(x2 ) = x1 , for all x2 ∈ X 2 , satisfying, p

p

μ A2 (x2 ) = μ A1 (g(x2 )), μnA2 (x2 ) = μnA1 (g(x2 )), for all x2 ∈ X 2 , p

p

μ B2 (x2 y2 ) ≤ μ B1 (g(x2 )g(y2 )), μnB2 (x2 y2 ) ≥ μnB1 (g(x2 )g(y2 )), for all x2 y2 ∈ E 2 . (1.8) The inequalities (1.7) and (1.8) hold on the finite sets X 1 and X 2 only when G 1 and G 2 have the same number of edges and the corresponding edges have same weight. Hence G 1 and G 2 are identical. To prove the transitivity, let f : X 1 → X 2 and g : X 2 → X 3 be weak isomorphisms from G 1 into G 2 and G 2 into G 3 , respectively. Then g ◦ f : X 1 → X 3 is a bijective map from X 1 to X 3 , where (g ◦ f )(x1 ) = g( f (x1 )), for all x1 ∈ X 1 . Since the map f : X 1 → X 2 defined by f (x1 ) = x2 , for x1 ∈ X 1 , is a weak isomorphism p

p

p

μ A1 (x1 ) = μ A2 ( f (x1 )) = μ A2 (x2 ), μnA1 (x1 ) = μnA2 ( f (x1 )) = μnA2 (x2 ), p

p

for all x1 ∈ X 1 ,

p

μ B1 (x1 y1 ) ≤ μ B2 ( f (x1 ) f (y1 )) = μ B2 (x2 y2 ),

μnB1 (x1 y1 ) ≥ μnB2 ( f (x1 ) f (y1 )) = μnB2 (x2 y2 ), for all x1 y1 ∈ E 1 .

(1.9) (1.10)

Since the map g : X 2 → X 3 defined by g(x2 ) = x3 , for x2 ∈ X 2 is a weak isomorphism, p

p

p

μ A2 (x2 ) = μ A3 (g(x2 )) = μ A3 (x3 ), μnA2 (x2 ) = μnA3 (g(x2 )) = μnA3 (x3 ), for all x2 ∈ X 2 , p

p

(1.11)

p

μ B2 (x2 y2 ) ≤ μ B3 (g(x2 )g(y2 )) = μ B3 (x3 y3 ), μnB2 (x2 y2 ) ≥ μnB3 (g(x2 )g(y2 )) = μnB3 (x3 y3 ), for all x2 y2 ∈ E 2 .

(1.12)

From (1.9), (1.11) and f (x1 ) = x2 , x1 ∈ X 1 , p

p

p

p

p

μ A1 (x1 ) = μ A2 ( f (x1 )) = μ A2 (x2 ) = μ A3 (g(x2 )) = μ A3 (g( f (x1 ))), μnA1 (x1 ) = μnA2 ( f (x1 )) = μnA2 (x2 ) = μnA3 (g(x2 )) = μnA3 (g( f (x1 ))), for all x1 ∈ X 1 . From (1.10) and (1.12),

1.3 Isomorphism of Bipolar Fuzzy Graphs p

39

p

p

μ B1 (x1 y1 ) ≤ μ B2 ( f (x1 ) f (y1 )) = μ B2 (x2 y2 ) p

p

≤ μ B3 (g(x2 )g(y2 )) = μ B3 (g( f (x1 ))g( f (y1 ))), μnB1 (x1 y1 ) ≥ μnB2 ( f (x1 ) f (y1 )) = μnB2 (x2 y2 ) ≥ μnB3 (g(x2 )g(y2 )) = μnB3 (g( f (x1 ))g( f (y1 ))), for all x1 y1 ∈ E 1 . Thus g ◦ f is a weak isomorphism between G 1 and G 3 . This completes the proof.  Definition 1.40 Let G = (A, B) be a bipolar fuzzy graph on X . The strength of vera bipolar fuzzy path x1 , e1 , x2 , e2 , x3 , . . . , er −1 , xr  between any two distinct  p tices x1 and xr is defined as a pair SG (x1 , xr ) = SG (x1 , xr ), SGn (x1 , xr ) , where p −1 p −1 n SG (x1 , xr ) = ∧ri=1 μ B (ei ) and SGn (x1 , xr ) = ∨ri=1 μ B (ei ). A strongest bipolar fuzzy path between any two distinct vertices is a bipolar fuzzy path with maximum positive strength and minimum negative strength. If P = {Pi | i = 1, 2, 3, . . .} is the collection of all bipolar fuzzy paths between x and y, then the strength bipo + of the strongest  − lar fuzzy path x − y is defined as the pair SG∞ (x, y) = SG∞ (x, y), SG∞ (x, y) = p (∨P SG (x, y), ∧P SGn (x, y)). It is referred as strength of connectedness between x and y. Definition 1.41 Let G = (A, B) be a bipolar fuzzy graph and x, y ∈ X be any two   distinct vertices of G. Let G = (A, B ) be a subgraph obtained by deleting the edge   x y from G, that is, B (x y) = (0, 0) and B = B for all other edges. Then x y is said to + + − be a bipolar fuzzy bridge in G if either SG∞ (x, y) < SG∞ (x, y) and SG∞ (x, y) ≥ − + + − − SG∞ (x, y) or SG∞ (x, y) ≤ SG∞ (x, y) and SG∞ (x, y) > SG∞ (x, y), for some x, y ∈ X. In other words, deleting an edge x y reduces the strength of connectedness between some pair of vertices. Thus, x y is a bipolar fuzzy bridge if and only if there exist u, v ∈ X such that x y is an edge of every strongest bipolar fuzzy path from u to v. Definition 1.42 An edge x y in a bipolar fuzzy graph G = (A, B) is said to be a + − p strong edge if μ B (x y) ≥ SG∞ (x, y) and μnB (x y) ≤ SG∞ (x, y). In this case, x, y are called strong neighbors. A vertex x is called a bipolar fuzzy end vertex of G if it has at most one strong neighbor in G. A bipolar fuzzy path consisting of only strong edges is called a strong bipolar fuzzy path in G. An edge with least positive membership value and greatest negative membership value is called a weakest bipolar fuzzy edge of G. Example 1.19 Consider a bipolar fuzzy graph on X = {a, b, c, d} as shown in Fig. 1.26. The strength of bipolar fuzzy path b − a − c is (0.3 ∧ 0.1, −0.2 ∨ −0.3) = (0.1, −0.2) and that of b − d − c is (0.2 ∧ 0.2, −0.3 ∨ −0.4) = (0.2, −0.3). The strength of connectedness between the vertices b and c is (0.2, −0.3). Example 1.20 Consider a bipolar fuzzy graph G on X = {a, b, c, d} as shown in Fig. 1.27. By routine computations, it is easy to see that

40

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs a(0.3, −0.3)

Fig. 1.26 Strength of connectedness

b(0.4, −0.3)

(0.1, −0.3)

(0.2, −0.3)

(0.3, −0.2)

(0.2, −0.4)

Fig. 1.27 Strongest bipolar fuzzy path

c(0.2, −0.4)

d(0.2, −0.4)

d(0.2, −0.4)

c(0.2, −0.5)

(0 .1 ,− 0. 4)

(0.2, −0.2)

(0.1, −0.4)

(0.2, −0.3)

(0.1, −0.3) a(0.1, −0.4)

1. 2. 3. 4.

b(0.3, −0.5)

a − d is a bipolar fuzzy path of strength (0.1, −0.4). The bipolar fuzzy path a − b − d has strength (0.1, −0.3). a − b − c − d is a bipolar fuzzy path of strength (0.2, −0.2). A strongest bipolar fuzzy path joining a and d is a − b − c − d with strength of connectedness (0.2, −0.2).

Theorem 1.6 Let G = (A, B) be a bipolar fuzzy graph, then G is connected if and only if S ∞ (x, y) = (0, 0), for all x, y ∈ X . Proof Assume that G is connected, then for every x, y ∈ X , there exists at least one bipolar fuzzy path such that p

⇒

Either SG (x, y) > 0 or SGn (x, y) < 0 p Either ∨P SG (x, y) > 0 or ∧P SGn (x, y) < 0

⇒ ⇒

(∨P SG (x, y), ∧P SGn (x, y)) = (0, 0) SG∞ (x, y) = (0, 0).

p

Conversely, suppose that for all x, y ∈ X , SG∞ (x, y) = (0, 0)

1.3 Isomorphism of Bipolar Fuzzy Graphs

41

p

⇒

(∨P SG (x, y), ∧P SGn (x, y)) = (0, 0)

⇒ ⇒

Either ∨P SG (x, y) > 0 or ∧P SGn (x, y) < 0 p Either SG (x, y) > 0 or SGn (x, y) < 0.

p

It follows that there exists at least one bipolar fuzzy path between each pair of distinct vertices x and y, that is, G is a connected bipolar fuzzy graph.  Theorem 1.7 Let G 1 and G 2 be two isomorphic bipolar fuzzy graphs, then G 1 is connected if and only if G 2 is connected. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let f : X 1 → X 2 be an isomorphism from G 1 to G 2 . Assume that G 1 is connected, then for any x1 , y1 ∈ X 1 , (0, 0) = SG∞1 (x1 , y1 ) = (∨P SG 1 (x1 , y1 ), ∧P SGn 1 (x1 , y1 ))   p = ∨P SG 2 ( f (x1 ), f (y1 )), ∧P SGn 2 ( f (x1 ), f (y1 )) p

= S2∞ ( f (x1 ), f (y1 )) ⇒ (0, 0) = S2∞ (x2 , y2 ). Conversely, let G 2 is connected, then SG∞2 (x2 , y2 ) = (0, 0). Since f −1 : X 2 → X 1 is an isomorphism from G 2 into G 1 , for every x2 , y2 ∈ X 2 (0, 0) = SG∞2 (x2 , y2 ) = (∨P SG 2 (x2 , y2 ), ∧P SGn 2 (x2 , y2 ))   p = ∨P SG 1 ( f −1 (x2 ), f −1 (y2 )), ∧P SGn 1 ( f −1 (x2 ), f −1 (y2 ))   = S1∞ f −1 (x2 ), f −1 (y2 ) ⇒ (0, 0) = SG∞1 (x1 , y1 ). p

Hence G 1 is connected.



Theorem 1.8 Let G 1 and G 2 be to isomorphic bipolar fuzzy graphs, then G 1 is strong if and only if G 2 is strong. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let f : X 1 → X 2 be an isomorphism from G 1 into G 2 . Assume that G 1 is strong, then for any x1 y1 ∈ E 1 and x2 y2 ∈ E 2 , B2 (x2 y2 ) = B2 ( f (x1 ) f (y1 )) = B1 (x1 y1 )  p  p = μ A1 (x1 ) ∧ μ A1 (y1 ), μnA1 (x1 ) ∨ μnA1 (y1 )  p  p = μ A2 ( f (x1 )) ∧ μ A2 ( f (y1 )), μnA2 ( f (x1 )) ∨ μnA2 ( f (y1 ))  p  p = μ A2 (x2 ) ∧ μ A2 (y2 ), μnA2 (x2 ) ∨ μnA2 (y2 ) . Hence G 2 is a strong bipolar fuzzy graph. Conversely, suppose that G 2 is a strong bipolar fuzzy graph, then for every x1 y1 ∈ E 1 ,

42

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

  B1 (x1 y1 ) = B1 f −1 (x2 ) f −1 (y2 ) = B2 (x2 y2 )  p  p = μ A2 (x2 ) ∧ μ A2 (y2 ), μnA2 (x2 ) ∨ μnA2 (y2 )  p  p = μ A1 ( f −1 (x2 )) ∧ μ A1 ( f −1 (y2 )), μnA1 ( f −1 (x2 )) ∨ μnA1 ( f −1 (y2 ))  p  p = μ A1 (x1 ) ∧ μ A1 (y1 ), μnA1 (x1 ) ∨ μnA1 (y1 ) . Thus G 1 is a strong bipolar fuzzy graph.



Theorem 1.9 Let G 1 and G 2 be isomorphic bipolar fuzzy graphs, then G 1 is complete if and only if G 2 is complete. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. Let g : X 1 → X 2 be an isomorphism from G 1 to G 2 . Assume that G 1 is complete, then for all x1 , y1 ∈ X 1 and x2 , y2 ∈ X 2 B2 (x2 y2 ) = B2 (g(x1 )g(y1 )) = B1 (x1 y1 ) p p = (μ A1 (x1 ) ∧ μ A1 (y1 ), μnA1 (x1 ) ∨ μnA1 (y1 )) p

p

= (μ A2 (g(x1 )) ∧ μ A2 (g(y1 )), μnA2 (g(x1 )) ∨ μnA2 (g(y1 ))) p

p

= (μ A2 (x2 ) ∧ μ A2 (y2 ), μnA2 (x2 ) ∨ μnA2 (y2 )). Hence G 2 is a complete bipolar fuzzy graph. Conversely, let G 2 is a complete bipolar fuzzy graph, then for every x1 , y1 ∈ X 1   B1 (x1 y1 ) = B1 g −1 (x2 )g −1 (y2 ) = B2 (x2 y2 )  p  p = μ A2 (x2 ) ∧ μ A2 (y2 ), μnA2 (x2 ) ∨ μnA2 (y2 )  p  p = μ A1 (g −1 (x2 )) ∧ μ A1 (g −1 (y2 )), μnA1 (g −1 (x2 )) ∨ μnA1 (g −1 (y2 ))  p  p = μ A1 (x1 ) ∧ μ A1 (y1 ), μnA1 (x1 ) ∨ μnA1 (y1 ) . Thus G 1 is a complete bipolar fuzzy graph.



1.4 Complex Bipolar Fuzzy Graphs The extension of crisp sets to fuzzy sets and then to bipolar fuzzy sets, in terms of membership functions, is mathematically comparable to the extension of the set of integers, Z , to the set of real numbers, R, and to the real plane, R2 . That is, expanding the range of the membership function from {0, 1} to [−1, 0] × [0, 1] is mathematically analogous to the extension of Z to R2 . Of course, the development of number system did not end with the set of real numbers. Historically, the introduction of real numbers was followed by their extension to the set of complex numbers, C. Based on this extension, the notion of a fuzzy set was extended to a complex fuzzy set, by Ramot et al. [37], as a fuzzy set characterized by a complex-valued membership function.

1.4 Complex Bipolar Fuzzy Graphs

43

Definition 1.43 A complex fuzzy set S defined on a universe of discourse X is characterized by a membership function that assigns any element x ∈ X , a complexvalued grade of membership in S. By definition, the values μ S (x) may receive all lying within the closed unit disk in√the complex plane, and are thus of the form μ S (x) = r S (x)eiwS (x) , where i = −1, r S (x)and w S (x) are both real numbers, r S (x) ∈ [0, 1] and w S (x) ∈ [0, 2π ]. The complex fuzzy set may be represented as the set of ordered pairs S=



  x, r S (x)eiwS (x) | x ∈ X .

The term r S (x) is called amplitude and w S (x) is called phase term of the complex grade of membership μ S (x). Note that the amplitude r S (x) is equal to | μ S (x) |. Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, thermodynamics, theory of relativity, and vibration analysis. The signals that change periodically are expressed in terms of sine or cosine function or a group of these together, that is, as a complex number eiθ = cos θ + i sin θ . However, adding a polarity perspective, we may represent signals as complex bipolar fuzzy sets for identifying a particular signal of interest out of a large number of signals received by a TV, radio, or digital receiver. A power factor correction uses a lot of complex numbers and phase vectors. Using a suitable method, complex bipolar fuzzy sets can be used to represent the power factors to find the optimal device to reduce power bills and save money. The notion of a complex bipolar fuzzy set can be implemented to represent the uncertainty and periodicity problems having bipolar information in complex geometry as shown in Figs. 1.28 and 1.29. The notion of complex bipolar fuzzy set represents bipolar behavior of uncertainty and periodicity semantics simultaneously by applying the notion of bipolar fuzzy set in complex geometry to illustrate various phenomena in

Fig. 1.28 Phase term in bipolar fuzzy set

w p (x) = 0 n

w (x) = 0

44

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

) p (x w2 w2n (x)

w1p (x)

) (x

wn 1 (x )

n

w3

wp 3 (x )

Fig. 1.29 Phase term in complex bipolar fuzzy set

different phases or levels. The bipolarity of uncertainty semantics can be represented in the form of amplitude terms as r p (x) and r n (x) and the bipolarity of periodicity semantics in the form of phase terms w p (x) and w n (x) corresponding to complex p n numbers r p (x)eiw (x) and r n (x)eiw (x) . The following definition illustrates the notion of complex bipolar fuzzy set. Definition 1.44 A complex bipolar fuzzy set A on a non-empty set X is an object of the form   p n p A = (x, r A (x)eiw A (x) , r An (x)eiw A (x) ) | x ∈ X , √ p p where i = −1, r A : X → [0, 1] and r An : X → [−1, 0] are mappings, w A (x) ∈ p [0, π ] and wnA (x) ∈ [π, 2π ] or w nA (x) ∈ [−π, 0]. For any element x ∈ X , r A (x) and p r An (x) are called amplitude terms, w A (x) and w nA (x) are called phase terms. The complex bipolar fuzzy set can also be written as   p n p A = (x, r A (x)ei arg A (x) , r An (x)ei arg A (x) ) | x ∈ X . p

1. If r A (x) = r An (x) = 0, then x is considered as neutral with no positive or negative satisfaction for A. p 2. If r A (x) = 0 and r An (x) = 0, it is the case when x is observed as having only p positive satisfaction for A effected by positive phase value w A (x). p n 3. If r A (x) = 0 and r A (x) = 0, it is the situation that x is not satisfying the property of A but satisfying the counter property to A with negative phase quality wnA (x). p 4. If r A (x) = 0 and r An (x) = 0, then the membership function of given property overlaps that of its counter property in some parts of X effected by both positive p and negative phase values w A (x) and w nA (x).

1.4 Complex Bipolar Fuzzy Graphs

45

Example 1.21 Every 11 years the sun undergoes a period of activity called the “solar maximum”, followed by a period of quiet called the “solar minimum”. During the solar maximum, there are many sunspots, solar flares, and coronal mass ejections, all of which can affect communications and weather on Earth. During the solar minimum, there are few sunspots. One way solar activity may be tracked is by observing sunspots. Sunspots are relatively cool areas that appear as dark blemishes on the face of the sun, and are sites where solar flares are observed to occur. Let A denote the complex bipolar fuzzy set with rising or decreasing solar activity in which each month is associated with a bipolar complex-valued grade of membership consisting of amplitude terms and phase terms. The role of amplitude terms is simple: it signifies the degree to which a particular month x has rising or decreasing solar activity. The phase term contains information regarding the position of the month under consideration in the solar cycle. Accordingly, any position between the solar minimum and the solar maximum is associated with a membership phase in the range (0, π ) if solar activity is on the rise, or a value in the range (π, 2π ) if solar activity is decreasing.   p n p Definition 1.45 Let A = (x, r A (x)eiw A (x) , r An (x)eiw A (x) ) | x ∈ X be a complex bipolar fuzzy set on a non-empty set X . A complex bipolar fuzzy relation D on A is a mapping D : A → A such that, for all x, y ∈ X , p

p

p

r D (x y) ≤ r A (x) ∧ r A (y) and r Dn (x y) ≥ r An (x) ∨ r An (y), (for amplitude terms) p p p w D (x y) ≤ w A (x) ∧ w A (y) and w nD (x y) ≥ w nA (x) ∨ w nA (y), (for phase terms), p

where μ D (x y) = r D (x y)eiw D (x y) , μnD (x y) = r Dn (x y)eiw D (x y) and i = p

p

n

√ −1.

 Note that D is also a complex bipolar fuzzy relation in X defined by D = (x y,  p n p p r D (x y)eiw D (x y) , r Dn (x y)eiw D (x y) ) | x y ∈ X × X , where r D : X × X → [0, 1] and p n n n r D : X × X → [−1, 0] are mappings,  w D ∈ [0, π ], w D ∈ [π, 2π  ] or w D ∈ [−π,  0]. p

p

n

p

p

n

We will use the notations A = r A eiw A , r An eiw A and D = r D eiw D , r Dn eiw D for a complex bipolar fuzzy set and a complex bipolar fuzzy relation, respectively.

Definition 1.46 A complex bipolar fuzzy graph on a non-empty set X is a pair p n p G = (A, D), where A = (r A ew A , r An ew A ) : X → {z | z ∈ C, |z| ≤ 1}2 is a complex n p wp bipolar fuzzy set on X and D = (r D e D , r Dn ew D ) : X × X → {z | z ∈ C, |z| ≤ 1}2 is a complex bipolar polar fuzzy relation in X such that, for all x, y ∈ X , p

p

p

r D (x y) ≤ r A (x) ∧ r A (y) and r Dn (x y) ≥ r An (x) ∨ r An (y), (for amplitude terms) p

p

p

w D (x y) ≤ w A (x) ∧ w A (y) and w nD (x y) ≥ w nA (x) ∨ w nA (y), (for phase terms), √ p p where i = −1, r A : X → [0, 1], r An : X → [−1, 0], r D : X × X → [0, 1] and r Dn : p p X × X → [−1, 0] are mappings, w A , w D ∈ [0, π ], w nA , w nD ∈ [π, 2π ] or w nA , w nD ∈ [−π, 0]. Note that D(x y) = (0, 0) for all x y ∈ X × X − E, where E ⊆ X × X is the set of edges. A is called a complex bipolar fuzzy vertex set of G and D is a complex

46

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

bipolar fuzzy edge set of G. A bipolar fuzzy relation D on X is symmetric if D(x y) = D(yx) for all x, y ∈ X . Notice that D(x y) = (0, 0) for x y ∈ / E. The following example illustrates the above definition. Example 1.22 Consider a complex bipolar fuzzy set A on the vertex set X = {x, y, z} as given in Table 1.7 and a complex bipolar fuzzy relation D in X as defined in Table 1.8. The complex bipolar fuzzy graph G = (A, D) on X is shown in Fig. 1.30. Definition 1.47 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two complex bipolar fuzzy graphs on non-empty sets X 1 and X 2 , respectively. The union of G 1 and G 2 is a pair G 1 ∪ G 2 = (A1 ∪ A2 , D1 ∪ D2 ) such that for all x, y ∈ X 1 ∪ X 2 ,

Table 1.7 Complex bipolar fuzzy set A A x1 p

μ A (x) μnA (x)

0.7e0.9πi 0.6e−0.8πi

x2

x3

0.5e0.5πi −0.7e−0.9πi

0.8e0.8πi −0.6e−0.9πi

Table 1.8 Complex bipolar fuzzy relation D D x y x y z

z

(0.4e0.5πi , −0.6e−0.7πi ) (0, 0) (0.5e0.5πi , −0.6e−0.8πi )

(0, 0) (0.4e0.5πi , −0.6e−0.7πi ) (0.7e0.8πi , −0.5e−0.7πi )

(0.7e0.8πi , −0.5e−0.7πi ) (0.5e0.5πi , −0.6e−0.8πi ) (0, 0)

−0.8π i ) 0.9π i , −0.6e

0.

e .7 (0

8π i −

5e 0. ,−

0. 7π i

)

(0.4e0.5πi , −0.6e−0.7πi )

x(0.7e

0.5 (0.5e −0.9π i ) 0.5πi , −0.7e

y(0.5e

Fig. 1.30 Complex bipolar fuzzy graph

.8π i ) πi , −0.6e−0

−0.9π i )

0.8πi , −0.6e z(0.8e

1.4 Complex Bipolar Fuzzy Graphs

47

1. If x ∈ X 1 \ X 2 , (A1 ∪ A2 )(x) = A1 (x), 2. If x ∈ X 2 \ X 1 , (A1 ∪ A2 )(x) = A2 (x), 3. If x ∈ X 1 ∩ X 2 , (A1 ∪ A2 )(x)       i w Ap (x)∨w Ap (x)  n  i wnA (x)∧wnA (x)  p p n 2 2 2 2 . , r A1 (x) ∧ r A2 (x) e = r A1 (x) ∨ r A2 (x) e If E 1 and E 2 are the sets of edges in G 1 and G 2 , respectively, then D1 ∪ D2 can be defined as 1. If x y ∈ E 1 \ E 2 , (D1 ∪ D2 )(x) = D1 (x y), y), 2. If x y ∈ E 2 \ E 1 , (D1 ∪ D2 )(x) = D  2 (x  p 3. If x ∈ E 1 ∩ E 2 , (D1 ∪ D2 )(x y) = μ D1 ∪D2 (x y), μnD1 ∪D2 (x y) , where 



 p  i w p (x y)∨w Dp (x y) p p 2 μ D1 ∪D2 (x y) = r D1 (x y) ∨ r D2 (x y) e D1 ,   n n   i w (x y)∧w D (x y) 2 μnD1 ∪D2 (x y) = r Dn 1 (x y) ∧ r Dn 2 (x y) e D1 . Definition 1.48 The intersection of two complex bipolar fuzzy graphs G 1 = (A1 , D1 ) and G 2 = (A2 , D2 ) is a defined as a pair G 1 ∩ G 2 = (A1 ∩ A2 , D1 ∩ D2 ), where A1 ∩ A2 and D1 ∩ D2 are defined as  p  1. For all x ∈ X 1 ∩ X 2 , (A1 ∩ A2 )(x) = μ A1 ∩A2 (x), μnA1 ∩A2 (x) such that 



 p  i w p (x)∧w Ap (x) p p 2 , μ A1 ∩A2 (x) = r A1 (x) ∧ r A2 (x) e A2     i wn (x)∨wnA (x) 2 . μnA1 ∩A2 (x) = r An 1 (x) ∨ r An 2 (x) e A2  p  2. For all x y ∈ E 1 ∩ E 2 , (D1 ∩ D2 )(x y) = μ D1 ∩D2 (x y), μnD1 ∩D2 (x y) such that 



 p  i w p (x y)∧w Dp (x y) p 2 = r D1 (x y) ∧ r D2 (x) e D2 ,   n n   i w (x y)∨w D (x y) 2 . μnD1 ∩D2 (x y) = r Dn 1 (x y) ∨ r Dn 2 (x) e D2

p μ D1 ∩D2 (x y)

Definition 1.49 The join of two complex bipolar fuzzy graphs G 1 = (A1 , D1 ) and G 2 = (A2 , D2 ) is a defined as a pair G 1 + G 2 = (A1 + A2 , D1 + D2 ) such that A1 + A2 = A1 ∪ A2 , for all x ∈ X 1 ∪ X 2 , and the membership values of the edges in G 1 + G 2 are defined as 1. D1 + D2 = D1 ∪ D2 , for all x y ∈ E 1 ∪ E 2 , 2. Let E  be the set of all edges joining the vertices of G 1 and G 2 , then for all x y ∈ E  , where x ∈ X 1 and y ∈ X 2 (D1 + D2 )(x y)

48

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

=

      p  i w p (x)∧w Ap (y)  n  i wn (x)∨wnA (y) p 2 2 . r A1 (x) ∧ r A2 (y) e A1 , r A1 (x) ∨ r An 2 (y) e A1

Proposition 1.21 If G 1 and G 2 are complex bipolar fuzzy graphs, then G 1 ∪ G 2 , G 1 ∩ G 2 and G 1 + G 2 are complex bipolar fuzzy graphs. Definition 1.50 The composition of two complex bipolar fuzzy graphs G 1 = (A1 , D1 ) and G 2 = (A2 , D2 ) is denoted by the pair G   1 p◦ G 2 = (A1 ◦ A2 , D1 ◦ D2 ), p where A1 ◦ A2 = μ A1 ◦A2 , μnA1 ◦A2 and D1 ◦ D2 = μ D1 ◦D2 , μnD1 ◦D2 are defined as  p  1. (A1 ◦ A2 )(x1 , x2 ) = μ A1 ◦A2 (x1 , x2 ), μnA1 ◦A2 (x1 , x2 ) 



 p  i w p (x )∧w p (x ) p p μ A1 ◦A2 (x1 , x2 ) = r A1 (x1 ) ∧ r A2 (x2 ) e A1 1 A2 2    n  i wnA (x1 )∨wnA (x2 ) n n 2 2 μ A1 ◦A2 (x1 , x2 ) = r A1 (x1 ) ∨ r A2 (x2 ) e , for all (x1 , x2 ) ∈ X 1 × X 2 ,  p  2. (D1 ◦ D2 )((x, x2 )(x, y2 )) = μ D1 ◦D2 ((x, x2 )(x, y2 )), μnD1 ◦D2 ((x, x2 )(x, y2 )) 



 p  i w p (x)∧w Dp (x2 y2 ) p 2 = r A1 (x) ∧ r D2 (x2 y2 ) e A1     i wn (x)∨wnD (x2 y2 ) 2 μnD1 ◦D2 ((x, x2 )(x, y2 )) = r An 1 (x) ∨ r Dn 2 (x2 y2 ) e A1 , p μ D1 ◦D2 ((x, x2 )(x, y2 ))

for all x ∈ X 1 , x2 y2 ∈ E 2 ,  p  3. (D1 ◦ D2 )((x1 , z)(y1 , z)) = μ D1 ◦D2 ((x1 , z)(y1 , z)), μnD1 ◦D2 ((x1 , z)(y1 , z)) 



 p  i w p (x y )∧w p (z) p p μ D1 ◦D2 ((x1 , z)(y1 , z)) = r D1 (x1 y1 ) ∧ r A2 (z) e D1 1 1 A2     i wn (x y )∨wn (z) μnD1 ◦D2 ((x1 , z)(y1 , z)) = r Dn 1 (x1 y1 ) ∨ r An 2 (z) e D1 1 1 A2 , for all x1 y1 ∈ E 1 , z ∈ X 2 ,  p 4. (D1 ◦ D2 )((x1 , x2 )(y1 , y2 )) = μ D1 ◦D2 ((x1 , x2 )(y1 , y2 )), μnD1 ◦D2 ((x1 , x2 ) (y1 , y2 ))) p

μ D1 ◦D2 ((x1 , x2 )(y1 , y2 ))    i w p (x y )∧w p (x )∧w p (y )  p p p = r D1 (x1 y1 ) ∧ r A2 (x2 ) ∧ r A2 (y2 ) e D1 1 1 A2 2 A2 2 μnD1 ◦D2 ((x1 , x2 )(y1 , y2 ))     i wn (x y )∨wn (x )∨wn (y ) = r Dn 1 (x1 y1 ) ∨ r An 2 (x2 ) ∨ r An 2 (y2 ) e D1 1 1 A2 2 A2 2 , for all x1 y1 ∈ E 1 , x2 , y2 ∈ X 2 .

1.4 Complex Bipolar Fuzzy Graphs

49

Definition 1.51 The Cartesian product of two complex bipolar fuzzy graphs G 1 = by the pair G 1 G 2 = (A1 A2 , D1 D2 ), (A1 , D1 ) and G 2 = (A2 , D2 ) is denoted  p p n where A1 A2 = μ A1 A2 , μ A1 A2 and D1 D2 = μ D1 D2 , μnD1 D2 are defined as   p 1. (A1 A2 )(x1 , x2 ) = μ A1 A2 (x1 , x2 ), μnA1 A2 (x1 , x2 ) 



 p  i w p (x )∧w p (x ) p = r A1 (x1 ) ∧ r A2 (x2 ) e A1 1 A2 2     i wn (x )∨wn (x ) μnA1 A2 (x1 , x2 ) = r An 1 (x1 ) ∨ r An 2 (x2 ) e A2 1 A2 2 , p μ A1 A2 (x1 , x2 )

for all (x1 , x2 ) ∈ X 1 × X 2 ,   p 2. (D1 D2 )((x, x2 )(x, y2 )) = μ D1 D2 ((x, x2 )(x, y2 )), μnD1 D2 ((x, x2 )(x, y2 )) 



 p  i w p (x)∧w Dp (x2 y2 ) p p 2 μ D1 D2 ((x, x2 )(x, y2 )) = r A1 (x) ∧ r D2 (x2 y2 ) e A1     i wn (x)∨wnD (x2 y2 ) 2 μnD1 D2 ((x, x2 )(x, y2 )) = r An 1 (x) ∨ r Dn 2 (x2 y2 ) e A1 , for all x ∈ X 1 , x2 y2 ∈ E 2 ,   p 3. (D1 D2 )((x1 , z)(y1 , z)) = μ D1 D2 ((x1 , z)(y1 , z)), μnD1 D2 ((x1 , z)(y1 , z)) 



 p  i w p (x y )∧w p (z) p p μ D1 D2 ((x1 , z)(y1 , z)) = r D1 (x1 y1 ) ∧ r A2 (z) e D1 1 1 A2     i wn (x y )∨wn (z) μnD1 D2 ((x1 , z)(y1 , z)) = r Dn 1 (x1 y1 ) ∨ r An 2 (z) e D1 1 1 A2 , for all x1 y1 ∈ E 1 , z ∈ X 2 , Definition 1.52 The direct product of two complex bipolar fuzzy graphs G 1 = (A1 , D1 ) and G 2 = (A2 , D2 ) is denoted by the pair G 1 × G 1×  2 p= (A1 × A2 , D p D2 ), where A1 × A2 = μ A1 ×A2 , μnA1 ×A2 and D1 × D2 = μ D1 ×D2 , μnD1 ×D2 are defined as  p  1. (A1 × A2 )(x1 , x2 ) = μ A1 ×A2 (x1 , x2 ), μnA1 ×A2 (x1 , x2 ) 



 p  i w p (x )∧w p (x ) p p μ A1 ×A2 (x1 , x2 ) = r A1 (x1 ) ∧ r A2 (x2 ) e A1 1 A2 2     i wn (x )∨wn (x ) μnA1 ×A2 (x1 , x2 ) = r An 1 (x1 ) ∨ r An 2 (x2 ) e A2 1 A2 2 , for all (x1 , x2 ) ∈ X 1 × X 2 ,  p 2. (D1 × D2 )((x1 , x2 )(y1 , y2 )) = μ D1 ×D2 ((x1 , x2 )(y1 , y2 )), μnD1 ×D2 ((x1 , x2 ) (y1 , y2 )))

50

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs 



 p  i w p (x y )∧w p (x y ) p p μ D1 ×D2 ((x1 , x2 )(y1 , y2 )) = r D1 (x1 y1 ) ∧ r D2 (x2 y2 ) e D1 1 1 D2 2 2    n  i wnD (x1 y1 )∨wnD (x2 y2 ) n n 1 2 μ D1 ×D2 ((x1 , x2 )(y1 , y2 )) = r D1 (x1 y1 ) ∨ r D2 (x2 y2 ) e , for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Definition 1.53 The strong product of two complex bipolar fuzzy graphs G 1 = (A1 , D1 ) and G 2 = (A2, D2 ) is denoted by the pair G 1  G 2 = (A1  A2 , D 1  p p D2 ), where A1  A2 = μ A1 A2 , μnA1 A2 and D1  D2 = μ D1 D2 , μnD1 D2 are defined as   p 1. (A1  A2 )(x1 , x2 ) = μ A1 A2 (x1 , x2 ), μnA1 A2 (x1 , x2 ) 



 p  i w p (x )∧w p (x ) p p μ A1 A2 (x1 , x2 ) = r A1 (x1 ) ∧ r A2 (x2 ) e A1 1 A2 2     i wn (x )∨wn (x ) μnA1 A2 (x1 , x2 ) = r An 1 (x1 ) ∨ r An 2 (x2 ) e A2 1 A2 2 , for all (x1 , x2 ) ∈ X 1 × X 2 ,   p 2. (D1  D2 )((x, x2 )(x, y2 )) = μ D1 D2 ((x, x2 )(x, y2 )), μnD1 D2 ((x, x2 )(x, y2 )) 



 p  i w p (x)∧w Dp (x2 y2 ) p p 2 μ D1 D2 ((x, x2 )(x, y2 )) = r A1 (x) ∧ r D2 (x2 y2 ) e A1     i wn (x)∨wnD (x2 y2 ) 2 μnD1 D2 ((x, x2 )(x, y2 )) = r An 1 (x) ∨ r Dn 2 (x2 y2 ) e A1 , for all x ∈ X 1 , x2 y2 ∈ E 2 ,   p 3. (D1  D2 )((x1 , z)(y1 , z)) = μ D1 D2 ((x1 , z)(y1 , z)), μnD1 D2 ((x1 , z)(y1 , z)) 



 p  i w p (x y )∧w p (z) p p μ D1 D2 ((x1 , z)(y1 , z)) = r D1 (x1 y1 ) ∧ r A2 (z) e D1 1 1 A2    n  i wnD (x1 y1 )∨wnA (z) n n 1 2 μ D1 D2 ((x1 , z)(y1 , z)) = r D1 (x1 y1 ) ∨ r A2 (z) e , for all x1 y1 ∈ E 1 , z ∈ X 2 ,  p 4. (D1  D2 )((x1 , x2 )(y1 , y2 )) = μ D1 D2 ((x1 , x2 )(y1 , y2 )), μnD1 D2 ((x1 , x2 ) (y1 , y2 ))) 



 p  i w p (x y )∧w p (x y ) p = r D1 (x1 y1 ) ∧ r D2 (x2 y2 ) e D1 1 1 D2 2 2     i wn (x y )∨wn (x y ) μnD1 D2 ((x1 , x2 )(y1 , y2 )) = r Dn 1 (x1 y1 ) ∨ r Dn 2 (x2 y2 ) e D1 1 1 D2 2 2 ,

p μ D1 D2 ((x1 , x2 )(y1 , y2 ))

for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 .

1.4 Complex Bipolar Fuzzy Graphs

51

Definition 1.54 A complex bipolar fuzzy graph G = (A, D) on a non-empty set X is known as a strong complex bipolar fuzzy graph if  p  p p p p p p μ D (x y) =r D (x y)eiw D (x y) = r A (x) ∧ r A (y) ei (w A (x)∧w A (y))   n n n μnD (x y) =r Dn (x y)eiw D (x y) = r An (x) ∨ r An (y) ei (w A (x)∨w A (y)) , for all x y ∈ E. Definition 1.55 A complex bipolar fuzzy graph G = (A, D) on a non-empty set X is known as a complete complex bipolar fuzzy graph if  p  p p p p p p μ D (x y) =r D (x y)eiw D (x y) = r A (x) ∧ r A (y) ei (w A (x)∧w A (y))   n n n μnD (x y) =r Dn (x y)eiw D (x y) = r An (x) ∨ r An (y) ei (w A (x)∨w A (y)) , for all x, y ∈ X. Definition 1.56 The complement of a strong complex bipolar fuzzy graph G = (A, D), on a non-empty set X , is a strong complex bipolar fuzzy graph G = (A, D) p on X such that A = A and D = (μ D , μnD ) is defined as p μ D (x y)

 p 0 if μ D (x y) = 0,  =  p p p p p r A (x) ∧ r A (y) ei (w A (x)∧w A (y)) if μ D (x y) = 0, 

μnD (x y)

0 if μnD (x y) = 0,  =  n n n r A (x) ∨ r An (y) ei (w A (x)∧w A (y)) if μnD (x y) = 0,

Definition 1.57 The complement of a complex bipolar fuzzy graph G = (A, D) is p a pair G = (A, D), where A = A and D = (μ D , μnD ) is defined as  p  p p p p p p μ D (x y) = r A (x) ∧ r A (y) − r D (x y) ei (w A (x)∧w A (y)−w D (x y)) ,   n n n μnD (x y) = r An (x) ∨ r An (y) − r Dn (x y) ei (w A (x)∨w A (y)−w D (x y)) . Example 1.23 Consider a complex bipolar fuzzy graph G = (A, D) on X = {a, b, c} as shown in Fig. 1.31. The complement of G is given in Fig. 1.32.

1.5 Bipolar Fuzzy Digraphs − → − → A directed graph, also known as digraph is a pair G ∗ = (X, E ), where X is the − → set of vertices and E is the set of directed edges. Note that the edges (or arcs) in − →∗ G have directions and arrows on these edges are used to encode the directional information: an arc from vertex x to vertex y indicates that one may move from x to y but not from y to x.

52

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs b(0.1ei0.5π , −0.4e−i0.5π ) c(0.2eiπ , −0.3e−i0.25π )

Fig. 1.31 Complex bipolar fuzzy graph G

(0.1ei0.5π , −0.3e−i0.25π )

(0.2ei0.3π , −0.3e−i0.25π )

a(0.3ei0.3π , −0.4e−i0.6π ) b(0.1ei0.5π , −0.4e−i0.5π ) c(0.2eiπ , −0.3e−i0.25π )

Fig. 1.32 Complement of G

(0.1ei0.3π , −0.4e−i0.5π )

a(0.3ei0.3π , −0.4e−i0.6π )

− → → In this section, we will write − x y ∈ E and say x and y are adjacent such that x is a starting node and y is an ending node. Definition 1.58 A bipolar fuzzy digraph on a non-empty set X is a pair G = − → − → p p n (A, B ), where A = (μ A , μnA ) is a bipolar fuzzy set on X and B = (μ− → , μ− → ) is a B B bipolar fuzzy relation in X such that p

p

p

n n n μ− → (x y) ≥ μ A (x) ∨ μ A (y), for all x, y ∈ X. → (x y) ≤ μ A (x) ∧ μ A (y) and μ− B

B

− → Note that B is not a symmetric bipolar fuzzy relation. Throughout this book, we − →∗ − → denote G a crisp digraph and G a bipolar fuzzy digraph. Example 1.24 Let A be a bipolar fuzzy set on X = {v1 , v2 , v3 , v4 }, given in − → Table 1.9, and B a bipolar fuzzy relation defined in Table 1.10. Figure 1.33 shows − → − → that G = (A, B ) is a bipolar fuzzy digraph. Table 1.9 Bipolar fuzzy set A on {v1 , v2 , v3 , v4 } A v1 v2 p

μA μnA

0.4 −0.2

0.6 −0.3

v3

v4

0.5 −0.2

0.5 −0.3

1.5 Bipolar Fuzzy Digraphs

53

− → Table 1.10 Bipolar fuzzy relation B in {v1 , v2 , v3 , v4 } − → v1 v2 B v1 v2 v3 v4

(0.1, −0.1) (0, 0) (0, 0) (0, 0)

(0, 0) (0, 0) (0, 0) (0.5, −0.1)

v3

v4

(0, 0) (0.2, −0.1) (0, 0) (0, 0)

(0, 0) (0, 0) (0.3, −0.1) (0, 0)

v2 (0.6, −0.3)

v1 (0.4, −0.2)

(0.1, −0.1)

(0.2, −0.1)

(0.4, −0.1) (0.3, −0.1) v4 (0.5, −0.3)

v3 (0.5, −0.2)

− → − → Fig. 1.33 Bipolar fuzzy digraph G = (A, B )

Definition 1.59 A bipolar fuzzy directed walk in a bipolar fuzzy digraph is an → → → e 1 , y1 , − e 2 , . . ., − e n−1 , yn−1 alternating sequence of vertices and directed edges y0 , − − → such that B (ek ) = (0, 0), for all 1 ≤ k ≤ n. Definition 1.60 A bipolar fuzzy directed path in a bipolar fuzzy digraph is a − → sequence of distinct vertices y1 , y2 , . . ., yn such that either B (yk yk+1 ) = (0, 0), for − → all 1 ≤ k ≤ n − 1. It is denoted by P n . If y1 = yn and A(y1 ) = A(yn ), the bipolar − → fuzzy directed path is known as a bipolar fuzzy directed cycle, denoted by C n . − → − → − → − → Definition 1.61 Let G 1 = (A1 , B 1 ) and G 2 = (A2 , B 2 ) be two bipolar fuzzy − → − → digraphs on non-empty sets X 1 and X 2 , respectively. The union of G 1 and G 2 is − → − → − → − → defined as a pair G 1 ∪ G 2 = (A1 ∪ A2 , B 1 ∪ B 2 ) such that for all x, y ∈ X 1 ∪ X 2 , p

p

1. (A1 ∪ A2 )(x) = (μ A1 (x) ∨ μ A2 (x), μnA1 (x) ∧ μnA2 (x)), − → → − → − → − → − → → 2. If − x y ∈ E 1, − xy ∈ / E 2 , then ( B 1 ∪ B 2 )(x y) = B 1 (x y), − → − → − → − → − → − → − → / E 2 , then ( B 1 ∪ B 2 )(x y) = B 2 (x y), 3. If x y ∈ E 1 , x y ∈ − → − → − → − → p p → n 4. If − x y ∈ E 1 ∩ E 2 , then ( B 1 ∪ B 2 )(x y) = (μ− → (x y) ∨ μ− → (x y), μ− → (x y) ∧ B1 B2 B1 n μ− → (x y)). B2

− → − → Example 1.25 Consider two bipolar fuzzy digraphs G 1 and G 2 as shown in − → − → Fig. 1.34. The union G 1 ∪ G 2 is also given in Fig. 1.34. − → − → Definition 1.62 The intersection of two bipolar fuzzy digraphs G 1 = (A1 , B 1 ) − → − → − → − → − → − → and G 2 = (A2 , B 2 ) is a pair G 1 ∩ G 2 = (A1 ∩ A2 , B 1 ∩ B 2 ), where A1 ∩ A2 − → − → and B 1 ∩ B 2 are defined as

54

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs G1 x(0.5, −0.3)

w(0.6, −0.1)

y(0.6, −0.2)

(0.4, −0.1)

(0.5, −0.2) G2

y(0.7, −0, 2)

w(0.5, −0.3)

(0.5, −0.2)

G1 ∪ G2 y(0.7, −0, 2)

x(0.5, −0.3)

(0.4, −0.1)

w(0.6, −0.3)

(0.5, −0.2)

(0.5, −0.2)

Fig. 1.34 Union of bipolar fuzzy digraphs G1 x(0.5, −0.3)

w(0.6, −0.2)

y(0.6, −0.2)

(0.4, −0.1)

(0.5, −0.2) G1 x(0.5, −0.3)

w(0.7, −0.1)

y(0.6, −0.2)

(0.4, −0.1)

(0.4, −0.1) G1 ∪ G2 y(0.6, −0, 2)

x(0.5, −0.3)

w(0.6, −0.1)

(0.4, −0.1)

− → − → Fig. 1.35 Intersection of bipolar fuzzy digraphs G 1 and G 2

p

p

for all x ∈ X 1 ∩ X 2 , 1. (A1 ∩ A2 )(x) = (μ A1 (x) ∧ μ A2 (x), μnA1 (x) ∨ μnA2 (x)), − → − → p p − → n n 2. ( B 1 ∩ B 2 )(x y) = (μ− → (x y) ∧ μ− → (x y), μ− → (x y) ∨ μ− → (x y)), for all x y ∈ B1 B2 B1 B2 − → − → E 1 ∩ E 2. − → − → Example 1.26 The intersection of two bipolar fuzzy digraphs G 1 and G 2 is shown in Fig. 1.35. − → − → − → Definition 1.63 The join of two bipolar fuzzy graphs G 1 = (A1 , B 1 ) and G 2 = − → − → − → − → − → (A2 , B 2 ) is defined as a pair G 1 + G 2 = (A1 + A2 , B 1 + B 2 ) such that A1 + A2 = A1 ∪ A2 , for all x ∈ X 1 ∪ X 2 , and the membership values of directed edges in − → − → G 1 + G 2 are defined as − → − → − → − → − → − → → for all − xy ∈ E ∪ E . 1. ]1.] B + B = B ∪ B , 1

2

1

2

1

2

1.5 Bipolar Fuzzy Digraphs x(0.5, −0.3)

55 x(0.5, −0.3)

z(0.7, −0.2)

(0.5, −0.2)

z(0.7, −0.2)

(0.5, −0.2)

3) 0. ,− .5 (0

(0 .6 ,− 0. 2)

(0.5, −0.2)

(0.5, −0.2)

(0.5, −0.2) y(0.6, −0.2) G1

y(0.6, −0.2)

w(0.5, −0.3) G2

(0.5, −0.2)

w(0.5, −0.3)

G1 + G2

− → − → Fig. 1.36 Join of bipolar fuzzy digraphs G 1 and G 2

− → − → 2. ]2.] Let E  be the set of all edges joining the vertices of G 1 and G 2 , − → − → p then for all x y ∈ E  where x ∈ X 1 and y ∈ X 2 , ( B 1 + B 2 )(x y) = (μ A1 (x) ∧ p n μ A2 (y), μ A1 (x) ∨ μnA2 (y)). − → − → Example 1.27 The join of two bipolar fuzzy digraphs G 1 and G 2 is shown in Fig. 1.36. − → − → Definition 1.64 The composition of two bipolar fuzzy digraphs G 1 = (A1 , B 1 ) − → − → − → − → − → − → and G 2 = (A2 , B 2 ) is denoted by the pair G 1 ◦ G 2 = (A1 ◦ A2 , B 1 ◦ B 2 ), where − → − → p p n A1 ◦ A2 = (μ A1 ◦A2 , μnA1 ◦A2 ) and B 1 ◦ B 2 = (μ− → − → , μ− → ) are defined as → − B 1◦ B 2

p (μ A1 (x1 )

p μ A2 (x2 ), μnA1 (x1 )

B 1◦ B 2

∧ ∨ for all (x1 , x2 ) ∈ 1. (A1 ◦ A2 )(x1 , x2 ) = X 1 × X 2, → − → − p p n n 2. ( B 1 ◦ B 2 )((x, x2 )(x, y2 )) = (μ A1 (x) ∧ μ− → (x 2 y2 ), μ A1 (x) ∨ μ− → (x 2 y2 )), B2 B2 − → − − → for all x ∈ X 1 , x2 y2 ∈ E 2 , − → − → p p n n 3. ( B 1 ◦ B 2 )((x1 , z)(y1 , z)) = (μ− → (x 1 y1 ) ∧ μ A2 (z), μ− → (x 1 y1 ) ∨ μ A2 (z)), B1 B1 → → − for all z ∈ X 2 , − x− 1 y1 ∈ E 1 , → − → − p p p n 4. ( B 1 ◦ B 2 )((x1 , x2 )(y1 , y2 )) = (μ A2 (x2 ) ∧ μ A2 (y2 ) ∧ μ− → (x 1 y1 ), μ A2 (x 2 ) ∨ B1 n n μ A2 (y2 ) ∨ μ− → (x 1 y1 )), B1 − → for all x , y ∈ X , − x−→ y ∈ E . 2

2

2

1 1

μnA2 (x2 )),

1

− → Definition 1.65 The Cartesian product of two bipolar fuzzy digraphs G 1 = − → − → − → − → − → − → (A1 , B 1 ) and G 2 = (A2 , B 2 ) is denoted by the pair G 1  G 2 = (A1 A2 , B 1  − → − → − → p p n B 2 ), where A1 A2 = (μ A1 A2 , μnA1 A2 ) and B 1  B 2 = (μ− → − → , μ− → ) are → − B 1 B 2 B 1 B 2 defined as

56

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs (0.1, −0.3)

(0.2, −0.3)

(0.1, −0.2) u(0.4, −0.4)

v(0.4, −0.5)

w(0.6, −0.5)

(u, a)(0.2, −0.3)

(v, a)(0.2, −0.3)

(w, a)(0.2, −0.3)

(0.2, −0.3)

(0.1, −0.2) (0.1, −0.3)

(u, b)(0.3, −0.4)

(x, a)(0.2, −0.3)

(0.1, −0.3)

(0.1, −0.3)

(0.1, −0.3)

(0.1, −0.3)

(v, b)(0.3, −0.5)

(w, b)(0.3, −0.5)

(x, b)(0.3, −0.5) (0.1, −0.3)

(0.1, −0.2)

(0.2, −0.3)

(0.2, −0.1)

(0.2, −0.1)

(0.2, −0.1)

(v, c)(0.4, −0.5)

(w, c)(0.4, −0.5)

(x, c)(0.4, −0.5)

(0.2, −0.1)

(u, c)(0.0, −0.4)

x(0.7, −0.6)

(0.1, −0.2)

(0.3, −0.1)

(0.3, −0.1)

(0.3, −0.1) (0.1, −0.3)

(0.2, −0.3)

(0.1, −0.2) (u, d)(0.3, −0.4)

(w, d)(0.3, −0.4)

(v, d)(0.3, −0.4)

a(0.2, −0.3) (0.1, −0.3)

b(0.3, −0.5) (0.2, −0.1) c(0.4, −0.5)

(0.1, −0.3)

(0.2, −0.3)

(0.3, −0.1)

P4

(x, d)(0.3, −0.4)

(0.3, −0.1) d(0.3, −0.4)

P4

− → − → Fig. 1.37 Cartesian product P 4  P 4

p

p

1. (A1 A2 )(x1 , x2 ) = (μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 )), for all (x1 , x2 ) ∈ X 1 × X 2, → − → − p p n n 2. ( B 1  B 2 )((x, x2 )(x, y2 )) = (μ A1 (x) ∧ μ− → (x 2 y2 ), μ A1 (x) ∨ μ− → (x 2 y2 )), B2 B2 − → for all x ∈ X 1 , x2 y2 ∈ E 2 , → − → − p p n n 3. ( B 1  B 2 )((x1 , z)(y1 , z)) = (μ− → (x 1 y1 ) ∧ μ A2 (z), μ− → (x 1 y1 ) ∨ μ A2 (z)), B1 B1 − → for all z ∈ X 2 , x1 y1 ∈ E 1 . Example 1.28 The Cartesian product of two bipolar fuzzy directed paths is shown in Fig. 1.37. − → − → Definition 1.66 The direct product of two bipolar fuzzy digraphs G 1 = (A1 , B 1 ) − → − → − → − → − → − → and G 2 = (A2 , B 2 ) is a pair G 1 × G 2 = (A1 × A2 , B 1 × B 2 ), where A1 × − → − → p p n A2 = (μ A1 ×A2 , μnA1 ×A2 ) and B 1 × B 2 = (μ− → − → , μ− → ) are defined as → − B 1× B 2

p (μ A1 (x1 )

B 1× B 2

p μ A2 (x2 ), μnA1 (x1 )

1. (A1 × A2 )(x1 , x2 ) = ∧ ∨ μnA2 (x2 )), for all (x1 , x2 ) ∈ X 1 × X 2, − → − → p p n n 2. ( B 1 × B 2 )((x1 , x2 )(x1 , y2 ))=(μ− → (x 1 y1 ) ∧ μ− → (x 2 y2 ), μ− → (x 1 y1 ) ∨ μ− → B1 B2 B1 B2 − → −−→ − → − − → (x y )), for all x y ∈ E , x y ∈ E . 2 2

1 1

1

2 2

2

Example 1.29 The direct product of two bipolar fuzzy digraphs is shown in Fig. 1.38.

1.5 Bipolar Fuzzy Digraphs

57 e(0.4, −0.5)

d(0.5, −0.8) (0.4, −0.5)

(0.3, −0.4)

(0.3, −0.4) (a, d)(0.2, −0.3) (0

.2 ,−

0. 3)

(0

f (0.6, −0.4) (0.4, −0.5)

(a, e)(0.2, −0.3) (0 .2 3) ,− . 0 0. − , 3) .2

(0

(b, d)(0.3, −0.4)

,− .2

(a, f )(0.2, −0.3)

0. 1) − .2 ,

1) 0.

(0

0. 1) −

,− .2 (0

.2 ,

a(0.2, −0.3)

(0.2, −0.3)

3) 0.

(b, f )(0.3, −0.4) b(0.3, −0.4)

(b, e)(0.3, −0.4)

(0

G

(0 .2 ,

(0.2, −0.1)

−

0. 1) c(0.2, −0.4)

(c, d)(0.2, −0.4)

(c, e)(0.2, −0.4)

(c, f )(0.2, −0.4)

P3

− → − → Fig. 1.38 Direct product G × P 3

− → − → Definition 1.67 The strong product of two bipolar fuzzy digraphs G 1 = (A1 , B 1 ) − → − → − → − → − → − → and G 2 = (A2 , B 2 ) is denoted by the pair G 1  G 2 = (A1  A2 , B 1  B 2 ) and defined as p

p

1. (A1  A2 )(x1 , x2 ) = (μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 )), for all (x1 , x2 ) ∈ X 1 × X 2, − → − → p p n n 2. ( B 1  B 2 )((x, x2 )(x, y2 ))=(μ A1 (x) ∧ μ− → (x 2 y2 ), μ A1 (x) ∨ μ− → (x 2 y2 )), for B2 B2 − → − − → all x ∈ X 1 , x2 y2 ∈ E 2 , − → − → p p n n 3. ( B 1  B 2 )((x1 , z)(y1 , z)) = (μ− → (x 1 y1 ) ∧ μ A2 (z), μ− → (x 1 y1 ) ∨ μ A2 (z)), B1 B1 → → − for all z ∈ X 2 , − x− 1 y1 ∈ E 1 , − → − → p p n n 4. ( B 1  B 2 )((x1 , x2 )(x1 , y2 )) = (μ− → (x 1 y1 ) ∧ μ− → (x 2 y2 ), μ− → (x 1 y1 ) ∨ μ− → B1 B2 B1 B2 − → −−→ − → − − → (x2 y2 )), for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Example 1.30 The strong product of two bipolar fuzzy directed paths is shown in Fig. 1.39. − → Definition 1.68 The lexicographic product of two bipolar fuzzy digraphs G 1 = − → − → − → − → − → (A1 , B 1 ) and G 2 = (A2 , B 2 ), denoted by G 1 • G 2 , is defined as a pair (A1 • − → − → A2 , B 1 • B 2 ) such that

58

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

v(0.4, −0.5)

w(0.6, −0.5)

(u, a)(0.2, −0.3)

(v, a)(0.2, −0.3)

(w, a)(0.2, −0.3)

(v, b)(0.3, −0.5)

(u, b)(0.3, −0.4)

(0.3, −0.1)

− 1, 0.

(0.3, −0.1)

(0.1, −0.2)

1) 0.

( (0.1, −0.2)

(u, d)(0.3, −0.4)

(w, c)(0.4, −0.5) (0.2, −0.3)

, 0.2

.1) −0

( (0.2, −0.3)

(v, d)(0.3, −0.4)

.3)

( (x, b)(0.3, −0.5) (0.1, −0.3) − 1, 0.

0.

1)

( (x, c)(0.4, −0.5)

(0.3, −0.1)

(u, c)(0.0, −0.4)

0.1

−0

(0.1, −0.3) ,− 0.1

) 0.1

( (0.1, −0.3)

(w, d)(0.3, −0.4)

(0.1, −0.3)

b(0.3, −0.5) (0.2, −0.1)

(0.3, −0.1)

. (0 (v, c)(0.4, −0.5)

)

− .2, (0

, 0.1

a(0.2, −0.3)

(0.2, −0.1)

1) 0.

(x, a)(0.2, −0.3)

(0.1, −0.3)

(0.2, −0.1)

(0.2, −0.1)

− 1,

(0.2, −0.3)

(0.2, −0.1)

(0.1, −0.2)

) 0.3 ,− 1 . (0 (w, b)(0.3, −0.5)

x(0.7, −0.6)

(0.1, −0.3)

2) 0.

(0.2, −0.3)

(0.1, −0.3)

,− .1 (0

(0.1, −0.3)

(0.1, −0.3)

(0.1, −0.2)

P4

(0.1, −0.3)

(0.2, −0.3)

(0.1, −0.2) u(0.4, −0.4)

(0.3, −0.1)

(x, d)(0.3, −0.4)

c(0.4, −0.5)

d(0.3, −0.4)

P4

− → − → Fig. 1.39 Strong product P 4  P 4 p

p

for all (x1 , x2 ) 1. (A1 • A2 )(x1 , x2 ) = (μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 )), ∈ X 1 × X 2, − → − → p p n n 2. ( B 1 • B 2 )((x, x2 )(x, y2 )) = (μ A1 (x) ∧ μ− → (x 2 y2 ), μ A1 (x) ∨ μ− → (x 2 y2 )), B2 B2 − → − − → for all x ∈ X 1 , x2 y2 ∈ E 2 , − → − → p p n n 3. ( B 1 • B 2 )((x1 , x2 )(x1 , y2 )) = (μ− → (x 1 y1 ) ∧ μ− → (x 2 y2 ), μ− → (x 1 y1 ) ∨ μ− → B1 B2 B1 B2 − → −−→ − → − − → (x2 y2 )), for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Example 1.31 The lexicographic product of two bipolar fuzzy directed paths is shown in Fig. 1.40. Theorem 1.10 The union, intersection, join, composition, Cartesian product, direct product, lexicographic product, and strong product of two bipolar fuzzy digraphs is also a bipolar fuzzy digraph. − → Definition 1.69 A bipolar fuzzy digraph G  = (A , B  ) is called a bipolar fuzzy − → − → − → − → subdigraph of a bipolar fuzzy digraph G = (A, B ), written as G  ⊆ G , if A ⊆ A − → − → and B ⊆ B , i.e., p p 1. μ A (x) ≤ μ A (x), μnA (x) ≥ μnA (x), for all x ∈ X, p p n n 2. μ− → (x y) ≤ μ− → (x y), μ− → (x y) ≥ μ− → (x y), for all x, y ∈ X. B

B

B

B

− → − → Definition 1.70 A bipolar fuzzy digraph G = (A, B ) is called strong if p

p

p

n n n μ− → (x y) = μ A (x) ∨ μ A (y), → (x y) = μ A (x) ∧ μ A (y) and μ− B

B

− → → for all − xy ∈ E .

1.5 Bipolar Fuzzy Digraphs

59

(f, a)(0.2, −0.3)

2, 0.

−

3) 0.

(

,−

(d, c)(0.2, −0.4)

)

,− .2 (0

(e, c)(0.2, −0.4)

1 0.

(0.2, −0.1)

.2 (0

1) 0.

(0.2, −0.1)

(e, b)(0.3, −0.4)

a(0.2, −0.3)

(0.2, −0.3)

(f , b)(0.3, −0.4)

(0.2, −0.3)

3) 0.

P3

f (0.6, −0.4)

(e, a)(0.2, −0.3)

(

(0.2, −0.1)

(d, b)(0.3, −0.4)

(0.2, −0.3)

(d, a)(0.2, −0.3)

− 2, 0.

(0.4, −0.5) e(0.4, −0.5)

(0.2, −0.3)

(0.4, −0.5) d(0.5, −0.8)

(f, c)(0.2, −0.4)

b(0.3, −0.4)

(0.2, −0.1)

c(0.2, −0.4) P3

− → − → Fig. 1.40 Lexicographic product P 3 • P 3

− → − → − → − → Proposition 1.22 If G 1 and G 2 are strong bipolar fuzzy digraphs, then G 1  G 2 , − → − → − → − → G 1 × G 2 and G 1  G 2 are strong bipolar fuzzy digraphs. Definition 1.71 Let Q(X → Y ) and R(Y → Z ) be two bipolar fuzzy relations. The max-min-max composition R ◦ Q(X → Z ) is the bipolar fuzzy relation defined by the membership functions p

μ R◦Q (x, z) =

 y

μnR◦Q (x, z) =



 p p μ Q (x, y) ∧ μ R (y, z) ,  μnQ (x, y) ∨ μnR (y, z) , for all (x, z) ∈ X × Z , y ∈ Y.

y

Definition 1.72 Let Q(X → Y ) and R(Y → Z ) be two bipolar fuzzy relations. The max-product−min-product composition R ◦ Q(X → Z ) is the bipolar fuzzy relation defined by the membership functions p

μ R◦Q (x, z) =

 p p (μ Q (x, y) · μ R (y, z)) y

μnR◦Q (x, z)

 = (μnQ (x, y) · μnR (y, z)), for all (x, z) ∈ X × Z , y ∈ Y. y

60

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making Decision-making refers to making decisions in the presence of multiple, usually conflicting criteria. Decision-making problems are common in everyday life. Graph theory is a useful tool in solving the combinatorial and decision-making problems in different areas of science and technology. To expand the base of applications, fuzzy graphs were introduced to model complex systems arising in computer science and operation research. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. In some cases, when the complex models are fuzzy in nature and contain bipolar information, it becomes necessary to make judgments with two opposite agreements. Decision-making situations can be categorized in different groups according to certain characteristics, sources of information and preference representations. 1. Single-criteria decision-making carries a situation, where we have only one source of information (or criteria) to solve a decision problem. In these situations, the solution to the problem comes directly and exclusively from the information provided. 2. In decision-making process, an expert mostly needs to compare a finite set of alternatives xi (i = 1, 2, . . . , n) and construct a preference relation. However, decision-making is not only the case for a single expert, whereas some decision problems have to be explained by a group of experts which work together to find the best alternative(s) from a set of feasible alternatives under certain characteristics. The nature of decision-making with multiple experts is called group decision-making (GDM) which is also known as multi-person decisionmaking. 3. Real-world decision-making problems are usually too complex and ill-structured to be considered through the examination of a single criterion, attribute, or point of view that will lead to the optimum decision. In fact, such a unidimensional approach is merely an oversimplification and can lead to unrealistic decisions. A more appealing approach would be the simultaneous consideration of all pertinent factors that are related to the problem. Multi-criteria decision-making (MCDM) is a discipline in its own right, which deals with decisions involving the choice of the best alternative from several potential candidates in a decision, subject to several criteria or attribute that may be concrete or vague. Using different criteria of decision-making, real-world applications of bipolar fuzzy sets and bipolar fuzzy digraphs in medical diagnosis, social networks, marketing, and decision support systems are discussed here in detail.

1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making

61

1.6.1 Bipolar Fuzzy Neurons in Medical Diagnosis Medical science has drawn much attention due to its recent advances in research in the past few years. Diseases are often misdiagnosed because the symptoms and their interrelationships are usually not considered during computations. Diseases can also be caused by the wrong medical treatments and the lack of information and ignorance of incompleteness of patient’s disease information. Bipolar fuzzy relations are one of the most fruitful and interesting mathematical techniques to study the relations of diseases and their causes. In the discrimination analysis for diagnosis of an illness, the symptoms are ranked according to the grade of discrimination of each disease by a particular symptom. In this application, the procedure for the diagnosis of diseases of patients with particular symptoms is discussed in detail. Let D = {d1 , d2 , . . . , dn } be the set of diseases, S = {s1 , s2 , . . . , sr } the set of symptoms and P = { p1 , p2 , . . . , pm } be the set of patients. Let Q : D → S and R : S → P be the bipolar fuzzy relations. The method for calculating the disease of each patient is given in Algorithm 1.6.1. Algorithm 1.6.1 Algorithm for the medical diagnosis of patients 01 Begin p p 02 Input the bipolar fuzzy relations Q = [(qi j , qinj )]n×r and R = [(ξ jk , ξ njk )]r ×m . 03 for i from 1 → n 04 for k from 1 → m p 05 tmax = 0 n =0 06 tmin 07 for j from 1 → r p p p 08 α j = qi j × ξ jk p p 09 α nj = qi j × ξ jk p p p 10 tmax = max{tmax , α j } n n n 11 tmax = max{tmax , α j } 12 end for p p 13 tik = tmax n n 14 tik = tmin 15 end for 16 end for 17 for i from 1 → n 18 Smax = 0 19 for k from1 → m p 20 dik = (tik )2 + (tikn )2 21 22 23 24 25 26

p

pik = tik + tikn × dik Smax = max{Smax , pik } if Smax = pik then print: Patient pk is suffering from disease di . end if end for

62

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

27 end for 28 End Description of Algorithm 1.6.1: Line 1 passes the input of bipolar fuzzy relations Q(diseases → symptoms) and R(symptoms → patients) with n number of diseases, r symptoms, and m patients. Lines 3−−16 calculate the max-product-min-product composition T = R ◦ Q. Lines 17−−26 calculate and print the symmetry index of diseases corresponding to each patient. Time complexity: Line 1 corresponds to the input of the algorithm, its time complexity is O(1). The for loop on line 7 runs r times, and therefore its complexity is O(r ). The nested loop on line 3 runs nmr times, and therefore its time complexity is O(nmr ). The time complexity of lines 17 and 19 is O(nm) and O(m), respectively. The net time complexity of the algorithm is O(nmr ). To illustrate Algorithm 1.6.1, consider the example of set of diseases D = {Diabetes, Dengue, Tuberculosis} and the set of symptoms S = {Temperature, Insulin, Blood pressure, Blood platelets, Cough}. The bipolar fuzzy relation Q(D → S) is shown in Table 1.11. Consider the set of patients as P = {Fayyaz, Amir, Aslam}. The bipolar fuzzy relation R(S → P) is given in Table 1.12. Using Definition 1.72, the max-product-min-product composition T = R ◦ Q is shown in Table 1.13. Table 1.11 Bipolar fuzzy relation Q(D → S) Q Temperature Insulin Diabetes Dengue Tuberculosis

(0.2, −0.8) (0.9, −0.1) (0.6, −0.2)

(0.9, −0.1) (0.0, −0.8) (0.0, −0.9)

Table 1.12 Bipolar fuzzy relation R(S → P) R Fayyaz Temperature Insulin Blood pressure Blood platelets Cough

(0.8, −0.1) (0.2, −0.6) (0.4, −0.4) (0.8, −0.1) (0.3, −0.4)

Blood pressure

Blood platelets

Cough

(0.1, −0.8) (0.8, −0.1) (0.4, −0.4)

(0.1, −0.8) (0.9, −0.1) (0.0, −0.8)

(0.1, −0.8) (0.1, −0.8) (0.9, −0.1)

Amir

Aslam

(0.6, −0.2) (0.9, −0.1) (0.1, −0.8) (0.2, −0.7) (0.5, −0.4)

(0.4, −0.4) (0.2, −0.7) (0.1, −0.7) (0.3, −0.6) (0.8, −0.2)

Table 1.13 Composition T = R ◦ Q(D → P) T Fayyaz Amir Diabetes Dengue Tuberculosis

(0.18, −0.06) (0.72, −0.01) (0.48, −0.02)

(0.81, −0.01) (0.54, −0.02) (0.45, −0.04)

Aslam (0.18, −0.07) (0.36, −0.04) (0.72, −0.02)

1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making

63

By applying Algorithm 1.6.1, it is identified that Fayyaz is suffering from Dengue, Amir is suffering from Diabetes, and Aslam is a patient of Tuberculosis.

1.6.2 Social Networking Social influence is a widespread mechanism to figure out a best-suited group economically, politically, educationally, etc. Social conflict and influence occur when individual opinions, decisions, and behaviors are influenced by others. In social networking problems, it is often observed that graphical models need to incorporate more structure than simply the adjacencies between vertices. In studies of group behavior, it is a common social issue that certain people can influence the thinking of others. A directed graph, called an influence graph, can be used to model this behavior. Each person of a group is represented by a vertex. There is a directed edge from vertex x to vertex y when the person represented by vertex x influences the person represented by vertex y. This digraph does not contain loops. In influence graph, the vertex (node) represents a power (authority) of a person and the edge represents the influence of a person on another person in the social group. The influence of the person has a fuzzy boundary, which gives a better representation in the fuzzy digraph. In a fuzzy influence graph, the degree of membership of nodes depicts the degree of power of a person belonging to the social group. The degree of membership can be interpreted as how much power a person posses in a group. The degree of membership of directed edges can be interpreted as the influence of one person onto another person. The influence of a person cannot always be positive. If two persons in a social group have some conflicts, then the influence between these persons can be negative. Similarly, if a person is not in the good books of the other person, the influence is negative on the other person. The coexistence of the positive and negative behavior of people can only be studied using a bipolar fuzzy digraph known as bipolar fuzzy influence graph. Consider the example of a bipolar fuzzy influence graph given in Fig. 1.41. In a bipolar fuzzy influence graph, the degree of membership of vertices is defined in terms of its positive and negative membership value. The positive degree of membership can be interpreted as how much power a person posses and negative membership value depicts as how much power a person losses, e.g., Tanzel has 70% power within the social group but he losses 60% power in the same group. The degree of membership of directed edges can be interpreted as the percentage of positive and negative influence, e.g., Amir follows 80% Naila’s suggestions but he does not follow 20% of her suggestions. Thus, bipolar fuzzy influence graphs are more compatible as compared to fuzzy influence graphs and can be used to study the influence and conflict among objects more precisely.

64 (0.5,-0.3)

Tanzel (0.7, -0.6)

Amir (0.9,-0.6)

.3)

-0

, 0.6

(

(0.2,-0.3)

(0.4,-0.4) (0.8,-0.2)

(0.4, -0.2)

Azhar (0.6, -0.5)

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Naila (0.8, -0.5)

(0.3,-0.2)

(0.7,-0.2)

Rajab (0.7, -0.4)

Fig. 1.41 Bipolar fuzzy influence graph

1.6.3 Bipolar Fuzzy Organizational Model Bipolar fuzzy influence graph can also be used to study the most influential person within an organization. In this bipolar fuzzy graphical model, vertices represent employees and directed edge represents the influence of an employee to another employee of the company. Such bipolar fuzzy digraphs have applications in modeling social structures, communication, and distributed computing. Consider the example of an organization having employees and their designation as shown in Table 1.14. For this organization, the set of employees is E = {B O D, M Q, T M, M Z , AK , R B, St}. The work environment within the organization is observed as follows: • Mujeeb Qayyum has worked with Munib Zia for over 10 years, and he values his input on strategic initiatives. • The board of directors is chaired by a long time associate of Munib Zia. Like Mujeeb, the chair of board also values Munib.

Table 1.14 Names of employees in the organization with their designations Name Designation Board of Director (BOD) Mujeeb Qayyum (MQ) Tahir Mahmood (TM) Munib Zia (MZ) Arif Kaleem (AK) Rizwan Bashir (RB) Staff (St)

Board of director CEO CTO Director of marketing Director of product development Director of human resources Staff

1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making

65

• For organization, the entire Marketing and HR team is very important. Rizwan Bashir is especially centered in the organization. • Tahir Mahmood and Rizwan Bashir have a history of conflict. • Tahir Mahmood has a great influence on the development team. Considering the above points, an influence graph can be constructed but such a directed graph cannot represent the power of employees within the organization and the degree of influence of employees on each other. As the power and influence have no defined boundaries, it is desired to represent them in the form of a fuzzy set. The fuzzy digraph represents the influence of employees on each other. To represent both power and conflict at the same time, we apply here the concept of bipolar fuzzy set, which is more precise about the influence and conflicts between the employees. The bipolar fuzzy set of the employees is shown in Table 1.15. The bipolar fuzzy relation among employees is shown in Table 1.16. The resultant bipolar fuzzy digraph is shown in Fig. 1.42. The positive membership value of a vertex in Fig. 1.42 represents the percentage of influence and negative membership value represents the percentage of conflict of one employee to the other, e.g., MQ possesses 90% power within the organization or MZ has 70% power and 20% conflicts. The positive degree of membership and negative degree of membership can also be interpreted as the percentage of positive and negative influence respectively, e.g., BOD is 60% influenced by MZ’s opinions but ignores 10% of his opinions. In Fig. 1.42, it is clear that MZ has influenced both BOD and MQ. He can influence both of them equally as the degree of membership in

Table 1.15 Bipolar fuzzy set A of employees A BOD MQ TM p

μA μnA

0.9 −0.1

0.9 0.0

0.8 −0.3

MZ

AK

RB

St

0.7 −0.2

0.6 −0.5

0.6 −0.4

0.5 −0.5

Table 1.16 Adjacency matrix corresponding to Fig. 1.42 Employees BOD MQ TM MZ

AK

RB

St

BOD MQ TM

(0, 0) (0, 0) (0, 0)

(0.8, 0.0) (0, 0) (0, 0)

(0, 0) (0.7, 0.0) (0, 0)

(0, 0) (0, 0) (0, 0)

(0, 0) (0, 0) (0, 0)

(0, 0) (0, 0) (0, 0)

MZ

(0.6, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

AK

(0.6, −0.1) (0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

RB

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0.5, −0.3) (0, 0)

St

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0) (0.5, 0.0) (0.4, −0.3) (0.4, −0.2) (0.5, −0.5) (0.5, −0.3) (0, 0)

66

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

(0.6, −0.1) BOD (0.9, −0.1)

(0.4, −0.2) MZ(0.7, −0.2)

St (0.5, −0.5)

(0.5, −0.3)

(0.5, −0.3)

(0.4, −0.3)

(0.8, 0.0)

(0.6, 0.0)

(0.5, −0.5)

(0.5, 0.0)

TM(0.8, −0.3)

MQ(0.9, 0.0)

RB(0.6, −0.4)

AK(0.6, −0.5)

(0.7, 0.0)

Fig. 1.42 Bipolar fuzzy organizational model

both cases is 0.6. But in the case of MQ, there is no negative degree of membership but in the case of BOQ there is a bit conflict which is 0.1. It is quite obvious that MZ is the most influential employee in the organization and possesses 70% power within the organization.

1.6.4 Bipolar Fuzzy Graphs in Marketability The marketability of science books is based on three criteria, i.e., pictures P, cost C, and examples E. It is known that if a book has more examples, low cost and a large number of pictures, the sale of the book improves. Here we assume that the sale of 60 percent of books in any stock is considered as a good sale. Consider the bipolar fuzzy set of criteria Cr = {(P, 0.6, −0.3), (C, 0.1, −0.8), (E, 0.6, −0.3)}. The degree of membership of E and P can be interpreted as 60% of books contain examples and pictures and 30% are theory based. The degree of membership of C depicts that 10% of the books have a low price but 80% are expensive. Cr can be written in the form of a matrix as   C = (0.6, −0.3) (0.1, −0.8) (E, 0.6, −0.3) To determine a better sale, we present it in bipolar fuzzy digraph given in Fig. 1.43 and apply Algorithm 1.6.2 to it. The bipolar fuzzy digraph in Fig. 1.43 shows a typical three-layered architecture of bipolar fuzzy neuron, i.e., input, hidden, and output layer. In a bipolar fuzzy neuron the input, hidden, and output weights are defined in terms of positive degree of membership and negative degree of non-membership. The aggregation, or activation,

1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making

67

Hidden Layer

Input Layer

Example (0.6, −0.3)

O1’(0.3, −0.64) (0.5, −0.5)

(0.5, −0.3)

Output Layer

(0.1, −0.8) (0.5, −0.5)

(0.5, −0.5) Cost (0.1, −0.8)

O2’(0.3, −0.09)

O1”(0.15, −0.32)

(0.1, −0.8)

(0.5, −0.3)

Pictures(0.6, −0.3)

(0.5, −0.5)

O3’(0.01, −0.64)

Fig. 1.43 Bipolar fuzzy digraph of marketability

of a neuron involves both positive and negative degrees of membership. A node in the input layer represents the criteria C of sales. A node in the hidden layer shows the aggregation/activation of the neuron, and the output layer shows the expected sales. The bipolar fuzzy relation I H between the input and hidden layers and the bipolar fuzzy relation H O between the hidden and output layers are given in Eq. (1.13). ⎡

⎤ (0.5, −0.5) (0.5, −0.3) (0.0, 0.0) I H = ⎣ (0.1, −0.8) (0.0, 0.0) (0.1, −0.8) ⎦ , (0.0, 0.0) (0.5, −0.3) (0.0, 0.0)

⎡

⎤ (0.5, −0.5) H O = ⎣ (0.5, −0.5) ⎦ . (0.5, −0.5) (1.13) The output on the hidden layer can be computed by taking composition between IH   and C, i.e., O  = C ◦ I H = (0.3, −0.64) (0.3, −0.09) (0.01, −0.64) . Similarly,   the final output  is calculatedby taking the composition of O and H O, i.e., O =  O ◦ H O = (0.15, −0.32) . The bipolar fuzzy digraph output layer in Fig. 1.43 shows that the sale is about 15 percent. The method to calculate the percentage sale of books is described in Algorithm 1.6.2. Algorithm 1.6.2 Algorithm to check the marketability of books 01 Begin 02 C := Criteria of sale 03 IH := Relation between input and hidden layer

68

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

04 HO := Relation between hidden and output layer 05 O  := C ◦ I H 06 O  := O  ◦ H O 07 if O  is not expected 08 Modify C using backpropagation and goto line 4 09 end if 10 print O  11 End Description of Algorithm 1.6.2: The first three lines of the algorithm set the required input. At lines 5 and 6, output on the hidden layer is calculated by taking the composition of bipolar fuzzy relations C and I H . The final output is calculated on line 5 by taking the composition of the output of hidden layer and bipolar fuzzy relation H O. Lines 7–10 check whether the results are in the desirable limits or not. If they are not within limits, the positive and negative membership degrees are modified using backpropagation. Time complexity of Algorithm 1.6.2: The net time complexity of the algorithm depends on lines 5 and 6 only. Line 5 is the composition of C and I H whose running time is cn, where c is the number of criteria and n is the number of activations in the hidden layer. Line 6 is the composition of O  and H O with n complexity. The complexity of algorithm is O(cn) or O(n). Since n is smaller than cn, the net complexity is O(cn).

1.6.5 Vulnerability Assessment of Gas Pipeline Networks The vulnerability assessment of a gas pipeline network can be categorized into structural components: reliability, connectivity reliability, flow performance reliability, and/or inter-dependent reliability. These reliabilities depend on the type of pipe and fittings used, their aging, and the connection between fittings and pipe. In most of the cases, we don’t know the exact age and condition of connectivity. We can present these factors as a bipolar fuzzy set. Any gas network can be represented as a bipo− → lar fuzzy digraph G = (F, P), where F is the bipolar fuzzy set of pipe fittings, presenting their ages and connectivity conditions as positive and negative degree of p membership μ F (x) and μnF (x), respectively, and P is a bipolar fuzzy relation of pipeline fittings. Consider the bipolar fuzzy set F of pipe fittings in Table 1.17 and

Table 1.17 Bipolar fuzzy set F A C1 C2 p

μA μnA

0.7 −0.1

0.5 −0.3

C3

C4

C5

C6

0.6 −0.3

0.7 −0.2

0.5 −0.4

0.5 −0.3

1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making Table 1.18 Bipolar fuzzy relation P P C1 C2 C1 C2 C3 C4 C5 C6

(0.5, −0.1) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0)

(0, 0) (0, 0) (0, 0) (0.6, −0.1) (0.6, −0.1) (0, 0)

C3

C4

C5

C6

(0, 0) (0.5, −0.3) (0, 0) (0, 0) (0, 0) (0, 0)

(0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0)

(0, 0) (0, 0) (0.5, −0.2) (0.5, −0.2) (0, 0) (0, 0)

(0, 0) (0, 0) (0.3, −0.3) (0.6, −0.2) (0, 0) (0,0)

C4

C5

C6

0 0 0 0 0 0

0 0 0.3923 0.3923 0 0

0 0 0.1727 0.4735 0 0

Table 1.19 Adjacency matrix of ranks of edges Weights C1 C2 C3 C1 C2 C3 C4 C5 C6

0 0 0 0.5392 0.5392 0

0.4490 0 0 0 0 0

0 0.3251 0 0 0 0

0.4490

0.5

C2

0.3251

C1

39

0.5392

2 0.3923

C5

C3

3

0.1727

92

0.3

C4

69

0.4735

C6

Fig. 1.44 Weighted digraph of a gas pipeline network

bipolar fuzzy relation P of the gas pipeline network is represented by the adjacency matrix in Table 1.18. Calculate the rank Si of each membership value of directed edge from bipolar fuzzy relation P as, Si = μi p + μi n ∗ di , where  di is the distance between membership poles of the directed edge, i.e., di = (μ p )2 i + (μn )2 i . The final adjacency

70

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

matrix of weights (ranks) of directed edges is shown in Table 1.19 and weighted digraph developed based on these weights is given in Fig. 1.44. This weighted digraph − → G w can be used for analyzing different kind of vulnerabilities in gas pipeline networks. The overall procedure to determine vulnerability of pipeline network in explained in Algorithm 1.6.3 with net complexity O( f 2 ). Algorithm 1.6.3 Algorithm to determine vulnerability of pipeline network Algorithm 01 Begin 02 Input the bipolar fuzzy set F = {C1 , C2 , . . . , C f } of f number of fittings. p 03 Input the bipolar fuzzy relation P = [(μi j , μinj ]n×n 04 for i from 1 → f 05 for j from 1to f 06

di j :=

p

Complexity f f2 f f2

(μi j )2 + (μinj )2

f2

+ μnP(i)

f2 f2 f f2

p μ P(i)

07 Si j := 08 end for 09 end for 10 Print W = [Si j ] f × f 11 End

∗ di

Description and complexity of Algorithm 1.6.3: The algorithm first takes a bipolar fuzzy set and bipolar fuzzy relation of pipeline fittings as an input. Lines 4–9 calculate the adjacency matrix of weights corresponding to bipolar fuzzy relation based on the degree of membership of directed edges. This weighted matrix is printed in line 10. Since nested loop on lines 5−8 runs f 2 times, the running time of Algorithm 1.6.3 is O( f 2 ).

1.6.6 Bipolar Fuzzy Digraph in Travel Time Communication graphs are used in many network models, such as transportation, as a natural mathematical tool to identify problems and, then calculating the most beneficial solution. These network models are mostly used to find the shortest path between end nodes. The optimality criteria is often evaluated in terms of weights of arcs/edges between two adjacent vertices in the network. In case of transportation and road networks, the travel time is mostly used as a weight. The travel time is a function of the traffic density on the road and/or the length of the road. The length of a road is a crisp quantity but the traffic density is fuzzy in nature. In a road network, we represent crossings as nodes and roads as directed edges. The traffic density is mostly calculated on the road between adjacent crossings and can be represented as bipolar fuzzy membership values. Figure 1.45 shows the model − → of a road network represented as a bipolar fuzzy digraph G = (C, L), where C = {(C1 , 0.8, −0.1), (C2 , 0.5, −0.3), (C3 , 0.6, −0.3), (C4 , 0.7, −0.2), (C5 , 0.5, −0.3)}

1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making (0.5,-0.1)

C2 (0.5,-0.3)

(0.

(0.6,-0.1)

6,0.1

)

(0.4

(0.4,0.3)

C1 (0.8,-0.1)

71

3) ,-0.

C3 (0.6,-0.3) (0.

2)

,-0.

C4 (0.7,-0.2)

(0.5

4,-

0.3

)

C5 (0.5,-0.3)

(0.6,-0.2)

Fig. 1.45 Bipolar fuzzy graph of a road network Table 1.20 Bipolar fuzzy relation on the set of crossings L C1 C2 C3 C1 C2 C3 C4 C5

(0, 0) (0, 0) (0.6, −0.1) (0.6, −0.1) (0, 0)

(0.5, −0.1) (0, 0) (0, 0) (0, 0) (0.4, −0.3)

(0, 0) (0.4, −0.3) (0, 0) (0.5, −0.2) (0, 0)

Table 1.21 Weights of directed edges in Fig. 1.45 L iw C1 C2 C3 C1 C2 C3 C4 C5

0 0 0.5392 0.5392 0

0.4490 0 0 0 0.25

0 0.2500 0 0.3923 0

C4

C5

(0, 0) (0, 0) (0, 0) (0, 0) (0.6, −0.2)

(0, 0) (0, 0) (0.4, −0.3) (0, 0) (0, 0)

C4

C5

0 0 0 0 0.4735

0 0 0.25 0 0

72

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs 0.4490

C1

0.5

0.25

C2

2

0.25

0.5392

39

C3

0.25

923

0.3 C4

0.4735

C5

Fig. 1.46 Weighted digraph of bipolar fuzzy road network

is a bipolar fuzzy set of crossings at which the traffic density is calculated and L is a bipolar fuzzy relation of crossings given in Table 1.20. The final weights of directed edges L i can be calculated by finding the rank , where di is the distance between membership poles of as L iw = μi p + μi n ∗ di directed edge, i.e., di = (μ p )2 i + (μn )2 i . The final adjacency matrix is shown in Table 1.21 and the weighted digraph is shown in Fig. 1.46. − → The above weighted adjacency matrix represents the final weighted digraph G w , which can be used to find the shortest/optimal path between two vertices by any of the known methods, including Dijkstra, A star, etc. Algorithm 1.6.4 generates the − → − → weighted digraph, G w , for the given bipolar fuzzy digraph G and can be used to calculate the optimal path from a source node to any other node.

1.6 Applications of Bipolar Fuzzy Digraphs to Decision Making

73

Algorithm 1.6.4 Computing adjacency matrix of weighted digraph of a road network Algorithm 01 Begin 02 C := Bipolar fuzzy set of crossings 03 R := Bipolar fuzzy relation (Adjacency matrix of C × C) 04 for each adjacent crossing x in C 05 for each adjacent crossing y in C 06 if (C(x) is adjacent to C(y)) p p p 07 μ R(x y) := min(μC(x) , μC(y) ) n n 08 μnR(x y) := max(μC(x) , μC(y) ) 09 end if 10 end for 11 end for 12 L := Bipolar fuzzy set of edges 13 W R := Weighted relation (Adjacency matrix of C × C) 14 for each bipolar fuzzy element i in L  15

p (μ L(i) )2 + (μnL(i) )2 p Si := μ L(i) + μnL(i) ∗ di W Rx y := Si

di :=

16 17 18 end for 19 print W R 20 Calculate Optimal path using W R 21 End

Complexity 1 1 c c2 c2 c2 c2 c2 c2 c 1 1 l l l l l 1

Description and complexity of Algorithm 1.6.4: This algorithm is quite similar to Algorithm 1.6.3. This algorithm initially sets a bipolar fuzzy set of crossings. Lines 3−6 calculate the membership values for roads which are assigned to bipolar fuzzy set of edges in line 7 and, then the adjacency matrix is prepared in line 8. Finally, a weighted adjacency matrix is calculated in lines 9−12 using rank techniques based on membership values. This weighted matrix, which is printed on line 19, can be used for calculating the shortest path using any known algorithm like Dijkstra or A star in line 20. On a similar argument as in Algorithm 1.6.3, this algorithm can also be divided into two blocks, lines 4−11 and lines 14−18. The blocks of lines 4−11 which are calculating positive membership function and negative membership function of edges are nested between two loops of size c each, where c is the number of crossings in the road network. The running time of this block is c2 . The other block of lines 14−18, calculating the weight of edges, consists of one loop of size l, where l is the number of roads. The running time of this block is l. The running time of the algorithm can be determined using these two blocks. Since O(c2 ) is greater than O(l), the running time of the algorithm is O(c2 ).

74

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

1.6.7 Comparison Analysis Decision-making is a procedure to make an ideal decision that has the highest level of achievement from a set of alternatives which are characterized regarding various conflicting criteria. The bipolar fuzzy graph is a conceptual framework to study and analyze the bipolar fuzzy units that are intensely or frequently connected in a network. These are used to study the mathematical structures and model various practical processes and pairwise bipolar fuzzy relations among objects in social, physical, information, and biological systems. In most of the real-world problems, there are situations when an object can travel from point x to point y but not from y to x. That is, the relation among objects is non-symmetric and bipolar in nature, for instance, 1. capacity of water in pipeline networks, 2. studying the various phenomena on roads networks such as density of traffic, severity of accidents at ongoing and incoming roads, dangers of robbery, etc., 3. modeling leadership and incapacity of candidates competing for different political seats, 4. communication network of server and client computers, 5. strength of predator-prey relations in ecological niches. Bipolar fuzzy digraphs play a vital role in studying such problems and representing non-symmetric bipolar fuzzy relations. These mathematical structures not only study multi-attribute data but also conflicting parameters of the data which can increase the accuracy of the results. Moreover, it is difficult to deal with data given in the form of symmetric bipolar fuzzy matrices as compared to non-symmetric bipolar fuzzy matrices in which most of the entries are (0, 0). Thus, the concept of bipolar fuzzy digraphs is more reliable in many decision-making problems because they decrease the calculation complexity and time consumption when the data is given in the form of higher order matrices.

1.7 Conclusions Mathematical modeling is an important aspect of apprehending the discrete and continuous physical systems. Bipolar uncertainty in data and information incorporates a significant role in various abstract and applied mathematical modeling and decision analysis. Graphical networks can be studied more precisely when two opposite linguistic properties are to be dealt with, emphasizing the need for a mathematical approach toward bipolar fuzzy information. Bipolar fuzzy graph has been introduced, using Zhang’s powerful bipolar fuzzy set, to overcome the limitations entailed in classical graph theory and fuzzy graph theory. In this chapter, we have illustrated the notion of bipolar fuzzy sets and presented several basic concepts concerning bipolar fuzzy graphs and bipolar fuzzy digraphs. We have discussed the isomorphism properties of bipolar fuzzy graphs and various methods of their construction. We have

1.7

Conclusions

75

studied certain types of bipolar fuzzy graphs, bipolar fuzzy directed walk, bipolar fuzzy bridge, connectedness in bipolar fuzzy graphs, weak and strong bipolar fuzzy edges, and properties of α−cuts in bipolar fuzzy graphs. We have also elaborated certain operations and properties of complex bipolar fuzzy graphs. We have discussed the importance of bipolar fuzzy graphical models in various real-world phenomenon with the help of composition of bipolar fuzzy relations, connectivity and weighted matrices.

Exercises 1 1. What do you mean by (1, −1), (1, 0), and (0, −1) in bipolar fuzzy environment? 2. If B is a bipolar fuzzy reflexive relation on A, then show that B(s,t) is a reflexive relation on A(s,t) , for s ∈ [0, 1], t ∈ [−1, 0]. 3. If B1 and B2 are bipolar fuzzy reflexive relations, then show that B1 ◦ B2 is bipolar fuzzy reflexive relation. 4. Discuss transitive closure of a bipolar fuzzy relation. 5. If B1 and B2 are bipolar fuzzy symmetric relations, then B1 ◦ B2 is a bipolar fuzzy symmetric relation if and only if B1 ◦ B2 = B2 ◦ B1 . 6. Give an example to show that bipolar fuzzy sets and intuitionistic fuzzy sets are not the same. 7. Determine whether the following bipolar fuzzy relation R is an equivalence relation. Also calculate R −1 . ⎡ ⎤ (1, −1) (1, −0.5) (0.6, −0.6) (0.4, −0.7) (0.2, −0.4) ⎢ (1, −0.7) (0.5, −0.5) (0.2, −0.6) (0.4, −0.4) (0.2, −0.5) ⎥ ⎢ ⎥ (1, 0) (0.6, −0.5) (0.6, −0.3) (0.5, −0.8) ⎥ R=⎢ ⎢ (0, −1) ⎥. ⎣ (0.1, −0.1) (1, −0.4) (0.2, −1) (0.1, −0.7) (0.2, 0) ⎦ (0, −1) (1, 0) (0.6, −0.5) (0.6, −0.3) (0.5, −0.8) 8. Consider the following set of diseases/diagnoses D, and set of symptoms S, D = {Diabetes, Dengue, Tuberculosis}, S = {Temperature, Insulin, Blood pressure, Blood platelets, Cough}. The bipolar fuzzy relations Q(D → S) and R(S → P) are shown in Tables 1.22 and 1.23. Calculate (a) Composition and (b) Disease of each person. 9. Let G and H be two isomorphic bipolar fuzzy graphs. (a)

If G = Pn is a bipolar fuzzy path of length n, then prove that H is also a bipolar fuzzy path of length n.

76

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

Table 1.22 Bipolar fuzzy relation Q(D → S) Q Temperature Insulin Diabetes Dengue Tuberculosis

(0.5, −0.8) (0.9, −0.5) (0.6, −0.5)

(0.9, −0.5) (0.4, −0.8) (0.5, −0.9)

Table 1.23 Bipolar fuzzy relation R(S → P) S Fayyaz Temperature Insulin Blood pressure Blood platelets Cough

(b)

(0.8, −0.6) (0.4, −0.6) (0.4, −0.6) (0.8, −0.5) (0.3, −0.4)

Blood pressure

Blood platelets

Cough

(0.4, −0.8) (0.8, −0.6) (0.4, −0.4)

(0.6, −0.8) (0.9, −0.3) (0.6, −0.8)

(0.61, −0.8) (0.4, −0.8) (0.9, −0.5)

Amir

Aslam

(0.6, −0.5) (0.9, −0.1) (0.5, −0.8) (0.5, −0.7) (0.5, −0.4)

(0.4, −0.4) (0.4, −0.7) (0.5, −0.7) (0.3, −0.6) (0.8, −0.2)

If G = Cn is a bipolar fuzzy cycle with n vertices, then prove that H is also a bipolar fuzzy cycle with n vertices.

10. The degree sequence of a bipolar fuzzy graph consists of degrees of vertices written in increasing order, with repeats where necessary. Classify whether the following statements are true or false. (a) Any two isomorphic bipolar fuzzy graphs have the same degree sequence. (b) Any two bipolar fuzzy graphs with same degree sequence are isomorphic. 11. Let G 1 be a disconnected bipolar fuzzy graph and φ be an isomorphism from G 1 onto a bipolar fuzzy graph G 2 , then prove that G 2 is also disconnected. If φ is a weak isomorphism, then what about G 2 ? 12. A null bipolar fuzzy graph is a bipolar fuzzy graph with no edges. Let f from G 1 into G 2 be a one-to-one homomorphism. If G 1 is a complete bipolar fuzzy graph, then prove that f (G c1 ) is a null bipolar fuzzy graph. Give an example to show that if f is not onto, then f (G c1 ) may not be a null bipolar fuzzy graph. (b) If G 1 is a null bipolar fuzzy graph, then prove that f (G c1 ) is a complete bipolar fuzzy graph. Give an example to show that if f is not onto, then f (G c1 ) may not be a complete bipolar fuzzy graph. (a)

13. If G 1 and G 2 are two complex bipolar fuzzy graphs, then prove that G 1 ∪ G 2 , G 1 ∩ G 2 , G 1 G 2 , G 1 × G 2 , G 1  G 2 , G 1 • G 2 , and G 1 ◦ G 2 are complex bipolar fuzzy graphs. 14. If G 1 and G 2 are two strong complex bipolar fuzzy graphs, then prove that G 1 G 2 , G 1 × G 2 , G 1  G 2 , G 1 • G 2 and G 1 ◦ G 2 are strong complex bipolar fuzzy graphs.

Exercises 1

77

15. Give an example to show that the union of strong complex bipolar fuzzy graphs may not be strong. 16. If G 1 G 2 is a strong complex bipolar fuzzy graph, then prove that at least one of G 1 or G 2 must be strong.     17. If G 1 , G 2 , G 1 , G 2 are bipolar fuzzy graphs and G 1 ∼ = G2, G1 ∼ = G 2 , then classify whether the following statements are true or false. (a) (b) (c) (d) (e) (f) (g)

  G 1 G 1 ∼ = G 2 G 2   G1  G1 ∼ = G2  G2   G1 + G1 ∼ = G2 + G2   (G 1 + G 1 )c ∼ = G c2 ∪ (G 2 )c   (G 1 + G 2 )c ∼ = (G 1 )c ∪ (G 2 )c   (G 1 ∪ G 1 )c ∼ = G c2 + (G 2 )c   c ∼ (G 1 ∪ G 2 ) = (G 1 )c + (G 2 )c

18. Let X , Y , and Z be three disjoint sets with |X | = p, |Y | = q and |Z | = r . If K p,q , K q,r , K p,r are complete bipartite bipolar fuzzy graphs on X ∪ Y , Y ∪ Z , X ∪ Z , respectively, and K p,q,r is tripartite bipolar fuzzy graph on X ∪ Y ∪ Z , then describe the conditions under which the following statements are true. (a) (b) (c) (d)

K p,q ∪ K q,r ∼ = K p,q,r K p,q ∪ K q,r  K p,q,r K p,q ∪ K r ∼ = K p,q,r K p,q,r is a null bipolar fuzzy graph.

Here K r is a complete bipolar fuzzy graph on Z . 19. Give an example of a bipolar fuzzy graph G such that

μ B (x y) =

1 p p (μ (x) ∧ μ A (y)), 2 x= y A

μnB (x y) =

1 n (μ (x) ∨ μnA (y)), 2 x= y A

p

x= y

x= y

but G is not self-complementary. 20. If G is a self-complementary strong bipolar fuzzy graph, then show that

p

μ B (x y) =

x= y

x= y



p

p

(μ A (x) ∧ μ A (y)),

x= y

μnB (x y) =



(μnA (x) ∨ μnA (y)),

x= y

do not hold. 21. Let G be a strong bipolar fuzzy graph such that

78

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs



p

μ B (x y) =

x= y

x= y



p

p

(μ A (x) ∧ μ A (y)),

x= y

μnB (x y) =



(μnA (x) ∨ μnA (y)).

x= y

Then G is not a self-complementary. 22. Let G 1 and G 2 be two strong bipolar fuzzy graphs, then show that G 1 ◦ G 2 is a strong bipolar fuzzy graph. Determine whether or not (G 1 ◦ G 2 )c = G 1 c ◦ G 2 c . 23. Let G 1 and G 2 be two bipolar fuzzy graphs. Give an example to show that (G 1 ◦ G 2 )c = (G 2 ◦ G 1 )c . 24. Give an example of a connected bipolar fuzzy graph G such that G − x y has at most two components, for any edge x y in G.

References 1. Akram, M.: Bipolar fuzzy graphs. Inf. Sci. 181(24), 5548–5564 (2011) 2. Akram, M.: Bipolar fuzzy graphs with applications. Knowl.-Based Syst. 39, 1–8 (2013) 3. Akram, M., Alshehri, N., Davvaz, B., Ashraf, A.: Bipolar fuzzy digraphs in decision support systems. J. Multiple-Valued Logic Soft. Comput. 27(5–6), 531–551 (2016) 4. Akram, M., Dudek, W.A.: Regular bipolar fuzzy graphs. Neural Comput. Appl. 21(supp 1), 197–205 (2012) 5. Akram, M., Farooq, A.: Bipolar fuzzy trees. New Trends Math. Sci. 4(3), 58–72 (2016) 6. Akram, M., Feng, F., Saeid, A.B., Leoreanu-Fotea, V.: A new multiple criteria decision-making method based on bipolar fuzzy soft graphs. Iranian J. Fuzzy Syst. 15(4), 73–92 (2018) 7. Akram, M., Karunambigai, M.G.: Metric in bipolar fuzzy graphs. World Appl. Sci. J. 14(12), 1920–1927 (2011) 8. Akram, M., Samanta, S., Pal, M.: Application of bipolar fuzzy sets in planar graphs. Int. J. Appl. Comput. Math. 3(2), 773–785 (2017) 9. Alkouri, A.U.M., Massa’deh, M.O., Ali, M.: On bipolar complex fuzzy sets and its application. J. Intell. Fuzzy Syst. (Preprint) (2020) 10. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 11. Bhattacharya, P.: Some remarks on fuzzy graphs. Pattern Recogn. Lett. 6(5), 297–302 (1987) 12. Bhutani, K.R.: On automorphisms of fuzzy graphs. Pattern Recogn. Lett. 9(3), 159–162 (1989) 13. Bhutani, K.R., Mordeson, J., Rosenfeld, A.: On degrees of end nodes and cut nodes in fuzzy graphs. Iranian J. Fuzzy Syst. 1(1), 57–64 (2004) 14. Bhutani, K.R., Rosenfeld, A.: Strong arcs in fuzzy graphs. Inf. Sci. 152, 319–322 (2003) 15. Chen, J., Li, S., Ma, S., Wang, X.: m−polar fuzzy sets: An extension of bipolar fuzzy sets. Sci. World J. (2014). https://doi.org/10.1155/2014/416530 16. Dubois, D., Kaci, S., Prade, H.: Bipolarity in reasoning and decision, an introduction. In: International Conference on Information Processing and Management of Uncertainty, IPMU04, pp. 959966 (2004) 17. Ghorai, G., Pal, M.: Certain types of product bipolar fuzzy graphs. Int. J. Appl. Comput. Math. 3(2), 605–619 (2017) 18. Harary, F.: Graph Theory. Addison-Wesley, Reading (1969) 19. Hwang, C.L., Yoon, K.: Multiple attribute decision-making-methods and applications. In: A State of the Art Survey. Springer, New York (1981) 20. Kaufmann, A.: Introduction la thorie des sous-ensembles flous l’usage des ingnieurs (Fuzzy sets theory). Masson, Paris (1973)

References

79

21. Lee, K.M.: Bipolar valued fuzzy sets and their basic operations. In: Proceedings of the International Conference, Bangkok, Thailand, pp. 307–317 (2000) 22. Ma, X., Zhan, J., Khan, M., Zeeshan, M., Anis, S., Awan, A.S.: Complex fuzzy sets with applications in signals. Comput. Appl. Math. 38(4), 150 (2019) 23. Mathew, S., Mordeson, J.N.: Fuzzy influence graphs. New Math. Nat. Comput. 13(3), 311–325 (2017) 24. Mathew, S., Sunitha, M.S.: Types of arcs in a fuzzy graph. Inf. Sci. 179(11), 1760–1768 (2009) 25. Mathew, S., Sunitha, M.: Strongest strong cycles and θ-fuzzy graphs. IEEE Trans. Fuzzy Syst. 21(6), 1096–1104 (2013) 26. Mathew, S., Sunitha, M.S., Anjali, N.: Some connectivity concepts in bipolar fuzzy graphs. Ann. Pure Appl. Math. 7(2), 98–108 (2014) 27. Mordeson, J.N., Nair, P.S.: Cycles and cocycles of fuzzy graphs. Inf. Sci. 90(1–4), 39–49 (1996) 28. Mordeson, J.N., Nair, P.S.: Successor and source of (fuzzy) finite state machines and (fuzzy)directed graphs. Inf. Sci. 95(1–2), 113–124 (1996) 29. Mordeson, J.N., Nair, P.S.: Fuzzy graphs and hypergraphs. Studies in Fuzziness and Soft Computing. Springer, Berlin (1998); 2nd edn (2001) 30. Mordeson, J.N., Chang-Shyh, P.: Operations on fuzzy graphs. Inf. Sci. 79(3–4), 159–170 (1994) 31. Nagoorgani, A.: Properties of μ−complement of a fuzzy graph. Int. J. Algorithms Comput. Math. 2(3), 73–83 (2009) 32. Nagoorgani, A., Chandrasekaran, V.T.: A First Look at Fuzzy Graph Theory. Allied Publishers, Chennai (2010) 33. Nagoorgani, A., Malarvizhi, J.: Isomorphism on fuzzy graphs. Int. J. Comput. Math. Sci. 2(4), 190–196 (2008) 34. Nagoorgani, A., Radha, K.: The degree of a vertex in some fuzzy graphs. Int. J. Algorithms Comput. Math. 2(3), 107–116 (2009) 35. Radha, K., Arumugam, S.: On strong product of two fuzzy graphs. Int. J. Sci. Res. Publ. 4(10), 1–6 (2014) 36. Radha, K., Arumugam, S.: On lexicographic products of two fuzzy graphs. Int. J. Fuzzy Math. Arch. 7(2), 169–176 (2015) 37. Ramot, D., Milo, R., Friedman, M., Kandel, A.: Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 10(2), 171–186 (2002) 38. Rao, R.V.: A decision-making framework model for evaluating flexible manufacturing systems using digraph and matrix methods. Int. J. Adv. Manuf. Technol. 30(11–12), 1101–1110 (2006) 39. Rashmanlou, H., Samanta, S., Pal, M., Borzooei, R.A.: A study on bipolar fuzzy graphs. J. Intell. Fuzzy Syst. 28(2), 571–580 (2015) 40. Rashmanlou, H., Samanta, S., Pal, M., Borzooei, R.A.: Bipolar fuzzy graphs with categorical properties. Int. J. Comput. Intell. Syst. 8(5), 808–818 (2015) 41. Rosenfeld, A.: Fuzzy graphs. In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds.) Fuzzy Sets and Their Applications, pp. 77–95. Academic, New York (1975) 42. Samanta, S., Pal, M.: Irregular bipolar fuzzy graphs. Int. J. Appl. Fuzzy Sets 2, 91–102 (2012) 43. Samanta, S., Pal, M.: Some more results on bipolar fuzzy sets and bipolar fuzzy intersection graphs. J. Fuzzy Math. 22(2), 253–262 (2014) 44. Sunitha, M.S., Vijayakumar, A.: Complement of a fuzzy graph. Indian J. Pure Appl. Math. 33(9), 1451–1464 (2002) 45. Talebi, A.A., Rashmanlou, H.: Complement and isomorphism on bipolar fuzzy graphs. Fuzzy Inf. Eng. 6, 505–522 (2014) 46. Yang, H.-L., Li, S.-G., Yang, W.-H., Lu, Y.: Notes on “Bipolar fuzzy graphs”. Inf. Sci. 242, 113–121 (2013) 47. Yaqoob, N., Gulistan, M., Kadry, S., Wahab, H.A.: Complex intuitionistic fuzzy graphs with application in cellular network provider companies. Mathematics 7(1), 35 (2019) 48. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 49. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3(2), 177–200 (1971) 50. Zadeh, L.A.: The concept of a lingusistic variable and its application to approximate reasoning. Inf. Sci. 8, 199–249 (1975)

80

1 Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

51. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 52. Zhang, W.R.: YinYang bipolar lattices and L−sets for bipolar knowledge fusion, visualization and decision. Int. J. Inf. Technol. Decis. Making 4(4), 621–645 (2005) 53. Zhang, W.-R.: Bipolar fuzzy sets. In: Proceedings of FUZZ-IEEE, pp. 835–840 (1998) 54. Zhang, W.-R., Zhang, L.: YinYang bipolar logic and bipolar fuzzy logic. Information Sciences. 165(3–4), 265–287 (2004) 55. Zhang, W.-R.: YinYang bipolar relativity: a unifying theory of nature, agents and causality with applications in quantum computing, cognitive informatics and life sciences. Information Sciences Reference, Hershey (2011) 56. Zhang, W.-R.: NPN fuzzy sets and NPN qualitative algebra: a computational framework for bipolar cognitive modeling and multiagent decision analysis. IEEE Trans. Syst. Man Cybern. Part B (Cybernetics). 26(4), 561-574 (1996) 57. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998) 58. Zhang, G., Dillon, T.S., Cai, K.Y., Ma, J., Lu, J.: Operation properties and δ-equalities of complex fuzzy sets. Int. J. Approx. Reason. 50(8), 1227–1249 (2009)

Chapter 2

Distance Measures in Bipolar Fuzzy Graphs

In this chapter, we discuss the notion of distance in bipolar fuzzy graphs and present certain properties concerning distance functions in complete bipolar fuzzy graphs, complete bipartite bipolar fuzzy graphs, and products of bipolar fuzzy graphs. We establish certain formulae of distance and degree of vertices of bipolar fuzzy graphs G 1 ∗ G 2 in terms of distance and degree of vertices of bipolar fuzzy graphs G 1 and G 2 , where ∗ is any given operation. We obtain a characterization of self-centered bipolar fuzzy graphs using eccentric vertices, peripheral vertices, and central vertices. We describe the notions of bipolar fuzzy path cover and edge cover of a bipolar fuzzy graph and, establish a necessary and sufficient condition for a complete bipolar fuzzy graph to have a bipolar fuzzy bridge. We discuss certain properties of self-centered, antipodal, and self-median bipolar fuzzy graphs when the bipolar fuzzy graph is complete or strong. Moreover, using the techniques of distance function, we describe the importance of bipolar fuzzy graphs with real-world applications in traveling and product manufacturing. This chapter is due to [4, 5, 18].

2.1 Introduction The mathematical field of graph theory is growing rapidly in recent times with a lot of research activities. In the past few decades, at the international level, one-third of the Mathematics research papers are from graph theory and combinatorics. Most of the real-world networks come with a spatial embedding, as they exist in threedimensional Euclidean space and have a natural notion of distance. The applications of distances in graph theory arise in various domains including drug discovery, robotic navigation, routing internet traffic, scheduling and planning, social networking and determining aircraft routes. The distance dG (x, y) between two vertices x and y in a graph G is the number of edges in the shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance [9]. If there is no path © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_2

81

82

2 Distance Measures in Bipolar Fuzzy Graphs

connecting the two vertices, i.e., if the vertices x and y belong to different connected components, then conventionally the distance is defined as infinite. Fuzzy set theory owes its remarkable origin to the work of Zadeh [22] in 1965. The theory of fuzzy sets has been introduced to solve real-life problems having uncertainty and vagueness. It has become a vigorous area of research in different disciplines. The advantage of replacing classical sets by Zadeh’s fuzzy sets is that it gives more accuracy and precision in theory and more efficiency and system compatibility in applications. In 1975, Rosenfeld [15] obtained fuzzy analogues of several basic graph-theoretic concepts like bridges, paths, cycles, trees, connectedness, and established some of their properties. The author has defined μ-length of any u − v fuzzy path P as the sum of reciprocals of edge weights in P and the distance between u and v, called μ-distance, is defined as the smallest μ-length of P. Based on this μ-distance, Bhattacharya [7] introduced the concepts of eccentricity and center in fuzzy graphs and the properties of this distance are further studied by Sunitha and Vijyakumar [19]. The concept of sum distance in fuzzy graphs is studied in [21] as the smallest sum of edge weights of any fuzzy path P and proved that it is a metric. The concept of length, distance, eccentricity, radius, diameter of a bipolar fuzzy graph, and selfcentered bipolar fuzzy graphs are studied in [4]. Bhutani and Rosenfeld introduced the concepts of strong arcs [8], fuzzy end nodes [11], and geodesics in fuzzy graphs [10]. Further studies based on g-distance are carried out by Sameena and Sunitha in [16] and [17]. The concepts of g-peripheral nodes, g-boundary nodes, and ginterior nodes based on g-distance were introduced by Linda and Sunitha [14]. In 1994, Zhang [23] introduced bipolar fuzzy set theory as an extension of fuzzy set theory in which the membership degree range is [−1, 1]. The main idea behind such characterization is linked with the existence of “bipolar information” (such as positive information and negative information) about the given set. Bipolar fuzzy sets not only have applications in mathematical theories but also in real-world problems [25, 26]. The notion of a bipolar fuzzy graph was introduced by Akram [1] in 2011 and, Akram and Dudek [3] discussed regular bipolar fuzzy graphs.

2.2 Distance in Bipolar Fuzzy Graphs In this section, the concept of distance function in bipolar fuzzy graphs is introduced and applied to various classes of bipolar fuzzy graphs. Definition 2.1 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X then the degree vertex x ∈ X is a pair = (deg p (x), degn (x)), where  of any  deg(x) p p n n deg (x) = u∈X μ B (xu) and deg (x) = u∈X μ B (xu). Definition 2.2 The Cartesian product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is denoted by the pair G 1 G 2 = (A1 A2 , B1 B2 ) , where p p A1 A2 = (μ A1 A2 , μnA1 A2 ) and B1 B2 = (μ B1 B2 , μnB1 B2 ) are defined as  p  p 1. (A1 A2 )(x1 , x2 ) = μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , for all (x1 , x2 ) ∈ X 1 × X 2,

2.2 Distance in Bipolar Fuzzy Graphs x1 (0.5, −0.3)

83 y1 (0.6, −0.2) G1

(0.5, −0.2) (x1 , x2 )(0.5, −0.2)

(y1 , x2 )(0.6, −0.2)

x2 (0.7, −0.2)

(0.5, −0.2)

G1 G2

(0.5, −0.2)

(0.5, −0.2)

(0.5, −0.2)

G2

(0.5, −0.2) (x1 , y2 )(0.5, −0.3)

(y1 , y2 )(0.5, −0.2)

y2 (0.5, −0.3)

Fig. 2.1 Cartesian product G 1 G 2

 p  p 2. (B1 B2 )((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 ), μnA1 (x) ∨ μnB2 (x2 y2 ) , for all x ∈ X 1 , x2 y2 ∈ E 2 ,  p  p 3. (B1 B2 )((x1 , z)(y1 , z)) = μ B1 (x1 y1 ) ∧ μ A2 (z), μnB1 (x1 y1 ) ∨ μnA2 (z) , for all z ∈ X 2 , x1 y1 ∈ E 1 . Example 2.1 The Cartesian product of two bipolar fuzzy graphs G 1 and G 2 is shown in Fig. 2.1. Definition 2.3 The direct product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is a pair G 1 × G 2 = (A1 × A2 , B1 × B2 ) , where A1 × A2 = p p (μ A1 ×A2 , μnA1 ×A2 ) and B1 × B2 = (μ B1 ×B2 , μnB1 ×B2 ) are defined as  p  p for all (x1 , x2 ) 1. (A1 × A2 )(x1 , x2 )= μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , ∈ X 1 × X 2,  p  p 2. (B1 × B2 )((x1 , x2 )(x1 , y2 ))= μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) , for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Example 2.2 The direct product of two bipolar fuzzy graphs G 1 and G 2 is shown in Fig. 2.2. Definition 2.4 The strong product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is denoted by the pair G 1  G 2 = (A1  A2 , B1  B2 ) and defined as  p  p 1. (A1  A2 )(x1 , x2 )= μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , for all (x1 , x2 ) ∈ X 1 × X 2,

84

2 Distance Measures in Bipolar Fuzzy Graphs x1 (0.5, −0.3)

y1 (0.6, −0.2) G1

(0.5, −0.2) (y1 , x2 )(0.6, −0.2)

(x1 , x2 )(0.5, −0.2)

(0 .5 ,

(x1 , y2 )(0.5, −0.3)

2) 0.

−

(0.5, −0.2)

(0

,− .5

x2 (0.7, −0.2)

0.

G2

2)

(y1 , y2 )(0.5, −0.2)

y2 (0.5, −0.3)

Fig. 2.2 Direct product G 1 × G 2

 p  p 2. (B1  B2 )((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 ), μnA1 (x) ∨ μnB2 (x2 y2 ) , for all x ∈ X 1 , x2 y2 ∈ E 2 ,  p  p 3. (B1  B2 )((x1 , z)(y1 , z)) = μ B1 (x1 y1 ) ∧ μ A2 (z), μnB1 (x1 y1 ) ∨ μnA2 (z) , for all z ∈ X 2 , x1 y1 ∈ E 1 ,  p  p 4. (B1  B2 )((x1 , x2 )(x1 , y2 ))= μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) , for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Example 2.3 The strong product of two bipolar fuzzy graphs G 1 and G 2 is shown in Fig. 2.3. Definition 2.5 The lexicographic product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ), denoted by G 1 • G 2 , is defined as a pair (A1 • A2 , B1 • B2 ) such that  p  p 1. (A1 • A2 )(x1 , x2 ) = μ A1 (x1 ) ∧ μ A2 (x2 ), μnA1 (x1 ) ∨ μnA2 (x2 ) , for all (x1 , x2 ) ∈ X 1 × X 2,  p  p 2. (B1 • B2 )((x, x2 )(x, y2 )) = μ A1 (x) ∧ μ B2 (x2 y2 ), μnA1 (x) ∨ μnB2 (x2 y2 ) , for all x ∈ X 1 , x2 y2 ∈ E 2 ,  p  p 3. (B1 • B2 )((x1 , x2 )(x1 , y2 )) = μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) , for all x1 y1 ∈ E 1 , x2 y2 ∈ E 2 . Definition 2.6 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on nonempty sets X 1 and X 2 , respectively. The union of G 1 and G 2 is defined as a pair G 1 ∪ G 2 = (A1 ∪ A2 , B1 ∪ B2 ) such that for all x, y ∈ X 1 ∪ X 2 , 1. If x ∈ X 1 , x ∈ / X 2 , then (A1 ∪ A2 )(x) = A1 (x),

2.2 Distance in Bipolar Fuzzy Graphs

85

x1 (0.5, −0.3)

y1 (0.6, −0.2) (0.5, −0.2)

G1

(y1 , x2 )(0.6, −0.2)

(x1 , x2 )(0.5, −0.2)

x2 (0.7, −0.2)

(0.5, −0.2) )

.5 ,

−

0. 2)

(0.5, −0.2)

(0

2 0.

(0.5, −0.2)

(0.5, −0.2)

. (0

− 5,

G2

(0.5, −0.2) (x1 , y2 )(0.5, −0.3)

(y1 , y2 )(0.5, −0.2)

y2 (0.5, −0.3)

Fig. 2.3 Strong product G 1  G 2

2. If x ∈ / X 1 , x ∈ X 2 , then (A1 ∪ A2 )(x) = A2 (x), p p 3. If x ∈ X 1 ∩ X 2 , then (A1 ∪ A2 )(x) = (μ A1 (x) ∨ μ A2 (x), μnA1 (x) ∧ μnA2 (x)). If supp(B1 ) = E 1 and supp(B2 ) = E 2 are the sets of edges in G 1 and G 2 , then B1 ∪ B2 can be defined as / E 2 , then (B1 ∪ B2 )(x y) = B1 (x y), 1. If x y ∈ E 1 , x y ∈ / E 2 , then (B1 ∪ B2 )(x y) = B2 (x y), 2. If x y ∈ E 1 , x y ∈ 3. If x y ∈ E 1 ∩ E 2 , then (B1 ∪ B2 )(x y) = (μ Bp 1 (x y) ∨ μ Bp 2 (x y), μnB1 (x y) ∧ μnB2 (x y)). Definition 2.7 The join of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is defined as a pair G 1 + G 2 = (A1 + A2 , B1 + B2 ) such that A1 + A2 = A1 ∪ A2 , for all x ∈ X 1 ∪ X 2 , and the membership values of the edges in G 1 + G 2 are defined as 1. B1 + B2 = B1 ∪ B2 , for all x y ∈ E 1 ∪ E 2 .  2. Let E  be the set of all edges joining the vertices  of G 1 and G 2 , then for all x y ∈ E , p p n n where x ∈ X 1 and y ∈ X 2 , (B1 + B2 )(x y)= μ A1 (x) ∧ μ A2 (y), μ A1 (x) ∨ μ A2 (y) . Definition 2.8 A bipolar fuzzy path in a bipolar fuzzy graph G = (A, B) is a p sequence of distinct vertices y1 , y2 , . . . , yn such that either μ B (yk yk+1 ) > 0 or μnB (yk yk+1 ) < 0, for all 1 ≤ k ≤ n − 1. It is denoted by Pn . If y1 = yn and A(y1 ) = A(yn ), the bipolar fuzzy path is known as a bipolar fuzzy cycle, denoted by Cn . A bipolar fuzzy graph G is connected if there exists a bipolar fuzzy path between each pair of distinct vertices. An example of a bipolar fuzzy path P5 is shown in Fig. 2.4.

86

2 Distance Measures in Bipolar Fuzzy Graphs

a1 (0.2, −0.5)

a2 (0.3, −0.4) (0.1, −0.3)

a4 (0.5, −0.1)

a3 (0.2, −0.5)

(0.2, −0.4)

(0.2, −0.1)

a5 (0.4, −0.3)

(0.1, −0.1)

Fig. 2.4 Bipolar fuzzy path P5

Definition 2.9 Let G = (A, B) be a bipolar fuzzy graph on X . For any bipolar fuzzy path R: y1 − y2 − ... − yn in G, the μ p −length of R is defined as the sum of μ p values of the edges and μn −length of R is defined as the sum of μn -values of the edges, that is, p

L (R) =

n 

n

p

μ B (y i−1 y i ),

L (R) =

i=2

n 

μnB (y i−1 y i ).

i=2

The length of bipolar fuzzy path R is represented by the ordered pair L(R) = p n (L (R), L (R)). For any two vertices x, y of G, let R = {Ri | Ri is an x − y bipolar fuzzy path, i = 1, 2, 3, ...} be the set of all bipolar fuzzy paths from x to y. Then the distance from x to y is denoted by d(x, y) or dG (x, y) and defined as the ordered p n pair d(x, y) = (d (x, y), d (x, y)), where  p p n d (x, y) = min L (R j ) | R j ∈ R, j = 1, 2, 3, . . . , d (x, y)  p = max L (R j ) | R j ∈ R, j = 1, 2, 3, . . . . Theorem 2.1 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs. If (x1 , x2 ) and (y1 , y2 ) are vertices of the Cartesian product G 1 G 2 , then p

p

p

n

n

n

d G 1 G 2 ((x1 , x2 ), (y1 , y2 )) ≤ d G 1 (x1 , y1 ) + d G 2 (x2 , y2 ), d G 1 G 2 ((x1 , x2 ), (y1 , y2 )) ≥ d G 1 (x1 , y1 ) + d G 2 (x2 , y2 ). Proof Assume that dG 1 (x1 , y1 ) and dG 2 (x2 , y2 ) are finite. Let R1 and R2 be the bipolar fuzzy paths in G 1 and Q 1 , Q 2 be bipolar fuzzy paths in G 2 , where Ri : x1 = xi1 , xi2 , . . . , xin = y1 is such that p

n

dG 1 (x1 , y1 ) = (L (R1 ), L (R2 )), and Q i : x2 = yi1 , yi2 , . . . , yin  = y2 be a bipolar fuzzy paths in G 2 such that p

n

dG 2 (x2 , y2 ) = (L (Q 1 ), L (Q 2 )). This establishes the following bipolar fuzzy paths in G 1 G 2 : Ri × {yi1 } = (xi1 , yi1 ), (xi2 , yi1 ), . . . , (xin , yi1 ) {xin } × Q i = (xin , yi1 ), (xin , yi2 ), . . . , (xin , yin  ),

i = 1, 2,

2.2 Distance in Bipolar Fuzzy Graphs

87

whose join are the bipolar fuzzy paths of length L(Ri × {y1 }) + L({xn } × Q i ). It is clear that p

p

p

n

n

d G 1 G 2 ((x1 , x2 ), (y1 , y2 )) ≤ L (R1 × {x2 }) + L ({y1 } × Q 1 ), n

dG

1 G 2

(2.1)

((x1 , x2 ), (y1 , y2 )) ≥ L (R2 × {x2 }) + L ({y1 } × Q 2 ).

p

p

p

p

(2.2)

p

L (R1 × {x2 }) = μ B1 (x11 x12 ) ∧ μ A2 (x2 ) + μ B1 (x12 x13 ) ∧ μ A2 (x2 ) ≤ p

L (R1 × {x2 }) ≤ p

p p + · · · + μ B1 (x1n−1 x1n ) ∧ μ A2 (x2 ) p p μ B1 (x11 x12 ) + μ B1 (x12 x13 ) + · · · + p p L (R1 ) = d G 1 (x1 , y1 )

p

(2.3) p

μ B1 (x1n−1 x1n ) = L (R1 ), (2.4)

p

⇒ L (R1 × {x2 }) ≤ d G 1 (x1 , y1 ).

(2.5)

By using similar argument, it can be proved that p

p

n

n

p

L ({y1 } × Q 1 ) ≤ L (Q 1 ) = d G 2 (x2 , y2 ), L (R2 × {x2 }) ≥ n

L ({y1 } × Q 2 ) ≥

(2.6)

n L (R2 ) = d G 1 (x1 , y1 ), n n L (Q 2 ) = d G 2 (x2 , y2 ). p

(2.7) (2.8) p

From (2.1), (2.4), and (2.6), we conclude that d G 1 G 2 ((x1 , x2 ), (y1 , y2 ))≤d G 1 (x1 , p y1 ) + d G 2 (x2 , y2 ). n n From (2.2), (2.7), and (2.8), it follows that d G 1 G 2 ((x1 , x2 ), (y1 , y2 )) ≥ d G 1 (x1 , n y1 ) + d G 2 (x2 , y2 ). Definition 2.10 Let G ∗1 • G ∗2 • . . . • G ∗k be any product of the graphs G ∗1 , G ∗2 ,. . ., G ∗k , where • represents any product including Cartesian product, direct product, strong product, or lexicographic product. The mapping f G i : G ∗1 • G ∗2 • · · · • G ∗k → G i∗ defined by f G i (x1 , x2 , . . . , xk ) = xi ,

xi ∈ X i , 1 ≤ i ≤ k,

is called the projection of G i∗ onto G ∗1 • G ∗2 • · · · • G ∗k . Theorem 2.2 Let S be a bipolar fuzzy path in G 1 G 2 and the membership values of both bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) have the folp p p p lowing relations: μ A1 ≥ μ B2 , μnA1 ≤ μnB2 , μ A2 ≥ μ B1 and μnA2 ≤ μnB1 , then L(S) = L( f G 1 (S)) + L( f G 2 (S)). Proof Let P : x1 − x2 − · · · − xn be a bipolar fuzzy path in G 1 and Q : y1 − y2 − · · · − yn  be a bipolar fuzzy path in G 2 . Let S be a bipolar fuzzy path in G 1 G 2

88

2 Distance Measures in Bipolar Fuzzy Graphs

which is established as follows: S : (x1 , y1 ) − (x2 , y1 ) − · · · − (xn , y1 ) − (xn , y2 ) − · · · − (xn , yn  ). p

n

Clearly, f G 1 (S) = P, f G 2 (S) = Q and L(S) = (L (S), L (S)). It follows that p

p

p

p

p

p

p

L (S) = μ B1 (x1 x2 ) ∧ μ A2 (y 1 ) + μ B1 (x2 x3 ) ∧ μ A2 (y 1 ) + · · · + μ B1 (xn−1 xn ) ∧ μ A2 (y 1 ) p

p

p

p

+ μ A1 (xn ) ∧ μ B2 (y 1 y 2 ) + · · · + μ A1 (xn ) ∧ μ B2 (yn  −1 yn  ) p

p

p

p

p

= μ B1 (x1 x2 ) + μ B1 (x2 x3 ) + · · · + μ B1 (xn−1 xn ) + μ B2 (y 1 y 2 ) + · · · + μ B2 (yn  −1 yn  ) p

p

p

p

= L (P) + L (Q) = L ( f G 1 (S)) + L ( f G 2 (S)). n

n

n

Similarly, it can be proved that L (S) = L ( f G 1 (S)) + L ( f G 2 (S)). Hence,   p n p p n n L(S) = (L (S), L (S)) = L ( f G 1 (S)) + L ( f G 2 (S)), L ( f G 1 (S)) + L ( f G 2 (S)) ,     p n p n L(S) = L ( f G 1 (S)), L ( f G 1 (S)) + L ( f G 2 (S)), L ( f G 2 (S)) = L( f G 1 (S)) + L( f G 2 (S)).

 Lemma 2.1 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs and (x1 , y1 ) and (x2 , y2 ) are vertices of the Cartesian product G 1 G 2 . If the membership p values of both bipolar fuzzy paths G 1 and G 2 have the following relations: μ A1 ≥ p p p n n n n μ B2 , μ A1 ≤ μ B2 , μ A2 ≥ μ B1 and μ A2 ≤ μ B1 , then dG 1 G 2 ((x1 , x2 ), (y1 , y2 )) = dG 1 (x1 , y1 ) + dG 2 (x2 , y2 ). Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be bipolar fuzzy graphs on X 1 and X 2 , respectively, and (x1 , y1 ), (x2 , y2 ) ∈ X 1 × X 2 . By Theorem 2.1, p

p

p

d G 1 G 2 ((x1 , x2 ), (y1 , y2 )) ≤ d G 1 (x1 , y1 ) + d G 2 (x2 , y2 ), n d G 1 G 2 ((x1 , x2 ), (y1 , y2 ))

≥

n d G 1 (x1 , y1 )

+

(2.9)

n d G 2 (x2 , y2 ).

(2.10)

Let R1 and R2 be the shortest bipolar fuzzy paths between vertices (x1 , x2 ) and p n (y1 , y2 ) such that dG 1 G 2 ((x1 , x2 ), (y1 , y2 )) = (L (S1 ), L (S2 )). The projections f G 1 (Si ) and f G 2 (Si ), i = 1, 2 are the bipolar fuzzy paths between the vertices x1 and y1 in G 1 , x2 , and y2 in G 2 . Consider p

p

p

p

p

d G 1 (x1 , y1 ) + d G 2 (x2 , y2 ) ≤ L ( f G 1 (S1 )) + L ( f G 2 (S1 )) = L (S1 ) p

= d G 1 G 2 ((x1 , x2 ), (y1 , y2 )). n

n

(2.11)

n

Similarly it can be proved that d G (x1 , y1 ) + d G (x2 , y2 ) ≥ d G G ((x1 , x2 ), (y1 , y2 )). 1 2 1 2

(2.12)

2.2 Distance in Bipolar Fuzzy Graphs (0.1, −0.2) u(0.4, −0.4)

(v, a)(0.2, −0.3)

(0.1, −0.2) (0.1, −0.3)

(u, b)(0.3, −0.4)

x(0.7, −0.6)

(w, a)(0.2, −0.3)

(x, a)(0.2, −0.3)

(0.2, −0.3)

(0.1, −0.3)

(0.1, −0.3)

(0.1, −0.3)

(v, b)(0.3, −0.5)

(w, b)(0.3, −0.5)

(x, b)(0.3, −0.5)

(0.1, −0.2)

(0.2, −0.3)

(0.1, −0.3)

(0.2, −0.1)

(0.2, −0.1)

(0.2, −0.1)

(v, c)(0.4, −0.5)

(w, c)(0.4, −0.5)

(x, c)(0.4, −0.5)

(0.1, −0.2)

(0.2, −0.3)

(0.3, −0.1) (u, d)(0.3, −0.4)

P4

(0.1, −0.3)

w(0.6, −0.5)

(0.1, −0.3)

(0.2, −0.1) (u, c)(0.4, −0.4)

(0.2, −0.3)

v(0.4, −0.5)

(u, a)(0.2, −0.3)

89

(0.3, −0.1)

(0.3, −0.1)

(v, d)(0.3, −0.4)

(w, d)(0.3, −0.4)

(x, d)(0.3, −0.4)

(0.2, −0.3)

(0.1, −0.3)

b(0.3, −0.5) (0.2, −0.1) c(0.4, −0.5)

(0.1, −0.3)

(0.3, −0.1)

(0.1, −0.2)

a(0.2, −0.3)

(0.1, −0.3)

(0.3, −0.1) d(0.3, −0.4)

P4

Fig. 2.5 Cartesian product P4 P4

By combining (2.9), (2.11), (2.10), and (2.12), the required result is obtained.



Example 2.4 Consider the Cartesian product of two bipolar fuzzy paths in Fig. 2.5. It can be easily seen that d P 4  P 4 ((u, a), (v, b)) = (0.2, −0.5) = d P 4 (u, v) + d P 4 (a, b), d P 4  P 4 ((u, a), (u, c)) = (0.3, −0.4) = d P 4 (u, u) + d P 4 (a, c). Similarly, for the other vertices. Theorem 2.3 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. If x1 ∈ X 1 , x2 ∈ X 2 and the membership values of G 1 and p p p p G 2 have the following relations: μ A1 ≥ μ B2 , μnA1 ≤ μnB2 , μ A2 ≥ μ B1 and μnA2 ≤ μnB1 , then degG 1 G 2 (x1 , x2 ) = degG 1 (x1 ) + degG 2 (x2 ). Proof By Definition 2.1, 

 p μ B1 B2 ((x1 , x2 )(y1 , y2 )), μnB1 B2 ((x1 , x2 )(y1 , y2 )) (y1 ,y2 )∈X 1 ×X 2    p μ B1 B2 ((x1 , x2 )(y1 , y2 )), μnB1 B2 ((x1 , x2 )(y1 , y2 )) = x 1 =y1 ,x 2 y2 ∈E 2    p μ B1 B2 ((x1 , x2 )(y1 , y2 )), μnB1 B2 ((x1 , x2 )(y1 , y2 )) + x 1 y1 ∈E 1 ,x 2 =y2    p p μ A1 (x1 ) ∧ μ B2 (x2 y2 ), μnA1 (x1 ) ∨ μnB2 (x2 y2 ) = x 1 =y1 ,x 2 y2 ∈E 2    p p μ B1 (x1 y1 ) ∧ μ A2 (x2 ), μnB1 (x1 y1 ) ∨ μnA2 (x2 ) + x 1 y1 ∈E 1 ,x 2 =y2       p p μ B2 (x2 y2 ), μnB2 (x2 y2 ) + μ B1 (x1 y1 ), μnB1 (x1 y1 ) =

degG 1 G 2 (x1 , x2 ) =



x 2 y2 ∈E 2

= degG 2 (x2 ) + degG 1 (x1 ).

x 1 y1 ∈E 1

90

2 Distance Measures in Bipolar Fuzzy Graphs

Theorem 2.4 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs p p such that membership values of edges have the following relations: μ B1 ≤ μ B2 n n and μ B1 ≥ μ B2 . Let x1 , y1 ∈ X 1 , x2 , y2 ∈ X 2 , and R1 , R2 be the shortest bipolar fuzzy paths between (x1 , x2 ) and (y1 , y2 ) such that dG 1 ×G 2 ((x1 , x2 ), (y1 , y2 )) = p n (L (R1 ), L (R2 )), then p

n

dG 1 ×G 2 ((x1 , x2 ), (y1 , y2 )) = (L ( f G 1 (R1 )), L ( f G 1 (R2 ))). Proof Assume that for i = 1, 2, Ri : (x1 , x2 ) = (x1(i1) , x2(i1) ) − (x1(i2) , x2(i2) ) − · · · − (x1(in) , x2(in) ) = (y1 , y2 ) are the shortest bipolar fuzzy paths between (x1 , x2 ) and (y1 , y2 ). Clearly, p

n

dG 1 ×G 2 ((x1 , x2 ), (y1 , y2 )) = (L (R1 ), L (R2 )).

(2.13)

If E is the set of edges in G 1 × G 2 , then p

  p (1 j) (1 j) (1 j+1) (1 j+1) (x μ , x )(x , x ) (1 j) (1 j) (1 j+1) (1 j+1) 1 2 1 2 (x1 ,x2 )(x1 ,x2 )∈E B1 ×B2      p (1 j) (1 j+1) p (1 j) (1 j+1) ∧ μ B2 x2 x2 μ B1 x1 x1 = (1 j) (1 j+1) (1 j) (1 j+1) x1 x1 ∈E 1 ,x2 x2 ∈E 2    p (1 j) (1 j+1) μ B1 x1 x1 = ( j) ( j+1) x1 x1 ∈E 1  p (2.14) = L f G 1 (R1 ) .

L (R1 ) =



It is easy to show that n

L (R2 ) = L

n



 f G 1 (R2 ) .

(2.15)

 p From (2.13), (2.14), and (2.15), dG 1 ×G 2 ((x1 , x2 ), (y1 , y2 )) = L ( f G 1 (R1 )),  n L ( f G 1 (R2 )) .  p

p

Remark 2.1 If membership values of edges have the relation: μ B2 ≤ μ B1 then Theorem 2.4 takes the form as  p  n dG 1 ×G 2 ((x1 , x2 ), (y1 , y2 )) = L ( f G 2 (R1 )), L ( f G 2 (R2 )) . Example 2.5 In Fig. 2.6, the shortest bipolar fuzzy path between the vertices (a, d) and (a, f ) is S : (a, d) − (b, e) − (a, f ). f P3 (S) = a − b − a = P. It can be easily seen that dP3 ×P3 ((a, d), (a, f )) = (0.4, −0.6) = L(P). For the vertices (a, d) and (c, f ), the shortest bipolar fuzzy path is S : (a, d) − (b, e) − (c, f ). Therefore, dP3 ×P3 ((a, d), (c, f )) = (0.4, −0.4). The projection of S in P3 is a − b − c, whose length is equal to dP3 ×P3 ((a, d), (c, f )).

2.2 Distance in Bipolar Fuzzy Graphs

91

(0.4, −0.5) d(0.5, −0.8)

e(0.4, −0.5)

(0 .

2, −

)

0. 3)

f (0.6, −0.4)

(a, e)(0.2, −0.3)

(a, d)(0.2, −0.3)

P3

(0.4, −0.5)

− 2, 0.

3 0.

(0 .

2, −

(

(a, f )(0.2, −0.3)

a(0.2, −0.3)

)

0. 3)

− 2, 0.

3 0.

(

(0.2, −0.3)

(b, f )(0.3, −0.4)

(b, d)(0.3, −0.4)

a(0.3, −0.4)

(b, e)(0.3, −0.4)

2, 0.

−

1) 0.

(0 .2 ,

(

(c, d)(0.2, −0.4)

−

0. 1)

2, 0.

−

1) 0.

(

(c, e)(0.2, −0.4)

(0

(0.2, −0.1)

.2 ,

−

0. 1)

(c, f )(0.2, −0.4)

c(0.2, −0.4)

P3

Fig. 2.6 Direct product P3 × P3

Theorem 2.5 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs p p such that the membership values of edges have the following relation: μ B1 ≤ μ B2 and n n μ B1 ≥ μ B2 . For any two vertices (x1 , x2 ) and (y1 , y2 ) of the direct product G 1 × G 2 , let k be a smallest positive integer such that G ∗1 has a x1 , y1 -walk of length k and G ∗2 has a x2 , y2 -walk of length k. The positive distance between (x1 , x2 ) and (y1 , y2 ) is the smallest μ p -length of any bipolar fuzzy walk between x1 and y1 , whose length in the crisp graph G ∗1 is k. The negative distance between (x1 , x2 ) and (y1 , y2 ) is the largest μn -length of any bipolar fuzzy walk between x1 and y1 , whose length in the crisp graph G ∗1 is k. Example 2.6 The bipolar fuzzy graph in Fig. 2.7 is the direct product of P3 and C3 . Take the vertices (a, d) and (c, e), the smallest a − c and d − e walks are of length 2. Therefore, the positive and negative distances between (a, d) and (c, e) must be the smallest and largest, respectively, lengths of a bipolar fuzzy walk in P3 whose length in crisp path P3 is 2. Such a bipolar fuzzy walk in P3 × C3 is (a, d) − (b, f ) − (c, e). Hence dP3 ×C3 ((a, d), (c, e)) = (0.4, −0.4). Theorem 2.6 Let G 1 and G 2 be two bipolar fuzzy graphs such that the membership p p degrees of edges have the following relation: μ B1 ≤ μ B2 and μnB1 ≥ μnB2 . If x1 ∈ X 1 and x2 ∈ X 2 , then

92

2 Distance Measures in Bipolar Fuzzy Graphs (0.5, −0.4) (0.4, −0.5)

(0.4, −0.5) d(0.5, −0.8)

e(0.4, −0.5)

(a, d)(0.2, −0.3)

(0 .2 ,−

,− 0.2

0 .3

)

3) 0.

,− .2 (0

(0 .2 ,

)

(0

) 0.3

(

3 0.

C3

(a, f )(0.2, −0.3)

(a, e)(0.2, −0.3)

(0. 2, −

,− .2

f (0.6, −0.4)

−

0

(0.2, −0.3)

0. 3)

(b, f )(0.3, −0.4)

(b, d)(0.3, −0.4)

(b, e)(0.3, −0.4) (0 .2 ,−

1) 0. − , .2 (0 (0 .2 ,− (0. 0. 2, − 1) 0.1 )

0. 1

)

− 2, 0.

0.

(

1)

2, − (0.

) 0.1

(c, d)(0.2, −0.4)

(c, e)(0.2, −0.4)

a(0.2, −0.3)

) .3

a(0.3, −0.4)

(0.2, −0.1)

c(0.2, −0.4)

(c, f )(0.2, −0.4)

P3

Fig. 2.7 Direct product P3 × C3

degG 1 ×G 2 (x1 , x2 ) = (number of vertices adjacent to x2 )degG 1 (x1 ). p

p

If for all the edges μ B2 ≤ μ B1 and μnB2 ≥ μnB1 , then degG 1 ×G 2 (x1 , x2 ) = (number of vertices adjacent to x1 )degG 2 (x2 ). p

p

Proof If membership values of the edges in G 1 and G 2 satisfy the relation: μ B1 ≤ μ B2 and μnB1 ≥ μnB2 then by Definition 2.1   p μ B1 ×B2 ((x1 , x2 )(y1 , y2 )), μnB1 ×B2 ((x1 , x2 )(y1 , y2 )) (y1 ,y2 )∈X 1 ×X 2    p μ B1 ×B2 ((x1 , x2 )(y1 , y2 )), μnB1 ×B2 ((x1 , x2 )(y1 , y2 )) = x 1 y1 ∈E 1 ,x 2 y2 ∈E 2    p p μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) =

degG 1 ×G 2 (x1 , x2 ) =



x 1 y1 ∈E 1 ,x 2 y2 ∈E 2

=



 x 1 y1 ∈E 1 ,x 2 y2 ∈E 2



p μ B1 (x1 y1 ), μnB1 (x1 y1 )

= (number of vertices adjacent to x2 )degG 1 (x1 ). p

p

If for all the edges μ B2 ≤ μ B1 and μnB2 ≥ μnB1 , then (2.16) takes the form

(2.16)

2.2 Distance in Bipolar Fuzzy Graphs

degG 1 ×G 2 (x1 , x2 ) =



93 p

x1 y1 ∈E 1 ,x2 y2 ∈E 2

(μ B2 (x1 y1 ), μnB2 (x1 y1 ))

= (number of vertices adjacent to x1 )degG 2 (x2 ). The proof is complete.



Theorem 2.7 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs such that the membership values of vertices and edges in G 1 and G 2 satisfy the p p p p p p following relations: μ A1 ≥ μ B2 , μnA1 ≤ μnB2 , μ A2 ≥ μ B1 , μnA2 ≤ μnB1 , μ B1 ≤ μ B2 and n n μ B1 ≥ μ B2 then the following conditions are satisfied: 1. If x2 = y2 , dG 1 G 2 ((x1 , x2 ), (y1 , y2 )) = dG 1 (x1 , y1 ). 2. If x1 = y1 and x2 = y2 or x1 = y1 and x2 = y2 . Then p p n n dG 1 G 2 ((x1 , x2 ), (y1 , y2 )) = (L (W ) ∧ L (T ), L (W ) ∨ L (T )), where (a) W is a bipolar fuzzy walk of smallest μ p −length and largest μn −length in G 1 × G 2 from (x1 , x2 ) to (y1 , y2 ) whose length in crisp strong product is the positive integer k such that k = dG ∗1 (x1 , y1 ) ∨ dG ∗2 (x2 , y2 ). p (b) L (T ) is the smallest μ p −length of any bipolar fuzzy walk T from x1 to y1 in G 1 such that the length of T ∗ is greater than k. n (c) L (T ) is the largest μn −length of any bipolar fuzzy walk T from x1 to y1 in G 1 such that the length of T ∗ is greater than k. Example 2.7 Consider the strong product of two bipolar fuzzy paths in Fig. 2.8, 1. dP3 P3 ((a, u), (c, u)) = (0.3, −0.7) = dP3 (a, c). 2. dP3 P3 ((a, u), (a, w)) = (0.4, −0.6) = L(W ), W : a − b − a. 3. dP3 P3 ((a, u), (c, w)) = (0.3, −0.7) = L(W ), here W = (a, u) − (b, v) − (c, w). It is clear that k = L(W ∗ ) = d P3 (a, c) ∨ d P3 (u, w).   4. dP3 P3 ((a, u), (b, w)) = (0.4, −0.9) = L 1 ( f P3 (W1 )), L 2 (W ) , where W = (a, u) − (b, v) − (b, w), d P3 (a, b) ∨ d P3 (u, w)=2 = L(W ∗ ) and similarly W1 =(a, u) − (b, v) − (c, w) − (b, w). Clearly, f P3 (W1 ) = a − b − c − b whose crisp length is 3 which is greater than k and L 1 ( f P3 (W1 )) < L 1 (W ). Theorem 2.8 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs such that the membership values of vertices and edges in G 1 and G 2 satisfy the p p p p p p following relations: μ A1 ≥ μ B2 , μnA1 ≤ μnB2 , μ A2 ≥ μ B1 , μnA2 ≤ μnB1 , μ B1 ≤ μ B2 and μnB1 ≥ μnB2 . If x1 ∈ X 1 and x2 ∈ X 2 , then degG 1 G 2 (x1 , x2 ) = degG 1 (x1 ) + degG 2 (x2 ) + r2 degG 1 (x1 ), where r2 is the number of vertices adjacent to x2 .

94

2 Distance Measures in Bipolar Fuzzy Graphs (0.3, −0.5) v(0.4, −0.6)

(0

−

(0 .2 ,

(0.2, −0.3)

,0 .3

,0 .1 )

3) 0.

,− .2

,− .1 (0

−

0. 3)

3 0.

(0.4, −0.6)

(0.3, −0.5)

(b, v)(0.4, −0.6) 0. 4)

4) 0.

(0

, .1

(0.3, −0.5)

(c, u)(0.3, −0.5)

−

0.

4)

(0

(0.4, −0.6) (c, v)(0.4, −0.6)

a(0.4, −0.6)

(0.4, −0.6)

)

(0

.1 ,−

(a, w)(0.4, −0.6)

.1 ,−

(0.1, −0.4)

, .2

(0.3, −0.5)

(0.1, −0.4)

(0

(0.1, −0.4)

(b, u)(0.3, −0.5)

(0.2, −0.3)

(0 .2

(a, v)(0.4, −0.6)

(0.2, −0.3)

(b, w)(0.5, −0.6)

(a, u)(0.3, −0.5)

P3

w(0.6, −0.7)

(0.2, −0.3)

u(0.3, −0.5)

(0.4, −0.6)

b(0.5, −0.6)

(0.1, −0.4)

0. 4)

(c, w)(0.6, −0.6)

c(0.6, −0.6) P3

Fig. 2.8 Strong product P3  P3

Proof By Definition 2.1,  p μ B1 B2 ((x1 , x2 )(y1 , y2 )), μnB1 B2 ((x1 , x2 )(y1 , y2 ))    p = μ B1 B2 ((x1 , x2 )(y1 , y2 )), μnB1 B2 ((x1 , x2 )(y1 , y2 )) x1 =y1 ,x2 y2 ∈E 2    p + μ B B ((x1 , x2 )(y1 , y2 )), μnB1 B2 ((x1 , x2 )(y1 , y2 )) 1 2 x1 y1 ∈E 1 ,x2 =y2    p + μ B1 B2 ((x1 , x2 )(y1 , y2 )), μnB1 B2 ((x1 , x2 )(y1 , y2 )) x1 y1 ∈E 1 ,x2 y2 ∈E 2    p p = μ A1 (x1 ) ∧ μ B2 (x2 y2 ), μnA1 (x1 ) ∨ μnB2 (x2 y2 ) x1 =y1 ,x2 y2 ∈E 2    p p + μ B1 (x1 y1 ) ∧ μ A2 (x2 ), μnB1 (x1 y1 ) ∨ μnA2 (x2 ) x1 y1 ∈E 1 ,x2 =y2    p p + μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) x1 y1 ∈E 1 ,x2 y2 ∈E 2       p p = μ B2 (x2 y2 ), μnB2 (x2 y2 ) + μ B1 (x1 y1 ), μnB1 (x1 y1 ) x2 y2 ∈E 2 x1 y1 ∈E 1    p + μ B1 (x1 y1 ), μnB1 (x1 y1 )

degG 1 G 2 (x1 , x2 ) =





(y1 ,y2 )∈X 1 ×X 2

x1 y1 ∈E 1 ,x2 y2 ∈E 2

= degG 2 (x2 ) + degG 1 (x1 ) + r2 degG 1 (x1 ).



2.2 Distance in Bipolar Fuzzy Graphs

95

Theorem 2.9 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs such that the membership values of vertices and edges in G 1 and G 2 follow the p p p p p p relations: μ A1 ≥ μ B2 , μnA1 ≤ μnB2 , μ A2 ≥ μ B1 , μnA2 ≤ μnB1 , μ B1 ≤ μ B2 and μnB1 ≥ n μ B2 . If x1 ∈ X 1 and x2 ∈ X 2 then the following conditions are satisfied: 1. If dG ∗1 (x1 , y1 ) = k, where k is even (or odd) and dG ∗2 (x2 , y2 ) is also even(or odd), p n then dG 1 •G 2 ((x1 , x2 ), (y1 , x2 )) = (L (W ), L (W )), where W is a bipolar fuzzy p n walk of smallest μ −length and largest μ −length in G 2 such that W ∗ is a walk of length k in G ∗2 . 2. If dG ∗1 (x1 , y1 ) = k, where k is even (or odd) and dG ∗2 (x2 , y2 ) is odd (or even), then p n dG 1 •G 2 ((x1 , x2 ), (y1 , x2 )) = (L (W ), L (W )), where W is a bipolar fuzzy walk p n of smallest μ −length and largest μ −length in G 2 such that W ∗ is a walk of length k + 1 in G ∗2 . Example 2.8 The lexicographic product of two bipolar fuzzy paths P3 and P3 is given in Fig. 2.9. In Fig. 2.9, 1. dP3 •P3 ((d, a), ( f, c)) = (0.4, −0.4) = L(W ), W = a − b − c. It is clear that L(W ∗ ) = d P3 (d, f ) because both d P3 (d, f ) and d P3 (a, c) are even. W = a − b − c − b. Here 2. dP3 •P3 ((d, a), ( f, b)) = (0.6, −0.5) = L(W ), L(W ∗ ) = d P3 (d, f ) + 1 because d P3 (d, f ) is even and d P3 (a, b) is odd. Theorem 2.10 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs such that the positive and negative degrees of membership of edges and vertices p p p p in G 1 and G 2 follow the relations: μ A1 ≥ μ B2 , μnA1 ≤ μnB2 , μ A2 ≥ μ B1 , μnA2 ≤ μnB1 , p p n n μ B1 ≤ μ B2 and μ B1 ≥ μ B2 . If x1 ∈ X 1 , and x2 ∈ X 2 , then degG 1 •G 2 (x1 , x2 ) = degG 2 (x2 ) + r2 degG 1 (x1 ), where r2 is the number of vertices adjacent to x2 . Proof By Definition 2.1, 

 p μ B1 •B2 ((x1 , x2 )(y1 , y2 )), μnB1 •B2 ((x1 , x2 )(y1 , y2 )) (y1 ,y2 )∈X 1 ×X 2    p μ B1 •B2 ((x1 , x2 )(y1 , y2 )), μnB1 •B2 ((x1 , x2 )(y1 , y2 )) = x1 =y1 ,x2 y2 ∈E 2    p μ B1 •B2 ((x1 , x2 )(y1 , y2 )), μnB1 •B2 ((x1 , x2 )(y1 , y2 )) + x1 y1 ∈E 1 ,x2 y2 ∈E 2    p p μ A1 (x1 ) ∧ μ B2 (x2 y2 ), μnA1 (x1 ) ∨ μnB2 (x2 y2 ) = x1 =y1 ,x2 y2 ∈E 2    p p μ B1 (x1 y1 ) ∧ μ B2 (x2 y2 ), μnB1 (x1 y1 ) ∨ μnB2 (x2 y2 ) + x1 y1 ∈E 1 ,x2 y2 ∈E 2       p p μ B2 (x2 y2 ), μnB2 (x2 y2 ) + μ B1 (x1 y1 ), μnB1 (x1 y1 ) =

degG 1 •G 2 ((x1 , x2 )) =



x2 y2 ∈E 2

x1 y1 ∈E 1 ,x2 y2 ∈E 2

= degG 2 (x2 ) + r2 degG 1 (x1 ).

The proof is complete.



96

2 Distance Measures in Bipolar Fuzzy Graphs (0.4, −0.5)

(0 −

(0.2, −0.3)

.2 ,

(e, a)(0.2, −0.3)

0. 3)

)

, .2 (0

−

3 0.

, .2 (0

(f, a)(0.2, −0.3) .2 ,

−

−

0. 3)

3) 0.

0.

1)

)

− 2, 0.

1 0.

(0

(

(d, c)(0.2, −0.4)

.2

,−

1) 0.

(0

(e, c)(0.2, −0.4)

.2 ,

−

(0.2, −0.1)

(e, b)(0.3, −0.4) .2 ,−

(0.2, −0.1)

(0

(0.2, −0.1)

(d, b)(0.3, −0.4)

(0.2, −0.3)

(0

f (0.6, −0.4)

a(0.2, −0.3)

(0.2, −0.3)

(f , b)(0.3, −0.4)

(d, a)(0.2, −0.3)

P3

(0.4, −0.5) e(0.4, −0.5)

(0.2, −0.3)

d(0.5, −0.8)

b(0.3, −0.4)

(0.2, −0.1)

0. 1)

c(0.2, −0.4)

(f, c)(0.2, −0.4)

P3

Fig. 2.9 Lexicographic product P3 ◦ P3

2.3 Metric in Bipolar Fuzzy Graphs In this section, the concept of metric, radius, diameter, and self-centered bipolar fuzzy graphs is discussed in detail. p

n

Theorem 2.11 For a bipolar fuzzy graph G = (A, B), d = (d , d ) : X × X → [0, 1] × [−1, 0] defines a metric on X with the following conditions: 1. 2. 3. 4.

p

n

d (x, y) ≥ 0 and d (x, y) ≤ 0, d(x, y) = (0, 0) ⇔ x = y, d(x, y) = d(y, x), p p p n n n d (x, z) ≤ d (x, y) + d (y, z) and d (x, z) ≥ d (x, y) + d (y, z) x, y, z ∈ X . p

n

for all

Proof It is clear from Definition 2.9 that L (x, y) ≥ 0 and L (x, y) ≤ 0 for all p n x, y ∈ X . Therefore, d (x, y) ≥ 0 and d (x, y) ≤ 0. p n If x = y, then d(x, y) = d(x, x) = (d (x, x), d (x, x)) = (0, 0). The inverse of any x − y bipolar fuzzy path is a y − x bipolar fuzzy path and vice versa of same distance. Let P1 , P2 be two x − y bipolar fuzzy paths and Q 1 , Q 2 be y − z bipolar fuzzy paths such that

2.3 Metric in Bipolar Fuzzy Graphs

97

p

n

p

n

p

n

d(x, y) = (d (x, y), d (x, y)) = (L (P1 ), L (P2 )), p

n

d(y, z) = (d (y, z), d (y, z)) = (L (Q 1 ), L (Q 2 )). P1 followed by Q 1 and P2 followed by Q 2 are x − z bipolar fuzzy walks, each of p which contains only one bipolar fuzzy path whose length cannot exceed d (x, y) + p n n p d (y, z) and d (x, y) + d (y, z), respectively. Thus, we can write d (x, z) ≤ p p n n n d (x, y) + d (y, z) and d (x, z) ≥ d (x, y) + d (y, z).  Definition 2.11 Let G = (A, B) be a bipolar fuzzy graph on X . The eccentricity of a vertex x in G is denoted by the ordered pair e(x) = (e p (x), en (x)) and defined as the distance to a vertex farthest from x, that is,  p n e(x) = max d (x, y), min d (x, y) . y∈X

y∈X

Definition 2.12 The radius of a bipolar fuzzy graph G, denoted by the ordered pair r (G) = (r p (G), r n (G)), is the minimum and maximum of all the positive and negative eccentricities of the vertices in G, respectively, that is,  r (G) = min e p (x), max en (x) . x∈X

x∈X

Definition 2.13 The diameter of a bipolar fuzzy graph G is defined as an ordered pair diam(G) = (diam p (G), diam n (G)), where diam p (G) and diam n (G) are the maximum and minimum values of all positive and negative eccentricities of the vertices in G, respectively, that is,  p n diam(G) = max e (x), min e (x) . x∈X

x∈X

Definition 2.14 A vertex y in a bipolar fuzzy graph G at a distance e(x) from x is called eccentric vertex of x. Definition 2.15 A vertex x in a bipolar fuzzy graph G is called a central vertex of G if e(x) = r (G). Definition 2.16 A bipolar fuzzy subgraph induced by the central vertices is known as center of a bipolar fuzzy graph G, denoted by C(G). If the center of G is G itself then G is called a self-centered bipolar fuzzy graph. Definition 2.17 A vertex x in a bipolar fuzzy graph G is called a peripheral vertex if e(x) = diam(G). Example 2.9 Consider a bipolar fuzzy graph as shown in Fig. 2.10. The distance between all the vertices is calculated as

98

2 Distance Measures in Bipolar Fuzzy Graphs a

(0.5, −0.2)

b

(0.9, −0.1)

c

(0

e

−0 .7)

(0.2, −0.6)

(0. 3,

(0.7, −0.1)

− .6 ,

)

0.3

d

Fig. 2.10 Radius and diameter of a bipolar fuzzy graph G

d(a, b) = (0.5, −0.2), d(a, c) = (0.8, −0.5), d(a, d) = (0.7, −0.2), d(a, e) = (0.9, −0.1), d(b, c) = (0.3, −0.7), d(b, d) = (0.2, −0.4), d(b, e) = (0.9, −0.3), d(c, d) = (0.5, −0.3), d(c, e) = (1.2, −0.4), d(d, e) = (0.7, 0.1, 0.1).

Therefore the eccentricities of the vertices using above distances are computed as follows: e(a) = (0.9, −0.5), e(b) = (0.9, −0.7), e(c) = (1.2, −0.7), e(d) = (0.7, −0.4), e(e) = (1.2, −0.4).

Hence r (G) = (0.7, −0.4), diam(G) = (1.2, −0.7), d is a central vertex and c is a peripheral vertex. Theorem 2.12 For any bipolar fuzzy graph G, radius and diameter satisfy the inequalities: r p (G) ≤ diam p (G) ≤ 2r p (G) and r n (G) ≥ diam n (G) ≥ 2r n (G). Proof By Definitions 2.12 and 2.13, we have r p (G) ≤ diam p (G) and r n (G) ≥ diam n (G). Let u, v be central vertices of G then e(u) = r (G) and e(v) = r (G). Let x be a peripheral vertex of G. Since d defines a metric, for some vertices y1 , y2 ∈ X , p p p diam p (G) = d (x, y1 ) ≤ d (x, u) + d (u, y1 ) ≤ 2r p (G), and n n n diam n (G) = d (x, y1 ) ≥ d (x, u) + d (u, y1 ) ≥ 2r n (G) for some u, v ∈ X.  The following theorem gives an absolute difference between the eccentricities of any two adjacent vertices in a bipolar fuzzy graph. Theorem 2.13 For any two adjacent vertices x and y in a bipolar fuzzy graph G,

2.3 Metric in Bipolar Fuzzy Graphs

99

|e(x) − e(y)| ≤ (1, 1). Proof Let G = (A, B) be a bipolar fuzzy graph on X . Without loss of generality, assume that e p (x) ≤ e p (y) and en (x) ≥ en (y). For any vertices z 1 , z 2 ∈ X , p

p

p

e p (y) = d (y, z 1 ) ≤ d (y, x) + d (x, z 1 ) ≤ 1 + e p (x). Hence − (e p (x) − en (y)) ≤ 1. Also

n

n

(2.17)

n

en (y) = d (y, z 2 ) ≥ d (y, x) + d (x, z 2 ) ≥ −1 + en (x). Thus en (x) − en (y) ≤ 1.

(2.18)

For any vertices u 1 , u 2 ∈ X p

p

p

e p (x) = d (x, u 1 ≤ d (x, y) + d (y, u 1 ) ≤ 1 + e p (y). So, e p (x) − en (y) ≤ 1.

(2.19)

Similarly, n

n

n

en (x) = d (x, u 2 ) ≥ d (x, y) + d (y, u 2 ) ≥ −1 + en (y). Hence, (en (x) − en (y)) ≤ 1.

(2.20)

Combining inequalities (2.17), (2.19), (2.18), and (2.20), the required result is obtained.  By assuming any two arbitrary vertices x and y in Theorem 2.13, we obtain the following result. Remark 2.2 For any two vertices x and y in a bipolar fuzzy graph G, p

n

|e p (x) − e p (y)| ≤ d (x, y) and |en (x) − en (y)| ≤ −d (x, y). Theorem 2.14 For any two adjacent vertices x and y in a bipolar fuzzy graph G |d(x, z) − d(y, z)| ≤ (1, 1) for every vertex z in G.

100

2 Distance Measures in Bipolar Fuzzy Graphs

Proof Let G be a bipolar fuzzy graph on X . Since d defines a metric on X , p

p

p

p

p

p

d (x, z) ≤ d (x, y) + d (y, z) ≤ 1 + d (y, z). Hence,

d (x, z) − d (y, z) ≤ 1. Similarly,

n

n

(2.21)

n

n

d (x, z) ≥ d (x, y) + d (y, z) ≥ −1 + d (y, z). Thus

n

n

− (d (x, z) − d (y, z)) ≤ 1. p

p

(2.22)

p

p

For the vertex y we have d (y, z) ≤ d (y, x) + d (x, z) ≤ 1 + d (x, z), which gives p p (2.23) − (d (x, z) − d (y, z)) ≤ 1. n

n

n

n

Analogously, d (y, z) ≥ d (y, x) + d (x, z) ≥ −1 + d (y, z), implies n

n

d (x, z) − d (y, z) ≤ 1.

(2.24)

The required result is clear from inequalities (2.21), (2.23), (2.22), and (2.24).



For any two arbitrary vertices x and y, Theorem 2.14 can be generalized in Remark 2.3. Remark 2.3 For any two vertices x and y in a bipolar fuzzy graph G, p

p

p

n

n

n

|d (x, z) − d (y, z)| ≤ d (x, y) and |d (x, z) − d (y, z)| ≤ −d (x, y). Theorem 2.15 If G is a self-centered bipolar fuzzy graph, then each vertex of G is an eccentric vertex. Proof Let y be an eccentric vertex of x, then e(x) = d(x, y). Since G is self-centered therefore, e(x) = e(y) = r (G). It follows that e(y) = d(x, y). It clearly shows that x is an eccentric vertex of y . Since x was taken to be arbitrary, the given statement is true for all vertices.  Remark 2.4 The converse of Theorem 2.15 is not true in general as it can be seen in Example 2.10. Example 2.10 Consider a bipolar fuzzy graph as shown in Fig. 2.11. From routine calculations, e(x) = (0.9, −0.5), e(y) = (0.9, −0.5), e(z) = (0.8, −0.5) e(w) = (0.8, −0.5), r (G) = (0.8, −0.5), diam(G) = (0.9, −0.5). All vertices of G are eccentric but G is not self-centered bipolar fuzzy graph because the center of G is shown in Fig. 2.12.

2.3 Metric in Bipolar Fuzzy Graphs

(0.4, −0.1)

(0 .7,

Fig. 2.12 Center of bipolar fuzzy graph G

y

(0.9, −0.6)

x

−0

.3)

, 0.5

−0

.4)

(

(0.6, −0.4)

Fig. 2.11 Eccentric vertices in bipolar fuzzy graph G

101

z

(0.8, −0.5)

w

z

(0.8, −0.5)

w

Theorem 2.16 Let G be a self-centered bipolar fuzzy graph. For every two vertices x, y ∈ X , if x ∈ Y ∗ then y ∈ X ∗ , where X ∗ is the set of all eccentric vertices of x and Y ∗ is the set of all eccentric vertices of y. Proof Since x is a eccentric vertex of y, e(x) = d(x, y) ⇒ x ∈ Y ∗ . It is given that G is a self-centered bipolar fuzzy graph so e(y) = e(x) = d(y, x), i.e., y is an eccentric  vertex of x. Hence y ∈ X ∗ . Remark 2.5 The converse of Theorem 2.16 is not true in general. In Example 2.10, x ∈ Y ∗ , y ∈ X ∗ , z ∈ W ∗ , and w ∈ Z ∗ but G is not self-centered. Theorem 2.17 Let G be a bipolar fuzzy graph then all peripheral vertices are eccentric vertices. Proof Let x be a peripheral vertex and y be its eccentric vertex, then diam(G) = e(x) = d(x, y) = d(y, x). It is only possible if diam(G) = d(y, x) = e(y). It shows that x is an eccentric vertex of y. Hence the proof.  Remark 2.6 The condition in Theorem 2.17 is not sufficient. In Example 2.10, z and w are eccentric vertices of each other but these are not peripheral vertices. p

Theorem 2.18 Let G be a complete bipolar fuzzy graph on X such that μ A (x1 ) ≤ p p p μ A (x2 ) ≤ μ A (x3 ) ≤ ... ≤ μ A (xn ) and μnA (x1 ) ≥ μnA (x2 ) ≥ μnA (x3 ) ≥ ... ≥ μnA (xn ), p then the distance between any two vertices xi and x j is d(xi , x j ) = μ B (xi x j )∧ p n n 2μ A (x1 ), μ B (xi x j ) ∨ 2μ A (x1 ) . Proof Let xi and x j be any two vertices of G then for some xk(1) , xk(2) ∈ X , the distance between xi and x j can be defined as p

n

d(xi , x j ) = (d (xi , x j ), d (xl , x j )),   p p p (1) (1) (2) (2) = min{μ B (xi x j ), μ B (xi xk ) + μ B (xk x j )}, max{μnB (xi x j ), μnB (xi xk ) + μnB (xk x j )} .

(2.25)

102

2 Distance Measures in Bipolar Fuzzy Graphs

Since G is a complete bipolar fuzzy graph, μ B (xi xk(1) ) = μ A (xi ) ∧ μ A (xk(1) ) and μnB (xi xk(1) ) = μnA (xi ) ∨ μnA (xk(1) ). Considering the conditions of membership values p p given in the statement, if we take xk(1) = xk(2) = x1 , then μ B (xi xk(1) ) = μ A (xk(1) ) = p (2) (2) n n n μ A (x1 ) and μ B (xi xk ) = μ A (xk ) = μ A (x1 ). Similarly, μ Bp (xk(1) x j ) = μ Ap (xk(1) ) = μ Ap (x1 ) and μnB (xk(2) x j ) = μnA (xk(2) ) = μnA (x1 ).  p p Equation (2.25) takes the form d(xi , x j ) = μ B (xi x j ) ∧ 2μ A (x1 ), μnB (xi x j ) ∨ 2μnA  (x1 ) .  p

p

p

Theorem 2.19 Let G = (A, B) be a complete bipartite bipolar fuzzy graph on X ∪ Y , where X = {x1 , x2 , . . . , xn } and Y = {y1 , y2 , . . . , yn  } such that p

p

p

p

μ A (x1 ) ≤ μ A (x2 ) ≤ μ A (x3 ) ≤ · · · ≤ μ A (xn ), μnA (x1 ) ≥ μnA (x2 ) ≥ μnA (x3 ) ≥ · · · ≥ μnA (xn ), p

p

p

p

μ A (y1 ) ≤ μ A (y2 ) ≤ μ A (y3 ) ≤ · · · ≤ μ A (yn  ), μnA (y1 ) ≥ μnA (y2 ) ≥ μnA (y3 ) ≥ · · · ≥ μnA (yn  ). p

p

p

p

If μ A (x1 ) ≤ μ A (y1 ), μnA (x1 ) ≥ μnA (y1 ), μ A (y1 ) ≤ μ A (x j ) and μnA (y1 ) ≥ μnA (x j ), for each 2 ≤ j ≤ n, then ⎧ ⎪ 2 A(x1 ), ⎪ ⎪ ⎪ ⎪ A(x1 ) + A(y1 ), ⎪ ⎪ ⎨ d(u, v) = 2 A(y1 ), ⎪ p p p ⎪ ⎪ μ B (uv) ∧ (2μ A (x1 ) + μ A (y1 )), ⎪ ⎪  ⎪ ⎪ ⎩ μn (uv) ∨ (2μn (x1 ) + μn (y1 )) B A A p

p

p

if u, v ∈ Y if u, v ∈ X, u = x1 or v = x1 if u, v ∈ X, u = x1 and v = x1 if u ∈ X and v ∈ Y. p

If μ A (y1 ) ≤ μ A (x1 ), μnA (y1 ) ≥ μnA (x1 ), μ A (x1 ) ≤ μ A (y j ) and μnA (x1 ) ≥ μnA (y j ),  for each 2 ≤ j ≤ n , then ⎧ ⎪ 2 A(y1 ), ⎪ ⎪ ⎪ ⎪ A(y1 ) + A(x1 ), ⎪ ⎪ ⎨ d(u, v) = 2 A(x1 ), ⎪ ⎪ ⎪ μ Bp (uv) ∧ (2μ Ap (y1 ) + μ Ap (x1 )), ⎪ ⎪  ⎪ ⎪ ⎩ μn (uv) ∨ (2μn (y1 ) + μn (x1 )) B A A p

p

if u, v ∈ X if u, v ∈ Y, u = y1 or v = y1 if u, v ∈ Y, u = y1 and v = y1 if u ∈ X and v ∈ Y. p

p

Proof Consider the case μ A (x1 ) ≤ μ A (y1 ), μnA (x1 ) ≥ μnA (y1 ), μ A (y1 ) ≤ μ A (x j ) and μnA (y1 ) ≥ μnA (x j ), for each 2 ≤ j ≤ n. Assume that u, v ∈ X are two arbitrary vertices of G then for some yk(1) , yk(2) ∈ X , the distance d(u, v) between u and v is given as   p n p p (1) p p (2) (1) (2) d(u, v) = (d (u, v), d (u, v)) = μ B (uyk ) + μ B (yk v), μ B (uyk ) + μ B (yk v) .

(2.26)

2.3 Metric in Bipolar Fuzzy Graphs

103

If u = x1 , then for yk(1) = yk(2) = y1 we have, μ B (uyk(1) ) = μ A (x1 ) and μnB (uyk(2) ) = p p p μnA (x1 ). Similarly, (μ B (yk(1) v), μnB (yk(2) v))=(μ A (yk(1) ), μnA (yk(2) ))=(μ A (y1 ), μnA (y1 )). Equation (2.26) takes the form as p

p

  p p d(u, v) = μ A (x1 ) + μ A (y1 ), μnA (x1 ) + μnA (y1 ) . If u, v = x1 , then for yk(1) = yk(1) = y1 , μ B (uyk(1) ) = μ A (yk(1) ) = μ A (y1 ) and p p p μnB (uyk(2) ) = μnA (yk(2) ) = μnA (y1 ). Similarly, μ B (yk(1) v) = μ A (yk(1) ) = μ A (y1 ) and μnB (yk(2) v) = μnA (yk(2) ) = μnA (y1 ). Equation (2.26) takes the form as d(u, v) = 2 A (y1 ). If u, v ∈ Y , there exist some xk(1) , xk(2) ∈ X such that p

p

n

p

(1)

p

p

(1)

p

(2)

(2)

d(u, v) = (d (u, v), d (u, v)) = (μ B (uxk ) + μ B (xk v), μnB (uxk ) + μnB (xk v)).

(2.27) p p (1) (2) n n (x ) ≤ μ (y ) and μ (x ) ≥ μ (y ), for x = x = x1 , Since μ 1 1 1 1 A A k k A A     p p (1) (2) (1) (2) n n μ B (uxk ), μ B (uxk ) = A(x1 ). Similarly, μ B (xk v), μ B (xk v) = A(x1 ). Equation (2.27) takes the form d(u, v) = 2 A(x1 ). If u ∈ X and v ∈ Y, then   p n d(u, v) = d (u, v), d (u, v)   p p p p = μ B (uv) ∧ (μ B (uy1 ) + μ B (y1 x1 ) + μ B (x1 v)), μnB (uv) ∧ (μnB (uy1 ) + μnB (y1 x1 ) + μnB (x1 v))   p p p = μ B (uv) ∧ (2μ A (x1 ) + μ A (y1 )), μnB (uv) ∧ (2μnA (x1 ) + μnA (y1 )) .



Similar argument can be used for other cases.

2.4 Self-centered Bipolar Fuzzy Graphs A notion of distance in bipolar fuzzy graphs is already given in Definition 2.9. In this section, self-centered bipolar fuzzy graphs are discussed with another notion of distance in Definition 2.18. Definition 2.18 Let G = (A, B) be a bipolar fuzzy graph. For any bipolar fuzzy path, P: y1 − y2 − ... − yn in G, the μ p −length of P is defined as the sum of reciprocals of μ p -values (positive membership values) of the edges and μn −length of P is determined as the sum of reciprocals of μn -values (negative membership values) of the edges, that is, p

l (P) =

n  i=2

 1 1 n , l (P) = . p n μ B (y i−1 y i ) μ B (y i−1 y i ) i=2 n

The length of bipolar fuzzy path P is represented by the ordered pair l(P) = p n (l (P), l (P)). For any two arbitrary vertices x, y ∈ X , let

104

2 Distance Measures in Bipolar Fuzzy Graphs 1 , − 31 ) d( 10

Fig. 2.13 Distance in bipolar fuzzy graph G

c( 15 , − 21 ) 1 ( 10 , − 51 )

1 , − 15 ) ( 10

1 ( 51 , − 10 )

)

1

1 , ( 10

5

−

1 ( 10 , − 51 )

b( 31 , − 21 )

1 a( 10 , − 13 )

P = {Pi | Pi is an x − y bipolar fuzzy path, i = 1, 2, 3, ...} be the set of all bipolar fuzzy paths from x to y. Then the distance from x to by δ(x,  y) or δG (x, y) and defined as the ordered pair δ(x, y) = y pis denoted n δ (x, y), δ (x, y) , where p

p

n

n

δ (x, y) = min{l (P j ) | P j ∈ P, j = 1, 2, 3, ...}, δ (x, y) = max{l (P j ) | P j ∈ P, j = 1, 2, 3, ...}.

Example 2.11 Consider a bipolar fuzzy connected graph G on the set X ={a, b, c, d} as shown in Fig. 2.13. By routine computations, it is easy to see that 1. a − d is a bipolar fuzzy path of length (10, −5), a − c − d is a bipolar fuzzy path of length (20, −10), a − b − c − d is a bipolar fuzzy path of length (25, −20). 2. The distances are calculated as p

p

p

p

n

n

p

p

δ (a, d) = 10, δ (a, b) = 10, δ (a, c) = 10, δ (b, c) = 5, δ (b, d) = 15, n

n

δ (c, d) = 10, δ (a, d) = −5, δ (a, b) = −5, δ (a, c) = −5, δ (b, c) = −10, n

n

δ (b, d) = −10, δ (c, d) = −5, δ(a, d) = (10, −5), δ(a, b) = (10, −5), δ(a, c) = (10, −5), δ(b, c) = (5, −10), δ(b, d) = (15, −10), δ(c, d) = (10, −5).

3. The eccentricity of each vertex is calculated as e(a) = (10, −5), e(b) = (15, −10), e(c) = (10, −10), e(d) = (15, −15). 4. The radius of G is (10, -5) and diameter of G is (15, -15). Example 2.12 Consider G = (A, B), a bipolar graph, on the set X = {a, b, c} as given in Fig. 2.14.

2.4 Self-centered Bipolar Fuzzy Graphs

105 c( 31 , − 16 )

Fig. 2.14 Self-centered bipolar fuzzy graph G

b( 71 , − 21 ) ( 71 , − 41 )

( 15 , − 14 )

1

1

(

,−

)

2

7

a( 15 , − 14 )

1. The distances between vertices are calculated as p

p

p

n

n

δ (a, b) = 7, δ (a, c) = 5, δ (b, c) = 7, δ (a, b) = −2, δ (a, c) = −4, n

δ (b, c) = −4, δ(a, b) = (7, −2), δ(a, c) = (5, −4), δ(b, c) = (7, −4). 2. The eccentricity of each vertex is (7, -4). 3. The radius of G is ( 7, -4). Hence G is a self-centered bipolar fuzzy graph (see Definition 2.16). Theorem 2.20 Every complete bipolar fuzzy graph G is a self-centered bipolar  fuzzy graph and r p (G) = μ p1(vi ) , r n (G) = μn 1(v ) , where μ p (vi ) and μn (vi ) are the i



least and greatest membership values, respectively, for some vertices vi and vi of G. Proof To prove that G is a self-centered bipolar fuzzy graph, it is to be shown that   every vertex of G is a central vertex. Let vi , vi ∈ X such that μ p (vi ) and μn (vi ) are the least and greatest membership values, respectively. Case 1: Consider all the vi − v j bipolar fuzzy paths P of length n in G, for all v j ∈ X. p 1. If n = 1 then μ p −length of P = l (P) = μ p1(vi ) . 2. If n > 1 then one of the edges of P possesses the positive membership value equal to μ p (vi ) and hence, μ p −length of a vi − v j bipolar fuzzy path will exceed p 1 . That is, l (P) > μ p1(vi ) . Hence, μ p (vi ) p

p

δ (vi , v j ) = min l (P) = P∈P

1 μ p (vi )

for all v j ∈ X.

(2.28)

Case 2: Let vk = vi ∈ X . Consider all vk − v j bipolar fuzzy paths Q of length n in G, for all v j ∈ X . p 1. If n = 1, then l (Q) = μ p (v1 v ) ≤ μ p1(vi ) . B

k

j

106

2 Distance Measures in Bipolar Fuzzy Graphs p

2. If n = 2, then l (Q) = p

3. If n > 2, then l (Q)

1 + μ p (v1 v ) p μ B (vk vk+1 ) B k+1 j ≤ μ pn(vi ) , since μ p (vi )

≤

2 , μ p (vi )

since μ p (vi ) is the least.

is the least. Thus,

p

p

δ (vk , v j ) = min l (Q) ≤ 1/μ p (vi ), for all vk , v j ∈ X. Q∈P

(2.29)

From (2.28) and (2.29), 1 , for all vi ∈ X. μ p (vi )

p

e p (vi ) = max δ (vi , v j ) = y j ∈X

(2.30)

N ow, r p (G) = min eμ+ (vi ) vi ∈X

1 , since by μ p (vi ) 1 . r p (G) = p μ (vi ) =



(2.34)



Choose a vertex vi ∈ X such that μn (vi ) is the greatest membership value of G. 

Case 1: Consider all the vi − v j bipolar fuzzy paths P of length n in G, for all v j ∈ X. n 1. If n = 1 then μn −length of P = l (P) = μn 1(v ) . i 2. If n > 1 then one of the edges of P possesses negative membership value equal n  to μ p (vi ) and hence l (P) > μn 1(v ) . Thus, i



p

1

n

δ (vi , v j ) = min l (P) = P∈P

 μn (vi )

, for all v j ∈ X.

(2.31)

Case 2: Let vk = vi ∈ X . Consider all vk − v j bipolar fuzzy paths paths Q of length n in G, for all v j ∈ X . n 1. If n = 1 then l (Q) = μn (v1k v j ) ≥ μn 1(v ) . B

n

2. If n = 2 l (Q) =

i

1

+

3. If n > 2 then l (Q) ≥

n  . μn (vi )

n

μnB (vk ,vk+1 )

n

1 μnB (vk+1 ,v j )

≥

2  , μn (vi )



since μn (vi ) is the greatest.

Thus, 

n

δ (vk , v j ) = min l (Q) ≥ 1/μn (vi ), for all vk , v j ∈ X. Q∈P

(2.32)

From (2.31) and (2.32), n

en (vi ) = min δ (vi , v j ) = v j ∈X

1  μn (vi )

, for all vi ∈ X.

(2.33)

2.4 Self-centered Bipolar Fuzzy Graphs

107 a( 15 , − 13 )

Fig. 2.15 Self-centered bipolar fuzzy graph

b( 51 , − 51 ) ( 61 , − 41 )

1 ) ( 15 , − 10

1 ( 51 , − 10 )

( 51 , − 51 ) d( 13 , − 61 )

c( 13 , − 16 )

N ow, r n (G) = min eμ− (vi ) vi ∈X

= r n (G) =

1  μn (vi )

1  μn (vi )

, since by (2.33) .

Equations (2.30) and (2.29) imply that every vertex of G is a central vertex. Hence G is a self-centered bipolar fuzzy graph.  The following example shows that the converse of Theorem 2.20 is not true in general. Example 2.13 Consider a bipolar fuzzy graph G on X = {a, b, c, d} as shown in Fig. 2.15. Clearly, G is a self-centered bipolar fuzzy graph but not complete. Lemma 2.2 A bipolar fuzzy graph G is a self-centered bipolar fuzzy graph if and only if r p (G) = diam p (G) and r n (G) = diam n (G). Definition 2.19 A bipolar fuzzy path cover of a bipolar fuzzy graph G is a set P of bipolar fuzzy paths such that every vertex of G is incident to some bipolar fuzzy path in P. If a bipolar fuzzy path between two vertices vi and v j belongs to P then it can be written as (vi , v j ) ∈ P. Definition 2.20 An edge cover of a bipolar fuzzy graph G is a set L of edges such that every vertex of G is incident to some edge in L. Definition 2.21 A maximal connected bipolar fuzzy subgraph of a bipolar fuzzy graph G is a bipolar fuzzy subgraph that is connected and not contained in any other connected bipolar fuzzy subgraph of G. The components of a disconnected bipolar fuzzy graph G are its maximal connected bipolar subgraphs.

108

2 Distance Measures in Bipolar Fuzzy Graphs

Theorem 2.21 Let G be a connected bipolar fuzzy graph with path covers P1 and P2 . The necessary and sufficient condition for a bipolar fuzzy graph to be self-centered is that p

δ (vi , v j ) = diam p (G), for all (vi , v j ) ∈ P1 n

δ (vi , v j ) = r (G), n

for all (vi , v j ) ∈ P2 .

(2.34) (2.35)

Proof Assume that G is a self-centered bipolar fuzzy graph. Suppose that (2.34) p and (2.35) do not hold then δ (vi , v j ) = diam p (G), for some (vi , v j ) ∈ P1 and n p δ (vi , v j ) = r n (G), for some (vi , v j ) ∈ P2 . Lemma 2.2 follows that δ (vi , v j ) = n r p (G), for some (vi , v j ) ∈ P1 and δ (vi , v j ) = r n (G), for some (vi , v j ) ∈ P2 . Then e p (vi ) = r p (G), en (vi ) = r n (G), for some vi ∈ X, which implies G is not a selfp centered bipolar fuzzy graph, a contradiction. Hence δ (vi , v j ) = diam p (G), for all n (vi , v j ) ∈ P1 and δ (vi , v j ) = r n (G), for all (vi , v j ) ∈ P2 . Conversely assume that (2.34) holds. It is to be shown that G is a self-centered p bipolar fuzzy graph. If (2.34) and (2.35) hold then e p (vi ) = δ (vi , v j ), for all p (vi , v j ) ∈ P1 and en (vi ) = δ (vi , v j ), for all (vi , v j ) ∈ P2 . It implies e p (vi ) = r p (G) and en (vi ) = r n (G), for all vi ∈ X . Hence G is a self-centered bipolar fuzzy graph.  Theorem 2.22 Let H = (A, B) be a connected self-centered bipolar fuzzy graph    on X then there exists a connected bipolar fuzzy graph G = (A , B ) on X such that C(G) is isomorphic to H . Also diam p (H ) = 2r p (G) and diam n (H ) = 2r n (G). Proof Given that H is a connected self-centered bipolar fuzzy graph. Let diam p (H ) = l and diam n (H ) = m. Then construct G from H as follows: p

1 Take two vertices vi , v j ∈ X with μ p (vi ) = μ A (v j ) = 1l , μnA (vi ) = μn (v j ) = 2m p p and join all the vertices of H to both vi and v j with μ B (vi vk ) = μ B (v j vk ) = 1l ,  p p 1 μnB (vi vk ) = μnB (v j vk ) = 2m for all vk ∈ X . Put μ A = μ A , μnA = μnA for all verp p tices in H and μ B = μ B  , μnB = μnB  for all edges in H . p

p

Claim: G is a bipolar fuzzy graph. First note that μ A (vi ) ≤ μ A (vk ), for all vk ∈ H . p p p If possible, let μ A (vi ) > μ A (vk ) for at least one vertex vk ∈ H . Then 1l > μ A (vk ), that is, l < μ p 1(v ) ≤ μ p (v1 v ) , where the last inequality holds for every vk ∈ X  , since A

k

B

k l

1 > l, for all vk ∈ H which contradicts p μ B (vk ,vl ) p p p p that diam (H ) = l. Therefore μ A (vi ) ≤ μ A (vk ) for all vk ∈ X  and μ B (vi vk ) ≤ p p p p p min(μ A (vi ), μ A (vk )) = 1l . Similarly, μ B (v j vk ) ≤ min(μ A (v j ), μ A (vk )) = 1l for all   vk ∈ X . Note that μnA (vi ) ≥ μnA (vk ), μnA (v j )≥ μnA (vk ), for all vk ∈ X , since p p p 1 n diam (H ) = m. Therefore μ B (vi vk ) ≥ max μ A (vi ), μ A (vk ) = 2m . Similarly, p p p 1 . Hence G is a bipolar fuzzy graph. μ B (v j vk ) ≥ max(μ A (v j ), μ A (vk )) = 2m  p Also, e (vk ) = l, for vk ∈ X and e p (vi ) = e p (v j ) = μ p (v1 v ) + μ p (v1 v ) = 2l, B i k B k l  r p (G) = l, diam p (G) = 2l. Next, en (vk ) = m for all vk ∈ X and en (vi ) = en (v j )

H is a bipolar fuzzy graph. That is,

2.4 Self-centered Bipolar Fuzzy Graphs

109



= μn (v1l vk ) = 2m for all vk ∈ X . Therefore, r n (G) = m, diam n (G) = 2m. Hence B C(G) is isomorphic to H .  p

Theorem 2.23 A bipolar fuzzy graph G is self-centered if and only if δ (vi , v j ) ≤ n r p (G) and δ (vi , v j ) ≥ r n (G) for all vi , v j ∈ X . Proof Assume that G is a self-centered bipolar fuzzy graph. That is, e p (vi ) = e p (v j ), en (vi ) = en (v j ), for all vi , v j ∈ X , r p (G) = e p (vi ), r n (G) = en (vi ), for all vi ∈ X. p n It is to be shown that δ (vi , v j ) ≤ r p (G), δ (vi , v j ) ≥ r n (G), for all vi , v j ∈ X . By Definition 2.11, p

n

δ (vi , v j ) ≤ e p (vi ) and δ (vi , v j ) ≥ en (vi ),

(2.36)

for all vi , v j ∈ X. This is possible only when e p (vi ) = e p (v j ), en (vi ) = en (v j ), for all vi , v j ∈ X. Since G is a self-centered bipolar fuzzy graph, (2.36) takes the form p n as δ (vi , v j ) ≤ r p (G), δ (vi , v j ) ≥ r n (G). p n Conversely, let δ (vi , v j ) ≤ r p (G), δ (vi , v j ) ≥ r n (G), for all vi , v j ∈ X . It is to be shown that G is a self-centered bipolar fuzzy graph. Suppose that G is not self-centered then e p (vi ) = r p (G) and en (vi ) = r n (G), for some vi ∈ X . Assume that e p (vi ) and en (vi ) are the least and greatest values among all other eccentricities, respectively. That is, r p (G) = e p (vi ), r n (G) = en (vi ),

(2.37)

where e p (vi ) < e p (v j ), en (vi ) < en (v j ) for some vi , v j ∈ X, and p

δ (vi , v j ) = e p (v j ) > e p (vi ),

(2.38)

δ (vi , v j ) = en (v j ) < en (vi ) for some vi , v j ∈ X.

(2.39)

n

p

n

Hence from (2.37), (2.38), and (2.39), δ (vi , v j ) > r p (G) and δ (vi , v j ) < r n (G), p n for some vi , v j ∈ X . A contradiction to the fact that δ (vi , v j ) ≤ r p (G), δ (vi , v j ) ≥  r n (G), for all vi , v j ∈ X . Hence G is a self-centered bipolar fuzzy graph. Definition 2.22 Let G = (A, B) be a bipolar fuzzy graph and x, y ∈ X be any two   distinct vertices of G. Let G = (A, B ) be a subgraph obtained by deleting the   edge x y from G, that is, B (x y) = (0, 0) and B = B for all other edges. Then x y + + is said to be a bipolar fuzzy bridge in G if either SG∞ (x, y) < SG∞ (x, y) and − − + + − − SG∞ (x, y) ≥ SG∞ (x, y) or SG∞ (x, y) ≤ SG∞ (x, y) and SG∞ (x, y) > SG∞ (x, y), for some x, y ∈ X. In other words, deleting an edge x y reduces the strength of connectedness between some pair of vertices. Thus, x y is a bipolar fuzzy bridge if and only if there exist u, v ∈ X such that x y is an edge of every strongest bipolar fuzzy path from u to v. Theorem 2.24 Let G be a bipolar fuzzy graph on X then the following conditions are satisfied.

110

2 Distance Measures in Bipolar Fuzzy Graphs

1. If B : X × X → [−1, 0] × [0, 1] is a constant function then G has no bipolar fuzzy bridge. 2. If B(x y) is not constant for all x y ∈ E, then G has at least one bipolar fuzzy bridge. Proof 1. Let B(vi v j ) = (c1 , c2 ), for all vi , v j ∈ X such that 0 ≤ c1 ≤ 1 and −1 ≤ c2 ≤ 0. Since each edge has the same membership value, deleting any edge does not reduce the strength of connectedness between any pair of vertices. Hence G has no bipolar fuzzy bridge. 2. Given that B(vi v j ) is not constant for all vi v j ∈ E. Choose an edge vx v y ∈ E such that p p μ B (vx v y ) = max{μ B (vi v j )|vi , v j ∈ X }, μnB (vx v y ) = min{μnB (vi v j )|vi , v j ∈ X }. p

p

There exists at least one edge vs vt distinct from vx v y such that μ B (vs vt ) < μ B (vx v y ) and μnB (vs vt ) > μnB (vx v y ). We claim that vx v y is a bipolar fuzzy bridge of G. For, if we delete the edge vx v y then the strength of connectedness between vx and v y   in the bipolar fuzzy subgraph G = (A, B ) thus obtained is decreased. In other + + − − words, SG∞ (x, y) < SG∞ (x, y) and SG∞ (x, y) > SG∞ (x, y). Hence by definition  of a bipolar fuzzy bridge, vx v y is a bipolar fuzzy bridge of G. Theorem 2.25 Let G = (A, B) be a bipolar fuzzy graph. If B : X × X → [−1, 0] × p [0, 1] is a constant function then an edge vi v j is a bipolar fuzzy bridge in G if μ B (vi v j ) n is maximum and μ B (vi v j ) is minimum in G. Definition 2.23 A bipolar fuzzy cycle C in a bipolar fuzzy graph G is called a strong bipolar fuzzy cycle if all edges of C are strong edges (see Definition 2.8). Theorem 2.26 If G is a bipartite bipolar fuzzy graph then it has no strong bipolar fuzzy cycle with odd number of vertices. Proof Let G be a bipartite bipolar fuzzy graph on X 1 ∪ X 2 . Suppose that it contains a strong bipolar fuzzy cycle say v1 , v2 , . . . , vn , v1 for some odd n. Without loss of generality, let v1 ∈ X 1 . Since (vi , vi+1 ) is strong for i = 1, 2, . . . , n − 1 and the vertices are alternatively in X 1 and X 2 , we have vn , v1 ∈ X 1 . It implies that vn v1 is an edge in X 1 , which contradicts the assumption that G is a bipartite bipolar fuzzy graph. Hence a bipartite bipolar fuzzy graph has no strong bipolar fuzzy cycle with odd number of vertices.  Theorem 2.27 If G = (A, B) is a complete bipolar fuzzy graph then there exists at + − p least one edge vi v j such that μ B (vi v j ) = SG∞ (vi , v j ) and μnB (vi v j ) = SG∞ (vi , v j ). p

Proof Consider a vertex vi whose positive membership value is μ A (vi ) and negative membership value is μnA (vi ). p

Case 1: Let μ A (vi ) be the least and μnA (v  i ) bep the greatest for the vertex vi ∈ p X . Let vi , v j ∈ X then μ B (vi v j ), μnB (vi v j ) = (μ A (vi ), μnA (vi )) and SG∞ (vi , v j ) =

2.4 Self-centered Bipolar Fuzzy Graphs

111

p

(μ A (vi ), μnA (vi )). The strength of all the edges which are incident with vertex vi is p (μ A (vi ), μnA (vi )) since G is a complete bipolar fuzzy graph. p Case 2: Let μ A (vi ) be the least and μnA (vk ) be the greatest, where vi = vk , then p p (μ B (vi vk ), μnB (vi vk )) = (μ A (vi ), μnA (vk )). Since it is a complete bipolar fuzzy graph, p there will be an edge between vi and vk such that SG∞ (vi , vk ) = (μ A (vi ), μnA (vk )). 

Theorem 2.28 Let G be a complete bipolar fuzzy graph with n vertices. Then G has a bipolar fuzzy bridge if and only if there exist an increasing sequence of real numbers p {t1 , t2 , . . . , tn−1 , tn } such that ti ∈ [0, 1] and ti = μ A (vi ), for every i = 1, 2, . . . , n and a decreasing sequence {s1 , s2 , . . . , sn−1 , sn } such that si ∈ [−1, 0] and si = μnA (vi ), for all i = 1, 2, . . . , n. Also the edge vn−1 vn is a bipolar fuzzy bridge in G. Proof Assume that G is a complete bipolar fuzzy graph and that G has a bipolar fuzzy p p p bridge vi v j . Without loss of generality, let tn−1 = μ A (vi ), tn = μ A (v j ) and μ A (vi ) ≤ p n n n n μ A (v j ), that is, tn−1 ≤ tn and, sn−1 = μ A (vi ), sn = μ A (v j ), μ A (vi ) ≥ μ A (v j ), that p p is, sn−1 ≥ sn . So μ B (vi v j ) = μ A (vi ), μnB (vi v j ) = μnA (vi ). p Assume on contrary that there is at least one vertex vi = vk such that μ A (vi ) ≤ p p p n μ A (vk ), that is, tn−1 ≤ tk , where k = n, tn−1 = μ A (vi ), tk = μ A (vk ), and μ A (vi ) ≥ μnA (vk ), sn−1 ≥ sk , where k = n, sn−1 = μnA (vi ), sk = μnA (vk ). Now consider the p p bipolar fuzzy cycle C : vi − v j − vk − vi in G such that μ B (vi v j ) =μ B (vi vk ) = p n n n μ A (vi ) and μ B (vi v j ) =μ B (vi vk ) = μ A (vi ).  p μ B (v j vk )

= 

μnB (v j vk )

=

p

p

p

p

p

p

μ A (v j ) if μ A (vi ) = μ A (v j ) or μ A (vi ) < μ A (v j ) ≤ μ A (vk ), p p p p μ A (vk ) if μ A (vi ) < μ A (vk ) ≤ μ A (v j ),

μnA (v j ) if μnA (vi ) = μnA (v j ) or μnA (vi ) > μnA (v j ) ≥ μnA (vk ), μnA (vk ) if μnA (vi ) > μnA (vk ) > μnA (v j ). p

In either case, it is seen that μ A (vi ) is the least and μnA (vi ) is the greatest among all other edges. But by Theorem 2.25, it is a contradiction to the fact that vi v j is a bipolar fuzzy bridge. p Conversely, let t1 < t2 < ... < tn−1 ≤ tn , ti = μ A (vi ) and s1 > s2 > ... > sn−2 > n sn−1 ≥ sn , si = μ1 (vi ). Claim: (vn−1 , vn ) is a bipolar fuzzy bridge of G. p p Now μ B (vn−1 vn ) = μ A (vn−1 ) and μnB (vn−1 , vn ) = μn1 (vn−1 ) and clearly by hypothp esis, all other edges of G will have μ p - strength strictly less than μ A (vn−1 ) and μn n strength strictly greater than μ A (vn−1 ). Hence the edge vn−1 vn is a bipolar fuzzy bridge by Theorem 2.25.  Theorem 2.29 Let G be a complete bipartite bipolar fuzzy graph then the complement of G is self-centered. Proof A bipolar fuzzy graph G = (A, B) on X 1 ∪ X 2 is said to be complete bipartite if

112

2 Distance Measures in Bipolar Fuzzy Graphs

1. B(x y) = (0, 0), if x, y ∈ X 1 or x, y ∈ X 2 , p p 2. B(x y) = (μ A (x) ∧ μ A (y), μnA (x) ∨ μnA (y)), if x ∈ X 1 and y ∈ X 2 or x ∈ X 2 and y ∈ X 1. By the above definition, the complement of G has two components and each component is a complete bipolar fuzzy graph, which are self-centered bipolar fuzzy graphs by Theorem 2.26.

2.5 Antipodal Bipolar Fuzzy Graphs In this section, the concept of antipodal bipolar fuzzy graphs is discussed with certain theoretical properties. Definition 2.24 Let G = (A, B) be a bipolar fuzzy graph. An antipodal bipolar fuzzy graph of G is defined as a pair A(G) = (C, F) such that 1. The bipolar fuzzy vertex set of G is taken as bipolar fuzzy vertex set of A(G), p p that is, μC (x) = μ A (x) and μCn (x) = μnA (x), for all x ∈ X . 2. If δ(x, y) = diam(G), then  p

μ F (x y) =  μnF (x y) =

p

if x and y are neighbors in G, μ B (x y), p p min(μ A (x), μ A (y)), if x and y are not neighbors in G, if x and y are neighbors in G, μnB (x y), max(μnA (x), μnA (y)), if x and y are not neighbors in G.

That is, two vertices in A(G) are made as neighborhood if the distance between them is diameter of G. Example 2.14 Consider a bipolar fuzzy graph G on X = {v1 , v2 , v3 } as shown p p p in Fig. 2.16. By routine calculations, δ (v1 , v2 ) = 6, δ (v1 , v3 ) = 3, δ (v2 , v3 ) = n n n p 3, δ (v1 , v2 ) = −5, δ (v1 , v3 ) = −4, δ (v2 , v3 ) = −2, e (v1 ) = 6, e p (v2 ) = 6, e p (v3 ) = 3, en (v1 ) = −5, en (v2 ) = −5, en (v3 ) = −4. Clearly, diam(G) = (6, −5), δ(v1 , v2 ) = (6, −5) = diam(G). Hence A(G) = (C, F), shown in Fig. 2.17, is an antipodal bipolar fuzzy graph of G. p

Theorem 2.30 Let G = (A, B) be a complete bipolar fuzzy graph where (μ A , μnA ) is constant function then G is isomorphic to A(G). p

Proof Given that G = (A, B) is a complete bipolar fuzzy graph with (μ A , μnA ) = (k1 , k2 ), where k1 and k2 are constants. It implies that δ(vi , v j ) = (l1 , l2 ), for all vi , v j ∈ X , where (l1 , l2 ) ∈ [0, 1] × [−1, 0]. Therefore, e(vi ) = (l1 , l2 ), for all vi ∈ X , that is, diam(G) = (l1 , l2 ). Hence δ(vi , v j ) = (l1 , l2 ) = diam(G), for all vi , v j ∈ X . Hence every pair of vertices are made as neighbors in A(G) such that

2.5 Antipodal Bipolar Fuzzy Graphs

113 v1 ( 31 , − 16 )

Fig. 2.16 Bipolar fuzzy graph G

(

,−

)

( 61 , − 51 )

1

1

4

3

( 13 , − 12 ) v3 ( 12 , − 15 )

v2 ( 31 , − 17 ) v1 ( 31 , − 61 )

Fig. 2.17 Antipodal bipolar fuzzy graph A(G)

( 61 , − 51 )

v3 ( 21 , − 51 )

v2 ( 31 , − 71 )

1. The bipolar fuzzy vertex set of G is taken as bipolar fuzzy vertex set of A(G), p p that is, μC (vi ) = μ A (vi ) and μCn (vi ) = μnA (vi ) for all vi ∈ X , p p 2. μ F (vi v j ) = μ B (vi v j ), since vi and v j are neighbors in G. μnF (vi v j ) = μnB (vi v j ), since vi and v j are neighbors in G. Hence G and A(G) have same number of vertices, edges and their degree of memberships are preserved. The degrees of vertices are also preserved. Hence G ∼ = A(G). Lemma 2.3 Let G = (A, B) be a connected bipolar fuzzy graph then the antipodal bipolar fuzzy graph of G is a spanning subgraph of G. Proof By Definition 2.24, A(G) contains all the vertices of G. That is, p

p

1. μC (x) = μ A (x) and μCn (x) = μnA (x) for all x ∈ X , and 2. if δ(x, y) = diam(G), then  p

μ F (x y) =

p

μ B (x y), if x and y are neighbors in G, p p min(μ A (x), μ A (y)), if x and y are not neighbors in G,

114

2 Distance Measures in Bipolar Fuzzy Graphs

 μnF (x y) =

if x and y are neighbors in G, μnB (x y), max(μnA (x), μnA (y)), if x and y are not neighbors in G. 

Hence A(G) is spanning subgraph of G.

Theorem 2.31 Let G = (A, B) be a bipolar fuzzy graph such that the crisp graph G ∗ is an even or odd cycle. If alternate edges have same membership values then G is self-centered bipolar fuzzy graph. Theorem 2.32 Let G be a bipolar fuzzy graph where crisp graph G ∗ is an even or odd cycle. If alternate edges have same positive and negative membership values then A(G) is the edge induced bipolar fuzzy subgraph of G c , whose end vertices (see Definition 1.42) are with maximum μ p −eccentricity and minimum μn −eccentricity in G. Proof If alternate edges have same positive and negative membership values then positive distance between nonadjacent vertices is greater than the adjacent vertices and negative distance between nonadjacent vertices is lesser than the adjacent verp p tices. Let μ B (vi v j ) be the least among all other edges then δ (vi , v j ) = μ p (v1 v ) . B

i

j

Claim (i): Neighbors in G are not neighbors in A(G). Consider an arbitrary bipolar fuzzy path connecting vk and vt such that / supp(B). vk vt ∈

(2.40)

If P is a bipolar fuzzy path of length 2 between vk and vt , then l p (P) ≥ p

Hence δ (vk , vt ) ≥ p

1 , p μ B (vi v j )

1

. p μ B (vi v j )

(2.41) p

by (2.40) and (2.41), which implies that δ (vi , v j ) < p

/ supp(G) and vi v j ∈ supp(G). That is, δ (vk , vt ) ≤ diam p (G), where δ (vk , vt ) ∈ p δ (vi , v j ) < diam p (G) if vi v j ∈ supp(B). Therefore, if vi v j ∈ supp(B) then vi and v j are not neighbors in A(G). Claim (ii): Edges in A(G) are edges in G c . p If vm vn ∈ supp(F), then by claim (i), vi v j ∈ / supp(B). So μ F (vm vn ) = p p min(μ A (vm ), μ A (vn )), by Definition 2.24, which implies that the edges in A p (G) are induced by the edges of G c whose end vertices are with maximum μ p −eccentricity in G. Consider the statement for negative membership values. Let vi v j ∈ supp(B) then n δ (vi , v j ) = k. Claim (i): Neighbors in G are not neighbors in A(G). Consider an arbitrary bipolar fuzzy path connecting vk and vt such that, / supp(B) vk vt ∈

(2.42)

2.5 Antipodal Bipolar Fuzzy Graphs

115

If Q is a bipolar fuzzy path of length 2 between vk and vt , then p

δ (P) ≤ k. n

n

(2.43) n

Hence δ (vk , vt ) ≤ δ (vi , v j ), by (2.42) and (2.43), which implies that δ (vk , vt ) ≥ n / supp(B) and vi v j ∈ supp(B). That is, δ (vi , v j ) ≥ diam n diam n (G) where vk vt ∈ (G), if (vi , v j ) ∈ supp(B). Therefore, if (vi , v j ) ∈ supp(B) then vi and v j are not neighbors in A(G). Claim (ii): Edges in A(G) are edges in G c . / supp(B). So μnF (vm vn ) = If vm vn ∈ supp(F), then by claim (i), vi v j ∈ n n max(μ A (vm ), μ A (vn )), by Definition 2.24, which implies that edges in An (G) are induced by the edges of G c whose end vertices are with minimum μn −eccentricity in G. Conclusively, the edges in A(G) are edges in G c . Hence A(G) is a bipolar fuzzy subgraph of G c induced by the edges of G c whose end vertices are with maximum μ p −eccentricity and minimum μn −eccentricity in G. Theorem 2.33 Let G be a bipolar fuzzy graph, where crisp graph G ∗ is an even or odd cycle. If alternate edges have same membership values then A(G) is a bipartite bipolar fuzzy graph. Theorem 2.34 Let G = (A, B) be a connected strong bipolar fuzzy graph where crisp graph G ∗ is an even or odd cycle such that A(vi ) = (k1 , k2 ), for all vi ∈ X . Then A(G) is the spanning bipolar fuzzy subgraph of G c induced by the edges of G c whose end vertices are with maximum μ p −eccentricity and minimum μn −eccentricity in G. p

Proof Let vi v j ∈ supp(B), δ (vi , v j ) = p

p

1 . k1

But for any vi v j ∈ / supp(B), δ(vk , vm )

≥ That is, δ (vi , v j ) = < ≤ δ (vk , vm ), where vi v j ∈ / supp(B). It follows that vi , v j are vertices in A(G) but these are not neighbors in A(G). Now, let vi v j ∈ n n n supp(B), δ (vi , v j ) > k11 . But for any vi v j ∈ / supp(B), δ (vi , v j ) ≥ δ (vk , vm ), / supp(B). It implies that vi , v j are vertices in A(G) but are not neighwhere vi v j ∈ bors in A(G). The remaining proof is similar to claim (ii) of Theorem 2.32.  2 . k1

1 k1

2 k1

Theorem 2.35 If G 1 and G 2 are isomorphic to each other then A(G 1 ) and A(G 2 ) are also isomorphic. Proof Let G = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 and, h : X 1 → X 2 be an isomorphism between G 1 and G 2 (see Definition 1.35). As G 1 and G 2 are isomorphic, membership degree of edges, length, and distance between corresponding vertices will be preserved. If the vertex vi has the maximum μ p −eccentricity and minimum μn −eccentricity in G 1 then h(vi ) has the maximum μ p −eccentricity and minimum μn −eccentricity in G 2 . So G 1 and G 2 will have the same diameter. If the distance between vi and v j is (k1 , k2 ) in G 1 , then h(vi ) and h(v j ) will also have their distance as (k1 , k2 ). The same mapping h itself is a bijection between A(G 1 ) and A(G 2 ) satisfying the isomorphism conditions.

116

1. 2. 3. 4. 5. 6.

2 Distance Measures in Bipolar Fuzzy Graphs p

p

p

p

μC1 (vi ) = μ A1 (vi ) = μ A2 (h(vi )) = μC2 (h(vi )), for all vi ∈ X 1 , μCn 1 (vi ) = μnA1 (vi ) = μnA2 (h(vi )) = μCn 2 (h(vi )), for all vi ∈ X 1 , p p μ F1 (vi v j ) = μ B1 (vi v j ), if vi and v j are neighbors in G 1 , p p p μ F1 (vi v j ) = min(μC1 (vi ), μC1 (v j )), if vi and v j are not neighbors in G 1 , μnF1 (vi v j ) = μnB1 (vi v j ), if vi and v j are neighbors in G 1 , μnF1 (vi v j ) = max(μCn 1 (vi ), μCn 1 (v j )), if vi and v j are not neighbors in G 1 .

As h : X 1 → X 2 is an isomorphism, μnF1 (vi v j ) = μnB2 (h(vi )h(v j )), if vi and v j are neighbors in G 1 , μnF1 (vi v j ) = max(μnA2 (vi ), μnA2 (v j )), if vi and v j are not neighbors in G 1 , μnF1 (vi v j ) = μnB2 (h(vi )h(v j )), if vi and v j are neighbors in G 1 , . μnF1 (vi v j ) = max(μnA2 (vi ), μnA2 (v j )), if vi and v j are not neighbors in G 1 . p

p

Hence μ F1 (vi v j ) = μ F2 (h(vi )h(v j )) and μnF1 (vi v j ) = μnF2 (h(vi )h(v j )). So the same  h is an isomorphism between A(G 1 ) and A(G 2 ). Theorem 2.36 If G 1 and G 2 are complete bipolar fuzzy graphs such that G 1 is co-weak isomorphic to G 2 then A(G 1 ) is co-weak isomorphic to A(G 2 ). Proof As G 1 = (A1 , B1 ) is co-weak isomorphic to G 2 = (A2 , B2 ), there exists a bijection h : X 1 → X 2 satisfying (see Definition 1.34), p

p

μ B (x1 y1 ) = μ B (h(x1 )h(y1 )), μnB1 (x1 y1 ) = μnB2 (h(x1 )h(y1 )), for all x1 , y1 ∈ X 1 . 1 2

If G 1 has n vertices v1 , v2 , . . . , vn then arrange the vertices of G 1 in such a way that p p p p μ A (v1 ) ≤ μ A (v2 ) ≤ μ A (v3 ) . . . μ A (vn ) and μnA (v1 ) ≥ μnA (v2 ) ≥ μnA (v3 ) . . . μnA (vn ). p As G 1 and G 2 are complete and co-weak isomorphic bipolar fuzzy graphs, μ B1 (vi v j ) p = μ B2 (h(vi )h(v j )), for all vi , v j ∈ X 1 . By Theorem 2.35 and Definition 2.24, p p A(G i ) contains all the vertices of G i , for i = 1, 2. That is, μCi (x) = μ Ai (x) and p p μCn i (x) = μnAi (x) for all x ∈ X i , i ∈ {1, 2}, and μ F1 (vk v j ) = μ F2 (h(vk )h(v j )), for  all vk , v j ∈ X 1 . So h is a co-weak isomorphism between A(G 1 ) and A(G 2 ). Remark 2.7 If G is a self-complementary bipolar fuzzy graph then its antipodal bipolar fuzzy graph may not be self- complementary as it can be seen in Example 2.15. Example 2.15 Consider a bipolar fuzzy graph G in Fig. 2.18. By routine calculations, δ(a, b) = (15, −4), δ(a, c) = (10, −2), δ(a, d) = (10, −2), δ(b, c) = (5, −2), δ(b, d) = (25, −6), δ(c, d) = (20, −4), e(a) = (15, −2), e(b) = (15, −2), e(c) = (20, −2), e(d) = (25, −2), diam(G) = (25, −2). Since diam(G) = δ(x, y) for all x, y ∈ X , A(G) is an antipodal bipolar fuzzy graph of G having same vertices as in G and no two vertices in A(G) are made as neighbors since the distance between them is not equal to the diameter of G (Fig. 2.19). Consider the complement G c of bipolar fuzzy graph G as given in Fig. 2.20.

2.5 Antipodal Bipolar Fuzzy Graphs Fig. 2.18 Bipolar fuzzy graph G

117 a( 18 , − 51 )

b( 51 , − 16 )

d( 18 , − 71 )

Fig. 2.19 Antipodal bipolar fuzzy graph A(G)

1 10

,−

2

1

)

( 51 , − 21 )

1 , − 12 ) ( 10

(

c( 31 , − 81 )

a( 81 , − 51 )

b( 15 , − 61 )

d( 18 , − 17 )

c( 31 , − 81 )

By routine calculations, δ(a, b) = (8, −5), δ(a, c) = (24, −18), δ(a, d) = (16, −11) δ(b, c) = (16, −13), δ(b, d) = (8, −6), δ(c, d) = (8, −7), e(a) = (24, −5), e(b) = (16, −5), e(c) = (24, −7), e(d) = (16, −6), d(G c ) = (24, −5). Since diam(G c ) = δ(x, y) for all x, y ∈ X in G c , A(G c ), in Fig. 2.21, is an antipodal bipolar fuzzy graph of G c having same vertices as in G c but no two vertices in A(G c ) are made as neighbors. Clearly, A(G) is not isomorphic to (A(G))c as shown in Fig. 2.22. Hence G is self-complementary but its antipodal bipolar fuzzy graph A(G) is not a selfcomplementary bipolar fuzzy graph. Example 2.16 Consider a bipolar fuzzy graph G in Fig. 2.23. By routine calculations, δ(a, b) = (5, −4), δ(a, c) = (11, −7), δ(a, d) = (17, −10), δ(b, c) = (6, −3), δ(b, d) = (12, −6), δ(c, d) = (6, −3), e(a) = (17, −4), e(b) = (12, −3), e(c) = (11, −3), e(d) = (17, −3), diam(G) = (17, −3). Since diam(G) = δ(x, y), for all x, y ∈ X . Hence A(G), in Fig. 2.24, is an antipodal bipolar fuzzy graph of G having same vertices as in G and no two vertices in A(G) are made as neighbors since the distance between them is not equal to the diameter of G.

118 Fig. 2.20 Complement G c

2 Distance Measures in Bipolar Fuzzy Graphs a( 18 , − 51 )

b( 51 , − 16 )

( 81 , − 51 )

1

1

(

,−

)

5

8

( 18 , − 17 )

Fig. 2.22 (A(G))c

c( 31 , − 81 )

a( 81 , − 51 )

b( 15 , − 61 )

d( 18 , − 17 )

c( 31 , − 81 )

a( 18 , − 15 )

b( 51 , − 61 ) (

8

1

,−

5

1

)

1

1

( d( 18 , − 71 )

( 81 , − 51 )

( 51 , − 61 )

( 81 , − 51 )

Fig. 2.21 Antipodal bipolar fuzzy graph A(G c )

d( 18 , − 71 )

,−

8

)

6

( 18 , − 17 ) c( 13 , − 18 )

2.5 Antipodal Bipolar Fuzzy Graphs b( 51 , − 14 )

c( 15 , − 51 )

( 51 , − 41 )

( 61 , − 31 )

( 51 , − 41 )

Fig. 2.23 Bipolar fuzzy graph G

119

d( 16 , − 31 )

a( 15 , − 41 )

Fig. 2.24 Antipodal bipolar fuzzy graph A(G)

Fig. 2.25 Complement G c

b( 15 , − 41 )

c( 51 , − 51 )

a( 51 , − 41 )

d( 16 , − 13 )

b( 51 , − 14 )

c( 15 , − 51 ) (

6

1

,−

3

1

)

1

1

( a( 15 , − 41 )

,−

8

)

6

( 16 , − 13 ) d( 16 , − 31 )

120 Fig. 2.26 Antipodal bipolar fuzzy graph A(G c )

Fig. 2.27 (A(G))c

2 Distance Measures in Bipolar Fuzzy Graphs b( 15 , − 41 )

c( 51 , − 51 )

a( 51 , − 41 )

d( 16 , − 13 )

b( 51 , − 41 )

c( 51 , − 15 ) (

6

1

,−

3

1

)

( 51 , − 41 )

( 61 , − 31 )

1

1

( a( 15 , − 41 )

( 51 , − 41 )

,−

8

)

6

( 16 , − 13 ) d( 16 , − 31 )

By routine calculations, δ(a, b) = (12, −6), δ(a, c) = (5, −4), δ(a, d) = (6, −3), δ(b, c) = (17, −10), δ(b, d) = (6, −3), δ(c, d) = (11, −7), e(a) = (12, −3), e(b) = (17, −3), e(c) = (17, −4), e(d) = (11, −3), d(G c ) = (17, −3). Since d(G c ) = δ(x, y), for all x, y ∈ X in G c . Hence A(G c ), in Fig. 2.26, is an antipodal bipolar fuzzy graph of G c , given in Fig. 2.25, having same vertices as in G c and no two vertices in A(G c ) are made as neighbors. Clearly A(G) is not isomorphic to (A(G))c as shown in Fig. 2.27, though A(G) is isomorphic to A(G c ). Hence G is a self-complementary bipolar fuzzy graph but A(G) is not a self-complementary bipolar fuzzy graph. Definition 2.25 Let G be a connected bipolar fuzzy The μ p -status of a vertex  graph. p p p vi is denoted by s (vi ) and defined as s (vi ) = v j ∈X δ (vi , v j ). The μn -status of  n a vertex vi is denoted by s n (vi ) and defined as s n (vi ) = v j ∈X δ (vi , v j ). The status p n of a vertex vi ∈ X is defined as s(G) = (s (G), s (G)). The minimum status of G is denoted by m[s(G)] and defined as m[s(G)] = (minvi ∈X s p (vi ), maxvi ∈X s n (vi )). The maximum status of G is denoted by M[s(G)] and defined as M[s(G)] = (maxvi ∈X s p (vi ), minvi ∈X s n (vi )). The total μ p -status of a bipolar fuzzy graph G

2.5 Antipodal Bipolar Fuzzy Graphs

121

 is denoted by t[s p (G)] and defined as t[s p (G)] = vi ∈X s p (vi ). The total μn status of a bipolar fuzzy graph G is denoted by t[s n (G)] and defined as t[s n (G)] =  n vi ∈X s (vi ). The total status of a bipolar fuzzy graph G is denoted by t[s(G)] and is defined as t[s(G)] = (t[s p (G)], t[s n (G)]). The median of a bipolar fuzzy graph G is denoted by M(G) and is defined as the set of vertices with minimum status. A bipolar fuzzy graph G is said to be self-median if all vertices have same status. In other words, G is self-median if and only if m[s(G)] = M[s(G)]. Example 2.17 Consider a bipolar fuzzy graph in Fig. 2.28. By routine calculap p p p p tions, δ (v1 , v2 ) = 3, δ (v1 , v3 ) = 6, δ (v1 , v4 ) = 3, δ (v2 , v3 ) = 3, δ (v2 , v4 ) = 6, p n n n n δ (v3 , v4 ) = 3, δ (v1 , v2 ) = −11, δ (v1 , v3 ) = −7, δ (v1 , v4 ) = −10, δ (v2 , v3 ) = n n p p p −10, δ (v2 , v4 ) = −7, δ (v3 , v4 ) = −11. s (v1 ) = 12, s (v2 ) = 12, s (v3 ) = 12, s p (v4 ) = 12, s n (v1 ) = −28, s n (v2 ) = −28, s n (v3 ) = −28, s n (v4 ) = −28. Therefore, s(v1 ) = (12, −28), s(v2 ) = (12, −28), s(v3 ) = (12, −28), s(v4 ) = (12, −28) and t[s(G)] = (48, −112). Here s(vi ) = (12, −28), for all vi ∈ X . Hence G is a self-median bipolar fuzzy graph. Theorem 2.37 Let G be a bipolar fuzzy graph where crisp graph G ∗ is an even cycle. If alternate edges have same membership values then G is a self-median bipolar fuzzy graph. Proof Let G = (A, B) be a bipolar fuzzy graph on X = {v1 , v2 , . . . , vn } such that the crisp graph G ∗ is an even cycle and alternate edges of G have same membership values. So δ(v1 , v2 ) = δ(v3 , v4 ) = δ(v1 , v2 ) = · · · = δ(vn−1 , vn ) and similarly, δ(v2 , v3 ) = δ(v4 , v5 ) = · · · = δ(vn , v1 ), δ(v1 , v3 ) = δ(v2 , v4 ) = δ(v3 , v5 ) = · · · = l and so on. Thus s p (vi ) = k and s n (vi ) = m, for all vi ∈ X . Hence G is a self-median bipolar fuzzy graph.  Remark 2.8 Let G be a bipolar fuzzy graph where crisp graph G ∗ is an odd cycle. If alternate edges have same membership values then G may not be self-median bipolar fuzzy graph.

v1 ( 21 , − 31 )

v2 ( 13 , − 14 ) ( 13 , − 31 )

( 31 , − 41 )

( 31 , − 41 )

Fig. 2.28 Self-median bipolar fuzzy graph

( 13 , − 31 ) v4 ( 12 , − 15 )

v3 ( 12 , − 31 )

122

2 Distance Measures in Bipolar Fuzzy Graphs

2.6 Applications of Bipolar Fuzzy Graphs In this section, the importance of bipolar fuzzy graphs is studied with example applications in decision support system and safe traveling.

2.6.1 Product Manufacturing A product can increase the profit of a company if it is sold in multiple areas. Before manufacturing a product, engineers and manufacturers test several important things in a product. Bipolar fuzzy graphs are used on a wide range in decision-making problems when it is necessary to make decisions with opposite agreements. Suppose a multinational enterprise (MNE) has to decide to manufacture a product among four products P1 , P2 , P3 , and P4 to market it in different countries. Every company considers various points before manufacturing a product. The most important thing is the demand for the product in the respective area. Here we consider the following two points in every product. 1. The product is appealing to people. 2. The product does not follow the mass market demands. We gather the above two points in a set as {Appealing, Not demanding}. Let the set of products be P = {P1 , P2 , P3 , P4 }. This phenomenon can be represented by a bipolar fuzzy graph G = (A, B) taking P as the set of vertices and A(P1 ) = (0.5, −0.8), A(P2 ) = (0.8, −0.2), A(P3 ) = (0.4, −0.6), A(P4 ) = (0.6, −0.8). The positive degree of membership of each product represents the percentage to which the product appeal the public and negative degree of membership shows the percentage that the product is not needed in the respective area. Take B(P1 P2 ) = (0.5, −0.5), B(P1 P3 ) = (0.1, −0.3), B(P1 P4 ) = (0.2, −0.3), B(P2 P3 ) = (0.3, −0.4), B(P2 P4 ) = (0.6, −0.2). The positive degree of membership of the edge between any two products shows that if these two products are manufactured then it would be beneficial for the company. The negative degree of membership depicts that it can harm company’s finance. The description of degree of membership of each edge can be gathered in a set as {Profit, Loss}. This means that the manufacturing of P1 and P2 can give 50% profit and 50% loss. It can be easily verified that it is a bipolar fuzzy graph as shown in Fig. 2.29. By observation, it is easy to see that the production of P2 has more demand and attraction. Also, its manufacturing with other products can be beneficial.

2.6.2 Safe Route and Shortest Path Problem Graphs are used as a common source to model communication networks such as transportation, calculating traffic density, intensity of accidents on various roads, or

2.6 Applications of Bipolar Fuzzy Graphs

123 P1 (0.5, −0.8)

Fig. 2.29 Bipolar fuzzy model of product manufacturing

P2 (0.8, −0.2)

(0.5, −0.5)

.3,

−0

.4)

P3 (0.4, −0.6)

.6 (0

0 ,−

.2) (0.6, −0.2)

(0.1, −0.3)

(0

P4 (0.6, −0.8)

to find the shortest paths between any two points of the network. Bipolar fuzzy graphs can be used to find the shortest paths when it is required to consider two opposite agreements. An agency wants to deliver a secret envelope from Astana Kazakhstan to British Columbia through delivery. The delivery car can only travel 4000 kilometers before refilling the tank. The agency requires to deliver the envelope with safety. From Astana to Columbia, there are 17 fuel stations, namely, Moscow Russia, London England, Barcelona Spain, Algiers Algeria, San Juan, Boston, Houston Texas, Helena Montana, Beijing China, Delhi India, Tokyo, Singapore, Australia, Honolulu, Alaska, and California. The agency demands to consider the following points during travel. 1. Refill the tank at a station having less danger of robbery. 2. Take into account the distance traveled (in kilometers) between two fuel stations. 3. Use a path which is safe and confidential. We represent the fuel stations by vertices. The negative membership value of each vertex represents the degree of safety at the fuel station and positive membership value shows the danger at the node. The degree of membership of edge between two stations represents the degree of danger and safety. As we are dealing with two opposite agreements, that is, danger and safety, this situation can be represented by a bipolar fuzzy graph in Fig. 2.30. Since in calculating the distance there is no use of membership value of vertices, we do not write these values in the bipolar fuzzy graph. It is necessary to travel through the bipolar fuzzy path with the least positive degree of membership and maximum negative degree of membership. For this, calculate the length of all bipolar fuzzy paths from Astana to British Columbia and travel through a bipolar fuzzy path with minimum danger. By routine calculations, it can be easily seen that the minimum danger between Astana and British Columbia is through the path Astana–Moscow–Barcelona–Boston–British Columbia. The same procedure can be used to find the shortest path between any two points on a bipolar fuzzy graph if the degrees of memberships are given in terms of distance traveled, traffic density, etc.

124

2 Distance Measures in Bipolar Fuzzy Graphs

(0.4, −0.45)

London, England

.5 ,−

Delhi,

0.

6)

India Brisbane, Australia

) 55 0.

(0

.7)

,− .6

(

, −0

(0

0.

Russsia

) 45

(0.6

(0.65, −0.45)

Moscow,

− 4, 0.

Astana, Kazakhstan

.45) , −0 (0.5

Beijing,

(0.65, −0.

China

55)

Singapore

−

0.7 5)

, .4 (0

(0. 8, −

.75) , −0 (0.8

5 0.8

,−

Tokyo, Japan

) 0.9

(0.8, −0.75)

5) 0.6

.5)

Anchorage, Alaska

,− ) 0.6

−0

0.5 )

. 3, (0

(0. 5, −

)

Helana, Montana

Houston, Taxas

California

0. 4

−0

.6 (0

. 5, (0

0.5)

7, − (0.

Boston, Massachuset

5, − (0.5

(

San Jaun, Puetro Rico

Honolulu

(0 .4 ,−

(0.7, −0.65)

6) 0.

)

. (0

,− 65

45 0.

Algiers, Algeria

Barcelona, Spain

.3)

(0.6, −0. 7)

British, Columbia

, (0.5

−0.4

5)

Fig. 2.30 Bipolar fuzzy model of shortest path problem

2.7 Conclusions The notion of distance function in bipolar fuzzy graphs plays a significant role to solve shortest path problems in communication networks in the presence of various conflicting criteria. In this chapter, we have illustrated different types of distance functions in bipolar fuzzy graphs and established formulae of degree and distance of higher order bipolar fuzzy graphs including complete bipolar fuzzy graphs, complete bipartite bipolar fuzzy graphs, and products of bipolar fuzzy graphs. Using eccentric vertices, peripheral vertices, and central vertices, we have described certain properties and characterizations of self-centered bipolar fuzzy graphs. We have studied relations of self-median, antipodal, and self-centered bipolar fuzzy graphs when the bipolar fuzzy graph is complete or strong. We have described the notions of bipolar fuzzy path cover and edge cover of a bipolar fuzzy graph and presented a necessary and sufficient condition for a complete bipolar fuzzy graph to have a bipolar fuzzy bridge. We have also discussed the importance of the distance function of bipolar fuzzy graphs with real-world applications in traveling and product manufacturing.

Exercises 2

125

Exercises 2 − → − → 1. Consider two bipolar fuzzy digraphs G 1 and G 2 as shown in Fig. 2.31. → ((u, a), (v, a)), d− → ((w, d), (w, c)) and → − → − Calculate the values of d− G 1 G 2 G 1 G 2 → − → ((u, d), (v, c)). d− G 1× G 2 2. If C2k+1 = (A, B) is an odd bipolar fuzzy cycle on X with A(x) = (0.5, −1) and B(x y) = (0.2, −0.5), for all x, y ∈ X . Determine whether C2k+1 × C2k+1 and C2k+1 C2k+1 are isomorphic or not? What happens if we replace odd bipolar fuzzy cycles by even bipolar fuzzy cycles? 3. Let G = (A, B) be a bipolar fuzzy graph, where A and B are constant functions. Calculate the value of dGG ((x, u), (y, v)), for all x, y, u, v ∈ X . 4. Let G 1 and G 2 be two bipolar fuzzy graphs, then show that G 1 G 2 is connected if and only if G 1 and G 2 are both connected. 5. Let G 1 , G 2 , and G 3 be three bipolar fuzzy graphs, then prove or disprove that G 1 × (G 2 + G 3 ) = G 1 × G 2 + G 1 × G 3 . 6. Determine whether or not the Cartesian product of antipodal bipolar fuzzy graphs is antipodal. n −  → G i is the Cartesian product or strong product of bipolar fuzzy 7. If G = i=1

− → − → digraphs G i , then show that G is connected is and only if each factor G i is connected. → − → − → − 8. Prove that the lexicographic product G = G 1 • G 2 • . . . • G n of nontrivial − → bipolar fuzzy digraphs is connected if and only if each factor G i is connected. p

p

k

p

p

9. Let (α1 , α1n ), (α2 , α2n ), . . . , (αk , αkn ) ∈ [0, 1] × [−1, 0], then max(αi , αin ) = i=1

k

p

k

(max αi , min αin ). Determine whether or not the diameter of strong product of i=1

i=1

k bipolar fuzzy graphs is the maximum of the diameters of the factors. p

p

k

p

p

10. Let (α1 , α1n ), (α2 , α2n ), . . . , (αk , αkn ) ∈ [0, 1] × [−1, 0], then min(αi , αin ) = i=1

k

p

k

(min αi , max αin ). Determine whether or not the radius of strong product of i=1

i=1

k bipolar fuzzy graphs is the minimum of the radii of the factors.

Fig. 2.31 Bipolar fuzzy graphs

126

2 Distance Measures in Bipolar Fuzzy Graphs

References 1. Akram, M.: Bipolar fuzzy graphs. Inf. Sci. 181(24), 5548–5564 (2011) 2. Akram, M.: Bipolar fuzzy graphs with applications. Knowl.-Based Syst. 39, 1–8 (2013) 3. Akram, M., Dudek, W.A.: Regular bipolar fuzzy graphs. Neural Comput. Appl. 21(supp 1), 197–205 (2012) 4. Akram, M., Karunambigai, M.G.: Metric in bipolar fuzzy graphs. World Appl. Sci. J. 14(12), 1920–1927 (2011) 5. Akram, M., Li, S.-G., Shum, K.P.: Antipodal bipolar fuzzy graphs. Ital. J. Pure Appl. Math. 31, 97–110 (2013) 6. Akram, M., Yousaf, M.M., Dudek, W.A.: Self centered interval-valued fuzzy graphs. Afr. Math. 26(5–6), 887–898 (2015) 7. Bhattacharya, P.: Some remarks on fuzzy graphs. Pattern Recogn. Lett. 6(5), 297–302 (1987) 8. Bhutani, K.R., Rosenfeld, A.: Strong arcs in fuzzy graphs. Inf. Sci. 152, 319–322 (2003) 9. Bouttier, J., Di Francesco, P., Guitter, E.: Geodesic distance in planar graphs. Nucl. Phys. B 663(3), 535–567 (2003) 10. Bhutani, K.R., Rosenfeld, A.: Geodesics in fuzzy graphs. Electron. Notes Discret. Math. 15, 49–52 (2003) 11. Bhutani, K.R., Rosenfeld, A.: Fuzzy end nodes in fuzzy graphs. Inf. Sci. 152, 323–326 (2003) 12. Dudek, W.A., Talebi, A.A.: Operations on level graphs of bipolar fuzzy graphs, Buletinul Academiei de Stiinte a Republici Moldova. Mathematica. 81(2), 107–126 (2016) 13. Hammack, R., Imrich, W., Kalvˇz ar, S.: Handbook of Product Graphs. Discrete Mathematics and its Applications, pp. 49–60. CRC Press, Taylor & Francis Group (2011) 14. Linda, J.P., Sunitha, M.S.: On g−eccentric nodes g−boundary nodes and g−interior nodes of a fuzzy graph. Int. J. Math. Sci. Appl. 2(2), 697–707 (2012) 15. Rosenfeld, A.: Fuzzy graphs. In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds.) Fuzzy Sets and Their Applications, pp. 77–95. Academic Press, New York (1975) 16. Sameena, K., Sunitha, M.S.: Characterisation of g−self centered fuzzy graphs. J. Fuzzy Math. 16, 787–791 (2008) 17. Sameena, K., Sunitha, M.S.: On g-distance in fuzzy trees. J. Fuzzy Math. 19, 787–791 (2011) 18. Sarwar, M., Akram, M.: Representation of graphs using m-polar fuzzy environment. Ital. J. Pure Appl. Math. 38, 291–312 (2017) 19. Sunitha, M.S., Vijayakumar, A.: Some metric aspects of fuzzy graphs. In: Proceedings of the Conference on Graph Connections. Cochin University of Science and Technology, Cochin, pp. 111–114 (1998) 20. Tom, M., Sunitha, M.: Strong sum distance in fuzzy graphs. SpringerPlus 4(1), 1–14 (2015) 21. Tom, M., Sunitha, M.: Sum distance in fuzzy graphs. Ann. Pure Appl. Math. 7(2), 73–89 (2014) 22. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 23. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 24. Zhang, W.-R., Zhang, L.: YinYang bipolar logic and bipolar fuzzy logic. Inf. Sci. 165(3–4), 265–287 (2004) 25. Zhang, W.-R.: YinYang bipolar relativity: A unifying theory of nature, agents and causality with applications in quantum computing, cognitive informatics and life sciences. Information Sciences Reference, Hershey (2011) 26. Zhang, W.-R.: NPN fuzzy sets and NPN qualitative algebra: A computational framework for bipolar cognitive modeling and multiagent decision analysis. IEEE Trans. Syst., Man, Cybern., Part B (Cybern.) 26(4), 561–574 (1996) 27. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998)

Chapter 3

Special Types of Bipolar Fuzzy Graphs

In this chapter, we discuss the concept of irregularity in bipolar fuzzy graphs and present isomorphism properties of regular, m-totally regular, neighborly irregular, totally irregular, highly irregular and neighborly totally irregular bipolar fuzzy graphs. We present certain characterizations under which, regular and totally regular bipolar fuzzy graphs, highly irregular and neighborly irregular bipolar fuzzy graphs are equivalent. We discuss certain formulae of order and size of m−totally regular bipolar fuzzy graphs. We study the concept of bipolar fuzzy line graphs and establish a necessary and sufficient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph. Using algebraic structures, we discuss the notions of regularity, vertex transitivity, and connectedness in Cayley bipolar fuzzy graphs. We describe the relations of connected, weakly connected, semiconnected, locally connected, and quasi-connected Cayley bipolar fuzzy graphs. The work in this chapter is from [1, 2, 9].

3.1 Introduction Graph theory was first initiated by Euler during the course of finding a solution to the problem of “Konigsberg ¨ bridge” in 1736. Graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network. Graphs can be classified on the basis of number of vertices, number of edges, order, size, connectivity, distance, degree, and their overall structure. Alavi [8] investigated several properties concerning the existence and enumeration of highly irregular graphs and extended this concept to k−path irregular graphs [7]. Most of the real-world problems ranging from engineering to medical and medical to social fields involve uncertainty in data. Fuzzy graphs occur when the uncertainty © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_3

127

128

3 Special Types of Bipolar Fuzzy Graphs

is observed in real-life situations. Structures of both crisp graphs and fuzzy graphs are similar but fuzzy graphs deal with uncertainty on vertices and/or edges. Several graph-theoretic concepts like paths, connectedness, bridges, cycles, and trees were obtained by Rosenfeld [16]. Nagoorgani and Radha [15] introduced regular fuzzy graphs, total Degree, and totally regular fuzzy graphs. Nagoorgani and Latha [13] initiated the concept of irregular fuzzy graphs, neighborly irregular fuzzy graphs, and highly irregular fuzzy graphs, and discussed isomorphism properties of these fuzzy graphs in [13]. An antipodal graph A(G) of a graph G, introduced by Smith [19], is a graph with the same set of vertices as of G, two vertices being adjacent if the distance between them is equal to the diameter of G. As a fuzzy analog to this, Nagoorgani [14] discussed the nature of antipodal fuzzy graph when the given fuzzy graph is complete or strong. In many everyday situations, a single-valued membership function fails to express the support evidence and counter-evidence simultaneously. In several practical situations, an appropriate level of information of the problem domain may not exist due to some uncertainty over the object consideration, which shows the characteristic of satisfaction and dissatisfaction. Zhang [20] extended fuzzy set theory to bipolar fuzzy sets and discussed the bipolar behavior of objects. The concept of bipolar fuzzy set can be observed as another approach, where existing information is not sufficient to define the uncertainty by using a fuzzy set. A fuzzy set gives the degree of membership of an element in a given set, while bipolar fuzzy set gives both a positive degree of membership and a negative degree of membership which are independent of each other; the only requirement is that one is opposite to the other. Bipolar fuzzy set can describe two-sided fuzzy characteristics of the objects more comprehensively, thus a more powerful and effective tool in dealing with fuzzy information. The concept of regular and totally regular bipolar fuzzy graphs was introduced by Akram and Dudek [2]. The same author Akram in collaboration with other researchers studied various types of irregular bipolar fuzzy graphs and bipolar fuzzy influence graphs in [1], antipodal and self-median bipolar fuzzy graphs in [5], and self-centered bipolar fuzzy graphs in [4]. Samanta and Pal [18] introduced the notion of bipolar fuzzy intersection graphs and proved that any bipolar fuzzy graph can be expressed as a bipolar fuzzy intersection graph of some bipolar fuzzy sets.

3.2 Regular and Irregular Bipolar Fuzzy Graphs The degree of a vertex x in a graph G is the number of edges incident to x. If each vertex in a graph G has the same degree then it is called a regular graph otherwise it is an irregular graph. In this section, different types of irregular bipolar fuzzy graphs are studied with certain interesting properties. Definition 3.1 The degree of a vertex x in a bipolar  fuzzy graph G is defined as a p p n p n (x), deg (x)), where deg (x) = pair deg(x) = (deg y∈X μ B (x y) and deg (x) =  n y∈X μ B (x y).

3.2 Regular and Irregular Bipolar Fuzzy Graphs

129

Table 3.1 Bipolar fuzzy set A on {a, b, c, d} A a b p μA μnA

0.5 −0.3

0.4 −0.2

Table 3.2 Bipolar fuzzy relation B in {a, b, c, d} B a b a b c d

(0.2, −0.1) (0, 0) (0.2, −0.1) (0, 0)

(0, 0) (0, 0) (0, 0) (0.4, −0.1)

c

d

0.7 −0.3

0.5 −0.5

c

d

(0, 0) (0.2, −0.1) (0, 0) (0.4, −0.1)

(0.4, −0.1) (0, 0) (0.4, −0.1) (0, 0)

a(0.5, −0.3)

Fig. 3.1 Regular bipolar fuzzy graph

b(0.4, −0.2)

(0.2, −0.1)

(0.4, −0.1)

(0.4, −0.1) (0.2, −0.1) d(0.5, −0.5)

c(0.7, −0.3)

Definition 3.2 Let G = (A, B) be a bipolar fuzzy graph on X . If all vertices of G have same degree then G is called a regular bipolar fuzzy graph , that is, deg(x) = deg(y), for all x, y ∈ X . Example 3.1 Let A be a bipolar fuzzy set on X = {a, b, c, d}, given in Table 3.1, and B be a bipolar fuzzy relation in X defined in Table 3.2. Routine computations show that the bipolar fuzzy graph G, in Fig. 3.1, is regular. Definition 3.3 The total degree of a vertex x in a bipolar fuzzy graph G is denoted by T deg(x) = (T deg p (x), T degn (x)) and defined as T deg p (x) =



p

p

p

μ B (x y) + μ A (x) = deg p (x) + μ A (x),

x y∈E

T degn (x) =



μnB (x y) + μnA (x) = degn (x) + μnA (x).

x y∈E

If each vertex of G has same total degree, say m = (m p , m n ), then G is called an m-totally regular bipolar fuzzy graph.

130

3 Special Types of Bipolar Fuzzy Graphs

Table 3.3 Bipolar fuzzy set on {v1 , v2 , v3 } A v1 p μA μnA

0.4 −0.1

v2

v3

0.8 −0.7

0.7 −0.6

Table 3.4 Bipolar fuzzy relation in {v1 , v2 , v3 } B v1 v2 v1 v2 v3

(0, 0) (0.3, −0.5) (0.4, −0.6)

(0.4, −0.6) (0, 0) (0, 0)

v1 (0.4, −0.1)

v2 (0.8, −0.7)

(0.3, −0.5) (0.4, −0.6)

Fig. 3.2 Totally regular bipolar fuzzy graph

v3

(0.3, −0.5) (0, 0) (0, 0)

v3 (0.7, −0.6) v1 (0.4, −0.1)

v2 (0.4, −0.1)

(0.3, −0.2) (0.3, −0.2)

Fig. 3.3 Regular and totally regular bipolar fuzzy graph G

2) 0. − , .3 (0

v3 (0.4, −0.1)

Example 3.2 Let A be a bipolar fuzzy set on X = {v1 , v2 , v3 } and B be a bipolar fuzzy relation in X as given in Table 3.3 and 3.4, respectively. Routine computations show that the bipolar fuzzy graph G, shown in Fig. 3.2, is totally regular but not regular. Example 3.3 Consider a bipolar fuzzy graph G on X = {v1 , v2 , v3 } as given in Fig. 3.3. It is easy to check that G is both regular and totally regular bipolar fuzzy graph.

3.2 Regular and Irregular Bipolar Fuzzy Graphs Table 3.5 Bipolar fuzzy set A A v1 p μA μnA

0.2 −0.6

Table 3.6 Bipolar fuzzy relation B B v1 v1 v2 v3

(0, 0) (0.1, −0.2) (0.1, −0.2)

131

v2

v3

0.2 −0.7

0.3 −0.4

v2

v3

(0.1, −0.2) (0, 0) (0.2, −0.3)

(0.1, −0.2) (0.2, −0.3) (0, 0)

Definition 3.4 The neighborhood of a vertex x in a bipolar fuzzy graph G = (A, B) is defined as a set N (x) = {y ∈ X | B(x y) = (0, 0)}. Definition 3.5 The neighborhood degree of a vertex x in G = (A, B) is defined p as a pair N deg(x) = (N deg p (x), N degn (x)), where N deg p (x) = y∈N (x) μ A (y) n n and N deg (x) = y∈N (x) μ A (y). Definition 3.6 Let G = (A, B) be a bipolar fuzzy graph on X . If there is a vertex which is adjacent to vertices with distinct neighborhood degrees then G is called an irregular bipolar fuzzy graph, that is, deg(x) = (r p , r n ), for all x ∈ X . Definition 3.7 The closed neighborhood degree of a vertex x in a bipolar fuzzy graph G is defined as a pair N deg[x] = (N deg p [x], N degn [x]) such that p

N deg p [x] = N deg p (x) + μ A (x), N degn [x] = N degn (x) + μnA (x). If there is a vertex which is adjacent to vertices with distinct closed neighborhood degrees then G is called a totally irregular bipolar fuzzy graph. Example 3.4 Let A be a bipolar fuzzy set on X = {v1 , v2 , v3 } as given in Table 3.5 and B be a bipolar fuzzy relation defined in Table 3.6. The bipolar fuzzy graph G = (A, B) is shown in Fig. 3.4. From Fig. 3.4,N deg(v1 ) = (0.5, −1.1), N deg(v2 ) = (0.5, −1.0), and N deg(v3 ) = (0.4, −1.3). It is clear that G is an irregular bipolar fuzzy graph. Example 3.5 Consider a bipolar fuzzy graph G on X = {v1 , v2 , v3 , v4 , v5 } as shown in Fig. 3.5. From usual calculations, N deg[v1 ] = (1.4, −2.4), N deg[v2 ] = (1.4, −2.4), N deg[v3 ] = (1.4, −2.4), N deg[v4 ] = (1.6, −2.9), and N deg[v5 ] = (0.6, −0.7). Clearly G is a totally irregular bipolar fuzzy graph.

132

3 Special Types of Bipolar Fuzzy Graphs v1 (0.2, −0.6) .1, (0 .2) −0

(0 .1 ,− 0. 2)

Fig. 3.4 Irregular bipolar fuzzy graph G

(0.2, −0.3)

v3 (0.3, −0.4)

Fig. 3.5 Totally irregular bipolar fuzzy graph

v2 (0.2, −0.7)

v1 (0.4, −0.6)

v2 (0.3, −0.5) (0.2, −0.3)

4) 0. − , .2 (0

(0.2, −0.2)

(0.1, −0.3)

v5 (0.2, −0.1) (0. 1, − 0.2 )

(0 .1 ,− 0. 4)

(0.3, −0.5) v3 (0.3, −0.7)

v4 (0.4, −0.6)

a2 (0.3, −0.4)

a1 (0.2, −0.5) (0.2, −0.4)

1) 0. − , .3 (0

(0 .2 ,− 0. 5)

(0.2, −0.4)

(0.2, −0.1)

Fig. 3.6 Complete bipolar fuzzy graph

(0.2, −0.1) a4 (0.5, −0.1)

a3 (0.2, −0.5)

Definition 3.8 If G = (A, B) is a bipolar fuzzy graph on  p set nX then  the a nonempty μ (x), μ (x) . The order of G is denoted by O(G)) and defined as O(G) = A  x∈X  pA size of G is denoted by S(G) and defined as S(G) = x,y∈X μ B (x y), μnB (x y) . Definition 3.9 A bipolar fuzzy graph G = (A, B) on a nonempty set X is known as a complete bipolar fuzzy graph if  p   p  p B(x y) = μ B (x y), μnB (x y) = μ A (x) ∧ μ A (y), μnA (x) ∨ μnA (y) , for all x, y ∈ X.

Example 3.6 Consider a bipolar fuzzy graph G on X = {a1 , a2 , a3 , a4 } as shown in Fig. 3.6. Routine calculations show that G is a complete bipolar fuzzy graph.

3.2 Regular and Irregular Bipolar Fuzzy Graphs

133 p

Theorem 3.1 Let G = (A, B) be a bipolar fuzzy graph of X . Then A = (μ A , μnA ) is a constant function if and only if the following statements are equivalent. (1) G is a regular bipolar fuzzy graph. (2) G is a totally regular bipolar fuzzy graph. p

Proof Suppose that A = (μ A , μnA ) is a constant function and A(x) = (c1 , c2 ), for all x ∈ X . (1) ⇒(2): Assume that G is an (n 1 , n 2 )-regular bipolar fuzzy graph, for all x ∈ X, then p

T deg p (x) = deg p (x) + μ A (x), T degn (x) = degn (x) + μnA (x) for all x ∈ X, ⇒ T deg p (x) = n 1 + c1 , T degn (x) = n 2 + c2 , for all x ∈ X. Hence G is a totally regular bipolar fuzzy graph. (2) ⇒ (1): Suppose that G is a totally regular bipolar fuzzy graph then for all x ∈ X , T deg p (x) = k1 , T degn (x) = k2 , p ⇒ deg p (x) + μ A (x) = k1 , degn (x) + μnA (x) = k2 , ⇒ deg p (x) + c1 = k1 , ⇒ deg p (x) = k1 − c1 ,

degn (x) + c2 = k2 , degn (x) = k2 − c2 .

Thus G is a regular bipolar fuzzy graph. Conversely, assume that G is both (n 1 , n 2 )-regular and (t1 , t2 )-totally regular bipolar fuzzy graph, then for all x ∈ X , T deg p (x) = t1 , T degn (x) = t2 , p

⇒ deg p (x) + μ A (x) = t1 , degn (x) + μnA (x) = t2 , p ⇒ n 1 + μ A (x) = t1 , n 2 + μnA (x) = t2 , p ⇒ μ A (x) = t1 − n 1 , μnA (x) = t2 − n 2 . p

It follows that A = (μ A , μnA ) is a constant function.



Proposition 3.1 If a bipolar fuzzy graph G is both regular and totally regular, then p A = (μ A , μnA ) is a constant function. Proof Let G be a regular and totally regular bipolar fuzzy graph, then deg p (x) = n 1 , degn (x) = n 2 , for all x ∈ X T deg p (x) = k1 , T degn (x) = k2 , for all x ∈ X. ⇒ T deg p (x) = k1 p ⇔ deg (x) + μ A (x) = k1 p

p

⇔ n 1 + μ A (x) = k1 p ⇔ μ A (x) = k1 − n 1 for all x ∈ X.

134

3 Special Types of Bipolar Fuzzy Graphs v1 (0.2, −0.6)

Fig. 3.7 Neighborly irregular bipolar fuzzy graph

v2 (0.3, −0.7)

(0.1, −0.2)

(0.1, −0.4)

(0.1, −0.4) (0.1, −0.2) v4 (0.5, −0.5)

v3 (0.4, −0.4)

p

Likewise, μnA (x) = k2 − n 2 , for all x ∈ X . Hence A = (μ A , μnA ) is a constant function.  Remark 3.1 The converse of Proposition 3.1 is not true in general. Let G = (A, B) be a bipolar fuzzy graph on X = {v1 , v2 , v3 } such that p

p

p

μ A (v1 ) = μ A (v2 ) = μ A (v3 ) = 0.4, μnA (v1 ) = μnA (v2 ) = μnA (v3 ) = −0.2, p

p

μ B (v1 v2 ) = 0.2, μ B (v1 v3 ) = 0.1, μnB (v1 v2 ) = −0.1, μnB (v1 v3 ) = −0.1. p

Clearly A = (μ A , μnA ) is a constant function but G is neither regular nor totally regular bipolar fuzzy graph. Definition 3.10 A connected bipolar fuzzy graph G is said to be a neighborly irregular bipolar fuzzy graph if every two adjacent vertices of G have distinct neighborhood degrees. Example 3.7 Consider a bipolar fuzzy graph G on X = {v1 , v2 , v3 , v4 } as shown in Fig. 3.7 such that N deg(v1 ) = (0.8, −1.2), N deg(v2 ) = (0.6, −1.0), N deg(v3 ) = (0.8, −1.2), and N deg(v4 ) = (0.6, −1.0). It is clear from calculations that G is neighborly irregular bipolar fuzzy graph. Definition 3.11 A connected bipolar fuzzy graph G is said to be neighborly totally irregular bipolar fuzzy graph if every two adjacent vertices of G have distinct closed neighborhood degrees. Example 3.8 Consider a bipolar fuzzy graph G on X = {v1 , v2 , v3 , v4 } as given in Fig. 3.8. From the figure, N deg[v1 ] = (1.2, −1.5), N deg[v2 ] = (1.4, −1.3), N deg[v3 ] = (1.6, −1.1) and N deg[v4 ] = (1.5, −1.2). It is easy to see that G is a neighborly totally irregular bipolar fuzzy graph. Definition 3.12 A connected bipolar fuzzy graph G is called highly irregular bipolar fuzzy graph if every vertex of G is adjacent to vertices with distinct neighborhood degrees.

3.2 Regular and Irregular Bipolar Fuzzy Graphs

135 v1 (0.3, −0.6)

Fig. 3.8 Neighborly totally irregular bipolar fuzzy graph

v2 (0.4, −0.5)

(0.1, −0.2)

(0.1, −0.2)

(0.1, −0.1) (0.1, −0.2)

(0. 2,

−0 .2)

v1 (0.2, −0.6)

v2 (0.4, −0.1) (0.1, −0.3)

.1) −0

.1, (0 v6 (0.3, −0.1)

(0.1, −0.1)

.2) , −0 2 . 0 (

v3 (0.7, −0.2)

(0.2, −0.1)

v5 (0.5, −0.5)

Fig. 3.9 Highly irregular bipolar fuzzy graph

v4 (0.5, −0.4)

(0.2, −0.1) v4 (0.3, −0.4)

v3 (0.7, −0.3)

Remark 3.2 A highly irregular bipolar fuzzy graph may not be a neighborly irregular bipolar fuzzy graph. There is no relation between highly irregular bipolar fuzzy graph and neighborly irregular bipolar fuzzy graph as explained in Example 3.9. Example 3.9 Consider a bipolar fuzzy graph G as shown in Fig. 3.9. Clearly N deg(v1 ) = (0.7, −0.5), N deg(v2 ) = (1.2, −1.0), N deg(v3 ) = (1.2, −1.0), N deg(v4 ) = (1.0, −0.7), N deg(v5 ) = (1.4, −1.4), and N deg(v6 ) = (0.4, −0.1). Consider a vertex v2 ∈ X which is adjacent to vertices v1 , v3 , and v6 with distinct neighborhood degrees but N deg(v2 ) = N deg(v3 ). So G is highly irregular but it is not a neighborly irregular bipolar fuzzy graph. Remark 3.3 A neighborly irregular bipolar fuzzy graph may not be a highly irregular bipolar fuzzy graph as shown in Example 3.10. Example 3.10 Consider a bipolar fuzzy graph G on set X = {v1 , v2 , v3 , v4 } given in Fig. 3.10 such that N deg(v1 ) = (0.6, −0.8), N deg(v2 ) = (0.8, −0.9), N deg(v3 ) = (0.6, −0.8), N deg(v4 ) = (0.8, −0.9). It is easy to see that every two adjacent vertices have distinct neighborhood degrees. Consider a vertex v2 which is adjacent to vertices v1 and v3 but N deg(v1 ) = N deg(v3 ). Hence G is neighborly irregular bipolar fuzzy graph but not a highly irregular bipolar fuzzy graph. Theorem 3.2 If all the vertices of a bipolar fuzzy graph G are of distinct neighborhood degrees, then G is highly irregular and neighborly irregular.

136

3 Special Types of Bipolar Fuzzy Graphs v1 (0.3, −0.4)

Fig. 3.10 Bipolar fuzzy graph G

v2 (0.2, −0.5)

(0.1, −0.2)

(0.1, −0.2)

(0.1, −0.1) (0.1, −0.2) v4 (0.4, −0.3)

v3 (0.5, −0.5)

Proof Assume that the neighborhood degrees of all the vertices of G are distinct. Claim: G is highly irregular and neighborly irregular bipolar fuzzy graph. Let deg(vi ) = (ki , li ), i = 1, 2, ..., n. Given that k1 = k2 = k3 = ... = kn and l1 = l2 = l3 = ... = ln , which implies that every two adjacent vertices have distinct neighborhood degrees, and to every vertex the adjacent vertices have distinct neighborhood degrees. It completes the proof.  Theorem 3.3 If G is highly irregular and neighborly irregular such that G ∗ is a complete graph, then all the vertices of a bipolar fuzzy graph G are of distinct neighborhood degrees. Proof Let G be a bipolar fuzzy graph with n-vertices v1 , v2 , . . . , vn . Assume that G is highly irregular and neighborly irregular bipolar fuzzy graph. Claim: The neighborhood degrees of all vertices of G are distinct. Let deg(vi ) = (ki , li ) for all i = 1, 2, . . . , n. The adjacent vertices of v1 are v2 , v3 , . . . , vn with neighborhood degrees (k2 , l2 ), (k3 , l3 ), . . . , (kn , ln ), respectively. As G is highly irregular so k2 = k3 = . . . = kn and l2 = l3 = . . . = ln . Also by assumption G is neighborly irregular, k1 = k2 = k3 = . . . = kn and l1 = l2 = l3 = . . . = ln . Hence the neighborhood degrees of all the vertices of G are distinct.  Theorem 3.4 Let G a bipolar fuzzy graph such that the underlying graph G ∗ is a cycle with 3 vertices then G is neighborly irregular and highly irregular bipolar fuzzy graph if and only if the positive membership and negative membership values of the vertices between every pair of vertices are all distinct. Proof Assume that positive membership and negative membership values of the vertices are all distinct. Claim: G = (A, B) is neighborly irregular and highly irregular bipolar fuzzy graph. p p p Let vi , v j , vk ∈ X . Given that, μ A (vi ) = μ A (v j ) = μ A (vk ) and μnA (vi ) = μnA (v j ) = μnA (vk ), which implies that  vi ∈N (x)

and

p

μ A (vi ) =

 v j ∈N (x)

p

μ A (v j ) =

 vk ∈N (x)

p

μ A (vk )

3.2 Regular and Irregular Bipolar Fuzzy Graphs

137 v1 (0.4, −0.6)

Fig. 3.11 Neighborly irregular bipolar fuzzy graph

v2 (0.3, −0.4)

(0.2, −0.2)

(0.3, −0.3)

(0.2, −0.1) (0.1, −0.3) v4 (0.4, −0.5)

 vi ∈N (x)

μnA (vi ) =

 v j ∈N (x)

μnA (v j ) =



v3 (0.4, −0.5)

μnA (vk ).

vk ∈N (x)

That is, N deg(vi ) = N deg(v j ) = N deg(vk ). Hence G is neighborly irregular and highly irregular bipolar fuzzy graph. Conversely, assume that G is neighborly irregular and highly irregular bipolar fuzzy graph. Claim: The positive membership and negative membership values of the vertices are all distinct. Let N deg(vi ) = (ki , li ), i = 1, 2, . . . , n. Suppose that positive membership and negative membership values of any two vertices are same. Let v1 , v2 ∈ X such that p p μ A (v1 ) = μ A (v2 ) and μnA (v1 ) = μnA (v2 ) then N deg(v1 ) = N deg(v2 ). Since G ∗ is cycle which is a contradiction to the fact that G is neighborly irregular and highly irregular bipolar fuzzy graph. Hence the positive membership and negative membership values of the vertices are all distinct.  Remark 3.4 A neighborly totally irregular bipolar fuzzy graph may not be neighborly irregular. Remark 3.5 A neighborly irregular bipolar fuzzy graph may not be neighborly totally irregular as it can be seen in Example 3.11. Example 3.11 Consider a bipolar fuzzy graph G as shown in Fig. 3.11. From the figure, N deg(v1 )=(0.7, −0.9), N deg(v2 ) = (0.8, −1.1), N deg(v3 ) = (0.7, −0.9), N deg(v4 ) = (0.8, −1.1) and N deg[v1 ] = (1.1, −1.5), N deg[v2 ] = (1.1, −1.5), N deg[v3 ] = (1.1, −1.4), N deg[v4 ] = (1.2, −1.6). It is easy to see that N deg[v1 ] = N deg[v2 ]. Hence G is neighborly irregular but not a neighborly totally irregular bipolar fuzzy graph. Example 3.12 Consider a bipolar fuzzy graph G as given in Fig. 3.13. Clearly N deg[v1 ] = (1.2, −1.5), N deg[v2 ] = (1.3, −1.4), N deg[v3 ] = (1.5, −1.2), N deg[v4 ] = (1.4, −1.3). But N deg(v1 ) = N deg(v2 ) = N deg(v2 ) = (0.9, −0.9). Hence G is neighborly totally irregular but not a neighborly irregular bipolar fuzzy graph (Fig. 3.12).

138

3 Special Types of Bipolar Fuzzy Graphs

Fig. 3.12 Neighborly totally irregular bipolar fuzzy graph

v1 (0.3, −0.6)

v2 (0.4, −0.5)

(0.1, −0.2)

(0.1, −0.2)

(0.1, −0.1) (0.1, −0.2) v4 (0.5, −0.4)

v3 (0.6, −0.3)

Proposition 3.2 Let G be a bipolar fuzzy graph. If G is neighborly irregular bipolar p fuzzy graph and (μ A , μnA ) is a constant function then G is a neighborly totally irregular bipolar fuzzy graph. Proof Assume that G is a neighborly irregular bipolar fuzzy graph, that is, the neighborhood degrees of every two adjacent vertices are distinct. Let vi , v j ∈ X , where vi and v j are adjacent vertices with distinct neighborhood degrees (k1 , l1 ) and (k2 , l2 ), respectively. That is, N deg(vi ) = (k1 , l1 ) and N deg(v j ) = (k2 , l2 ), where k1 = k2 , l1 = 12 . Let us assume that A(vi ) = A(v j ) = (c1 , c2 ), where c1 , c2 are constant and (c1 , c2 ) ∈ [0, 1] × [−1, 0]. Therefore, N deg p [vi ] = N deg p (vi ) + μ p (vi ) = k1 + N degn [vi ] = N degn (vi ) + μn (vi ) = l1 + c2 , N deg p [v j ] = c1 , p n n p n N deg (v j ) + μ (v j ) = k2 + c1 , and deg [v j ] = deg (v j ) + μ (v j ) = l2 + c2 . Claim: N deg p [vi ] = N deg p [v j ] and N degn [vi ] = N degn [v j ]. On contrary, assume that N deg p [vi ] = N deg p [v j ] and N degn [vi ] = N degn [v j ]. Consider N deg p [vi ] = N deg p [v j ] k1 + c1 = k2 + c1 k1 − k2 = c1 − c1 = 0 k1 = k2 , which is a contradiction. So, N deg p [vi ] = N deg p [v j ]. Similarly, N degn [vi ] = N degn [v j ] l1 + c2 = l2 + c2 l1 − l2 = c2 − c2 = 0 l1 = l2 , which is a contradiction. Thus N degn [vi ] = N degn [v j ]. Hence G is a neighborly totally irregular bipolar fuzzy graph.  Theorem 3.5 Let G be a bipolar fuzzy graph. If G is neighborly totally irregular p and (μ A , μnA ) is a constant function, then G is a neighborly irregular bipolar fuzzy graph.

3.2 Regular and Irregular Bipolar Fuzzy Graphs

139

Proof Assume that G is a neighborly totally irregular bipolar fuzzy graph. That is, the closed neighborhood degree of every two adjacent vertices are distinct. Let vi , v j ∈ X and N deg[vi ] = (k1 , l1 ) and N deg[v j ] = (k2 , l2 ), where k1 = k2 and l1 = l2 . Assume that A(vi ) = A(v j ) = (c1 , c2 ), where (c1 , c2 ) ∈ [0, 1] × [−1, 0] and N deg[vi ] = N deg[v j ]. Claim: N deg(vi ) = N deg(v j ). Given that N deg[vi ] = N deg[v j ] which implies that N deg p [vi ] = deg p [v j ] and degn [vi ] = degn [v j ]. Now, N deg p [vi ] = N deg p [v j ] N degn [vi ] = N degn [v j ] k1 + c1 = k2 + c1 l1 + c2 = l2 + c2 k1  = k2 . l1  = l2 . Hence the neighborhood degrees of every pair of adjacent vertices are distinct in G.  Remark 3.6 If G is neighborly irregular bipolar fuzzy graph then bipolar fuzzy subgraph H = (A , B ) of G may not be neighborly irregular bipolar fuzzy graph. Example 3.13 Consider a bipolar fuzzy graph G = (A, B) as given in Fig. 3.13, where A is a bipolar fuzzy set on X = {v1 , v2 , v3 , v4 , v5 } and B is a bipolar fuzzy relation in X such that supp(B) = E = {v1 v2 , v2 v3 , v3 v1 , v3 v4 , v4 v1 , v4 v5 , v5 v1 }. Consider the bipolar fuzzy subgraph, in Fig. 3.14, H = (A , B ) of G such that X = {v1 , v2 , v3 } and supp(B ) = E = {v1 v2 , v2 v3 , v3 v1 }. 1. For G: By routine computations, N deg(v1 ) = (2.0, −1.4), N deg(v2 ) = (0.7, −0.8), N deg(v3 ) = (1.9, −0.6), N deg(v4 ) = (1.3, −1.2) and N deg (v5 ) = (1.3, −0.4). 2. For H : By routine computations, N deg(v1 ) = (0.7, −0.8), N deg(v2 ) = (0.7, −0.8) and N deg(v3 ) = (1.2, −0.4). It is easy to see that v1 and v2 are adjacent vertices with the same neighborhood degrees in H . Hence, H is not a neighborly irregular bipolar fuzzy graph but G is neighborly irregular bipolar fuzzy graph. Remark 3.7 If G is totally irregular then the bipolar fuzzy subgraph H = (A , B ) of G may not be totally irregular bipolar fuzzy graph. Example 3.14 Consider a bipolar fuzzy graph G = (A, B), shown in Fig. 3.15, on set X ={v1 , v2 , v3 , v4 } such that supp(B) = E={v1 v2 , v2 v3 , v2 v4 , v3 v4 , v3 v1 , v4 v1 }. Consider the bipolar fuzzy subgraph, in Fig. 3.16, H = (A , B ) of G such that X = {v1 , v2 , v3 } and supp(B ) = E = {v1 v2 , v2 v3 , v3 v1 }. 1. For G: By routine computations, N deg[v1 ] = (1.4, −2.4), N deg[v2 ] = (1.0, −1.8), N deg[v3 ] = (1.4, −2.4), N deg[v4 ] = (1.1, −1.9). Here there is a vertex v3 which is adjacent to v1 , v2 and v4 , where N deg[v1 ] = N deg[v2 ]N = N deg[v4 ].

140

3 Special Types of Bipolar Fuzzy Graphs v2 (0.6, −0.2)

1, (0.

−0 .1)

Fig. 3.13 Neighborly irregular bipolar fuzzy graph

0.1) .6, − v 3(0

v1 (0 .6, − 0.2)

(0. 2,

.1) −0

(0.1, −0.1)

(0.2, −0.1)

(0.1, −0.1)

(0 .1 ,− 0. 1)

(0.2, −0.2) v5 (0.6, −0.4)

v4 (0.7, −0.2)

v2 (0.6, −0.2)

.1) −0

(0. 2,

1, (0.

−0 .1)

Fig. 3.14 Bipolar fuzzy subgraph H which is not neighborly irregular

(0.1, −0.1) v1 (0.6, −0.2)

v3 (0.6, −0.1)

2. For H : By routine computations, N deg[v1 ] = (1.0, −1.8), N deg[v2 ] = (1.0, −1.8) and N deg[v3 ] = (1.0, −1.8). Here, there is a vertex v1 which is adjacent to the vertices v2 and v3 with same closed neighborhood degrees. Also, v2 which is adjacent to the vertices v1 and v3 with same closed neighborhood degrees and v3 which is adjacent to the vertices v1 and v2 with same closed neighborhood degrees. Hence H is not a totally irregular bipolar fuzzy graph but G is totally irregular bipolar fuzzy graph.

3.3 Bipolar Fuzzy Line Graphs v1 (0.4, −0.6)

v2 (0.3, −0.5)

(0.2, −0.2) (0 .1 ,− 0. 3)

(0.2, −0.2)

(0.1, −0.3)

Fig. 3.15 Totally irregular bipolar fuzzy graph G

141

(0.2, −0.3) v4 (0.4, −0.6)

Fig. 3.16 Bipolar fuzzy subgraph which is not totally irregular

v3 (0.3, −0.7)

v1 (0.4, −0.6)

v2 (0.3, −0.5) (0.2, −0.4)

(0.1, −0.2)

(0 .1 ,− 0. 4)

v3 (0.3, −0.7)

3.3 Bipolar Fuzzy Line Graphs The line graph of a simple graph G ∗ is a graph L(G ∗ ) that represents the adjacencies between edges of G ∗ . Let G ∗ = (X, E) be a graph then its line graph L(G ∗ ) is such that 1. Each vertex of L(G ∗ ) represents an edge of G ∗ . 2. Two vertices of L(G ∗ ) are adjacent if and only if their corresponding edges share a common endpoint in G ∗ . Definition 3.13 Let G = (A, B) be a bipolar fuzzy graph on X and supp(B) = {Sx y | x, y ∈ X }. The bipolar fuzzy line graph L(G) = (A L , B L ) of G is a bipolar fuzzy graph, where A L = B and there is an edge between two vertices Sx y and S yz in L(G) if Sx y and S yz share a common vertex in y in G. The degree of membership of vertices and edges in L(G) is defined as 1. 2. 3. 4.

p

p

μ A L (Sx y ) = μ B (x y), μnA L (Sx y ) = μnB (x y), p p p μ BL (Sx y S yz ) = μ B (x y) ∧ μ B (yz), μnBL (Sx y S yz ) = μnB (x y) ∨ μnB (yz).

142

3 Special Types of Bipolar Fuzzy Graphs

Table 3.7 Bipolar fuzzy set A on X A v1 p μA μnA

0.2 −0.5

v2

v3

v4

0.3 −0.4

0.4 −0.5

0.2 −0.3

x3

x4

0.1 −0.2

0.1 −0.2

Table 3.8 Bipolar fuzzy relation B in X B x1 x2 p

μB μnB

0.1 −0.2

0.2 −0.3

v1 (0.2, −0.5)

Fig. 3.17 Bipolar fuzzy graph G

v2 (0.3, −0.4)

(0.1, −0.2)

(0.2, −0.3)

(0.1, −0.2) (0.1, −0.2) v4 (0.2, −0.3)

v3 (0.4, −0.5)

Example 3.15 Consider the set of vertices X = {v1 , v2 , v3 , v4 } and the set of edges E = {x1 = v1 v2 , x2 = v2 v3 , x3 = v3 v4 , x4 = v4 v1 }. Let A be a bipolar fuzzy set on X , in Table 3.7, and B a bipolar fuzzy set on E defined in Table 3.8. It is easy to see, from Fig. 3.17, that G is a bipolar fuzzy graph. The bipolar fuzzy line graph L(G) = (A L , B L ) of G is shown in Fig. 3.18, where A L = B and the membership values of edges in L(G) are computed as p

p

p

p

μ A L (Sx1 ) = 0.1, μ A L (Sx2 ) = 0.2, μ A L (Sx3 ) = 0.1, μ A L (Sx4 ) = 0.1, μnA L (Sx1 ) = −0.2, μnA L (Sx2 ) = −0.3, μnA L (Sx3 ) = −0.2, μnA L (Sx4 ) = −0.2. p

p

p

p

μ BL (Sx1 Sx2 ) = 0.1, μ BL (Sx2 Sx3 ) = 0.1, μ BL (Sx3 Sx4 ) = 0.1, μ BL (Sx4 Sx1 ) = 0.1, μnB L (Sx1 Sx2 ) = −0.2, μnB L (Sx2 Sx3 )= − 0.2, μnB L (Sx3 Sx4 )= − 0.2, μnB L (Sx4 Sx1 ) = −0.2.

L(G) is a regular bipolar fuzzy graph but not totally regular bipolar fuzzy graph since Tdeg(Sx2 ) = Tdeg(Sx4 ).

3.3 Bipolar Fuzzy Line Graphs

143 Sx1 (0.1, −0.2)

Fig. 3.18 Bipolar fuzzy line graph L(G)

Sx2 (0.2, −0.3)

(0.1, −0.2)

(0.1, −0.2)

(0.1, −0.2) (0.1, −0.2) Sx4 (0.1, −0.2)

Sx3 (0.1, −0.2)

Proposition 3.3 If L(G) is a bipolar fuzzy line graph of a bipolar fuzzy graph G, then L(G ∗ ) is the line graph of G ∗ . Proof Let G = (A, B) be a bipolar fuzzy graph and L(G) = (A L , B L ). Let (Z , W ) p p be a crisp graph corresponding to L(G) = (A L , B L ). Clearly, μ A (Sx y ) = μ B (x y), n n μ A (Sx y ) = μ B (x y), for all x y ∈ E and so Sx y ∈ Z ⇔ x ∈ E. By definition of bipolar fuzzy line graph, p p p μ BL (Sx y S yz ) = μ B (x y) ∧ μ B (yz) μnBL (Sx y S yz ) = μnB (x y) ∨ μnB (yz), for all Sx y , S yz ∈ Z , and so W = {Sx y S yz |Sx y ∩ S yz = ∅, x, y ∈ E, x = y}. It completes the proof.  Proposition 3.4 L(G) is a bipolar fuzzy line graph of some bipolar fuzzy graph G if and only if p

p

p

μ BL (Sx y S yz ) = μ A L (Sx y ) ∧ μ A L (S yz ), for all Sxy Syz ∈ W,

(3.1)

μnBL (Sx y S yz )

(3.2)

=

μnA L (Sx y )

p

∨

μnA L (S yz ),

p

for all Sxy Syz ∈ W.

p

Proof Assume that μ BL (Sx y S yz ) = μ A L (Sx y ) ∧ μ A L (S yz ), for all Sx y S yz ∈ W , then p

p

p

p

p

μ BL (Sx y S yz ) = μ A L (Sx y ) ∧ μ A L (S yz ) = μ B (x y) ∧ μ B (yz), μnBL (Sx y S yz ) = μnA L (Sx y ) ∨ μnA L (S yz ) = μnB (x y) ∨ μnB (yz). p

A bipolar fuzzy set A = (μ A , μnA ) that yields the property p

p

p

μ B (x y) ≤ μ A (x) ∧ μ A (y), μnB (x y) ≥ μnA (x) ∨ μnA (y) will suffice. Thus (A L , B L ) is a bipolar fuzzy line graph of G = (A, B).

144

3 Special Types of Bipolar Fuzzy Graphs

Conversely, suppose that L(G) = (A L , B L ) is a bipolar fuzzy line graph of G = p p p p (A, B), then μ BL (Sx y S yz ) = μ B (x) ∧ μ B (y), hence consequently, μ BL (Sx y S yz ) = p p μ A L (Sx y ) ∧ μ A L (S yz ). Similarly, μnBL (Sx y S yz ) = μnB (x) ∨ μnB (y) implies  μnBL (Sx y S yz ) = μnA L (Sx y ) ∨ μnA L (S yz ). Proposition 3.5 L(G) is a bipolar fuzzy line graph of G if and only if L(G ∗ ) is a line graph of G ∗ and p

p

p

μ BL (uv) = μ A L (u) ∧ μ A L (v) μnBL (uv) = μnA L (u) ∨ μnA L (v), for all uv ∈ W. Definition 3.14 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. A homomorphism of bipolar fuzzy graphs G 1 and G 2 is a mapping f : X 1 → X 2 such that p

p

1. μ A1 (x1 ) ≤ μ A2 ( f (x1 )), μnA1 (x1 ) ≥ μnA2 ( f (x1 )), for all x1 ∈ X 1 , p p 2. μ B1 (x1 y1 ) ≤ μ B2 ( f (x1 ) f (y1 )), μnB1 (x1 y1 ) ≥ μnB2 ( f (x1 ) f (y1 )), for all x1 , y1 ∈ X 1. Definition 3.15 A weak isomorphism of bipolar fuzzy graphs G 1 and G 2 is a bijective homomorphism f : X 1 → X 2 that satisfies p

p

μ A (x1 ) = μ A ( f (x1 )), μnA1 (x1 ) = μnA2 ( f (x1 )), for all x1 ∈ X 1 . 1 2

Definition 3.16 A co-weak isomorphism of bipolar fuzzy graphs G 1 and G 2 is a bijective homomorphism f : X 1 → X 2 that satisfies p

p

μ B (x1 y1 ) = μ B ( f (x1 ) f (y1 )), μnB1 (x1 y1 ) = μnB2 ( f (x1 ) f (y1 )), for all x1 , y1 ∈ X 1 . 1 2

Definition 3.17 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively. An isomorphism of bipolar fuzzy graphs G 1 and G 2 , written as G 1 ∼ = G 2 , is a bijective mapping f : X 1 → X 2 that satisfies p

p

1. μ A1 (x1 ) = μ A2 ( f (x1 )), μnA1 (x1 ) = μnA2 ( f (x1 )), for all x1 ∈ X 1 , p p 2. μ B1 (x1 y1 ) = μ B2 ( f (x1 ) f (y1 )), μnB1 (x1 y1 ) = μnB2 ( f (x1 ) f (y1 )), for all x1 , y1 ∈ X 1. Proposition 3.6 Let G 1 and G 2 be bipolar fuzzy graphs. If f is a weak isomorphism from G 1 into G 2 , then f is an isomorphism from G ∗1 into G ∗2 . Theorem 3.6 Let L(G) = (A L , B L ) be a bipolar fuzzy line graph corresponding to bipolar fuzzy graph G = (A, B). If the crisp graph G ∗ = (X, E) corresponding to G is connected, then 1. There exists a week isomorphism from G into L(G) if and only if G ∗ is a cycle p p and, A = (μ A , μnA ) and B = (μ B , μnB ) are constant functions. 2. If f is a weak isomorphism from G into L(G) then f is an isomorphism.

3.3 Bipolar Fuzzy Line Graphs

145

Proof Assume that f is a weak isomorphism of L(G) into G. From Proposition 3.6, it follows that G ∗ = (X, E) is a cycle. Let X = {v1 , v2 , . . . , vn } and E = {x1 = v1 v2 , x2 = v2 v3 , . . . , xn = vn v1 }, where v1 v2 v3 . . . vn v1 is a cyclic. Define bipolar fuzzy sets A and B as p μ A (vi ) = si , μnA (vi ) = s´i p

μ B (vi vi+1 ) = ri , μnB (vi vi+1 ) = r´i , i = 1, 2, . . . , n, vn+1 = v1 . Then for sn+1 = s1 , s´n+1 = s´1 , 

ri ≤ si ∧ si+1 , r´i ≥ s´i ∨ s´i+1 , i = 1, 2, . . . , n.

(3.3)

It follows that Z = {Sx1 , Sx1 , Sx2 , . . . , Sxn } and W = {Sx1 Sx2 , Sx2 Sx3 , . . . , Sxn Sx1 }. Also for rn+1 = r1 , p

p

p

μ A L (Sxi ) = μ A (xi ) = μ A (vi vi+1 ) = ri , μnA L (Sxi ) = μnA (xi ) = μnA (vi vi+1 ) = r´i , p

p

μ BL (Sxi Sxi+1 ) =

p

μ B (xi ) ∧ μ B (xi+1 ) p

p

= μ B (vi vi+1 ) ∧ μ B (vi+1 vi+2 ) = ri ∧ ri+1 , μnBL (Sxi Sxi+1 ) =

μnB (xi ) ∨ μnB (xi+1 )

= μnB (vi vi+1 ) ∨ μnB (vi+1 vi+2 ) = r´i ∨ r´i+1 for i = 1, 2, . . . , n, vn+1 = v1 , vn+2 = v2 . Since f is an isomorphism from G ∗ into L(G ∗ ), f maps X one-to-one and onto Z . Also f preserves adjacency. Hence f induces a permutation π of {1, 2, . . . , n} such that f (vi ) = Sxπ(i) = Sxπ(i) Sxπ(i+1) xi = vi vi+1 → f (vi ) f (vi+1 ) = Svπ(i) Svπ(i+1) Svπ(i+2) , i = 1, 2, . . . , n − 1. p

p

p

si = μ A (vi ) ≤ μ A L ( f (vi )) = μ A L (Svπ(i) vπ(i+1) ) = rπ(i) , s´i = μnA (vi ) ≥ μnA L ( f (vi )) = μnA L (Svπ(i) vπ(i+1) ) = r´π(i) ,

146

3 Special Types of Bipolar Fuzzy Graphs p

p

ri = μ B (vi vi+1 ) ≤ μ BL ( f (vi ) f (vi+1 )) p

= μ BL (Svπ(i) Svπ(i)+1 Svπ(i+1)+1 ) p

p

= μ B (vπ(i) vπ(i)+1 ) ∧ μ B (vπ(i)+1 vπ(i+1)+1 ) = rπ(i) ∧ rπ(i+1) , r´i = μnB (vi vi+1 ) ≥ μnBL ( f (vi ) f (vi+1 )) = μnBL (Svπ(i) Svπ(i)+1 Svπ(i+1)+1 = μnB (vπ(i) vπ(i)+1 ) ∨ μnB (vπ(i)+1 vπ(i+1)+1 )) = r´π(i) ∨ r´π(i+1) for i = 1, 2, . . . , n. That is, si ≤ rπ(i) , s´i ≥ r´π (i) and, 

ri ≤ rπ(i) ∧ rπ(i+1) , r´i ≥ r´π(i) ∨ r´π(i+1) .

(3.4)

From (3.4), we get ri ≤ rπ(i) , r´i ≥ r´π(i) , for i = 1, 2, . . . , n and so rπ (i) ≤ rπ(π(i)) , r´π (i) ≤ r´π(π(i)) , for i = 1, 2, . . . , n. Continuing this process, ri ≤ rπ(i) ≤ . . . ≤ rπ j (i) ≤ ri , r´i ≥ r´π(i) ≥ . . . ≥ r´π j (i) ≥ r´i . So ri = rπ(i) , r´i = r´π(i) , i = 1, 2, . . . , n, where π j+1 is the identity map. From (3.4), ri ≤ rπ(i+1) = ri+1 , i = 1, 2, . . . , rn+1 = r1 , r´i ≥ r´π(i+1) = r´i+1 , i = 1, 2, . . . , r´n+1 = r´1 . From Eqs. (3.3) and (3.4), r 1 = . . . = r n = s 1 = . . . = sn , r´1 = . . . = r´n = s´1 = . . . = s´n . It is not only proved that A and B are constant functions, but condition (2) is also satisfied.  Theorem 3.7 Let G and H be two bipolar fuzzy graphs such that the corresponding crisp graphs G ∗ and H ∗ are connected. Suppose that it is not the case that one of G ∗ and H ∗ is a complete graph K 3 and the other is complete bipartite graph K 1,3 . If L(G) and L(H ) are isomorphic then G and H are co-weak isomorphic.

3.4 Cayley Bipolar Fuzzy Graphs

147

3.4 Cayley Bipolar Fuzzy Graphs In this section, the concept of Cayley bipolar fuzzy graphs is discussed with connectedness properties. Some interesting properties of bipolar fuzzy graphs are presented in terms of algebraic structures. Let G be a finite group and S be a minimal generating set of G. A Cayley graph (G, S) has elements of G as its vertices, the edge-set is given by {(g, gs ) | g ∈ G, s ∈ S}. Two vertices g1 and g2 are adjacent if g2 = g1 .s, where s ∈ S. Note that a generating set S is minimal if S generates G but no proper subset of S does. All Cayley graphs are vertex transitive. Definition 3.18 Let (V, ∗) be a group and A be any subset of V . Then the Cayley graph induced by (V, ∗, A) is the graph G = (V, R), where R = {(x, y) | x −1 y ∈ A}. Definition 3.19 Let A and B be two bipolar fuzzy sets on X and Y , respectively. A R from A to B is a mapping R : A → B defined as R =  bipolarp fuzzy relation x, y, μ R (x, y), μnR (x, y) | (x, y) ∈ X × Y such that p

p

p

μ R (x, y) ≤ μ A (x) ∧ μ B (y) and μnR (x, y) ≥ μnA (x) ∨ μnB (y). R is also a bipolar fuzzy relation in X × Y defined by the mapping R : X × Y : [0, 1] × [−1, 0]. A bipolar fuzzy relation R in X is defined by a mapping R : X × X : [0, 1] × [−1, 0]. Definition 3.20 Let R be a bipolar fuzzy relation in X . Then R is called a bipolar fuzzy equivalence relation on X if it satisfies the following conditions: 1. R is bipolar fuzzy reflexive, i.e., R(x, x) = (1, −1), for each x ∈ X . 2. R is bipolar fuzzy symmetric, i.e., R(x, y) = R(y, x), for any x, y ∈ X . 3. R is bipolar fuzzy transitive, i.e., R(x, z) ≥ y (R(x, y) ∧ R(y, z)). Definition 3.21 Let R be a bipolar fuzzy relation in X . Then R is called a bipolar fuzzy partial order relation on X if it satisfies the following conditions: 1. R is bipolar fuzzy reflexive, i.e., R(x, x) = (1, −1), for each x ∈ X . 2. R is bipolar fuzzy anti-symmetric, i.e., R(x, y) = R(y, x), for any x, y ∈ X . 3. R is bipolar fuzzy transitive, i.e., R(x, z) ≥ y (R(x, y) ∧ R(y, z)). Definition 3.22 Let R be a bipolar fuzzy relation in X . Then R is called a bipolar fuzzy linear order relation on X if it satisfies the following conditions: 1. R is bipolar fuzzy partial order relation. 2. R(x, y) = (0, 0) or R(y, x) = (0, 0), for all x, y ∈ X .

148

3 Special Types of Bipolar Fuzzy Graphs

Table 3.9 R(a, b) for Cayley bipolar fuzzy graph a 0 0 0 1 1

1

2

2

2

b 0 a −1 + 0 b R(a,b) (0.5, −0.4)

1 1

2 2

0 2

1 0

2 1

0 1

1 2

2 0

(0.3, −0.2)

(0.3, −0.2)

(0.3, −0.2)

(0.5, −0.4)

(0.3, −0.2)

(0.3, −0.2)

(0.3, −0.2)

(0.5, −0.4)

2

Definition 3.23 Let G = (A, B) be a bipolar fuzzy digraph. The in-degree of p p a vertex x in G is defined by ind(x) = (indμ (x), indμn (x)), where indμ (x) =   p n n y=x μB (x y) and indμ (x) = y=x μB (x y). Similarly, the out-degree of a verp p tex x in G is defined by outd(x) = (outdμ (x), outdμn (x)), where outdμ (x) =   p n n y=x μB (x y) and outdμ (x) = y=x μB (x y). A bipolar fuzzy digraph in which each vertex has same out-degree r is called an out-regular digraph with index of out-regularity r . In-regular digraphs are defined similarly. p

Definition 3.24 Let (V, ∗) be a group and A = (μ A , μnA ) be the bipolar fuzzy subset on V . The bipolar fuzzy relation R defined on V by R(x, y) = (μ R (x −1 ∗ y), μnR (x −1 ∗ y)), for all x,y ∈ V p

induces a bipolar fuzzy graph G = (V, R) called Cayley bipolar fuzzy graph, p denoted by by (V, ∗, μ R , μnR ). p

Definition 3.25 Let (V, ∗) be a group and A = (μ A , μnA ) be an bipolar fuzzy subset on V . Then the bipolar fuzzy relation R on V defined by R(x, y) = {(μ R (x −1 ∗ y), μnR (x −1 ∗ y)) | x, y ∈ V } p

induces a bipolar fuzzy graph G = (V, R), called Cayley bipolar fuzzy graph. It is denoted by (V, ∗, A). p

Example 3.16 Consider the group Z 3 and take V = {0, 1, 2}. Define μ A : V → p p p [0, 1] and μnA : V → [−1, 0] by μ A (0) = μ A (1) = μ A (2) = 0.5, μnA (0) = μnA (1) = n μ A (2) = −0.4. Then the Cayley bipolar fuzzy graph G = (V, R) induced by (Z 3 , +, A) is given by the following Table 3.9 and Fig. 3.19. We see that Cayley bipolar fuzzy graphs are actually bipolar fuzzy digraphs. Furthermore, the relation R in the above definition describes the strength of each directed edge. Theorem 3.8 The Cayley bipolar fuzzy graph G is vertex transitive.

3.4 Cayley Bipolar Fuzzy Graphs

149

Fig. 3.19 Cayley bipolar fuzzy graph

0 (0.5, -0.4) (0.3, -0.3)

(0.3, -0.3)

(0.3, -0.2)

(0.3, -0.2)

(0.5, -0.4) (0.3, -0.2)

(0.5, -0.4)

1

(0.3, -0.2)

2

Proof Let a, b ∈ V . Define ψ : V → V by ψ(x) = ba −1 x ∀x ∈ V . Clearly, ψ is a bijective map. For each x, y ∈ V , R(ψ(x), ψ(y)) = (Rμ p (ψ(x), ψ(y)), Rμn (ψ(x), ψ(y))). Now Rμ p (ψ(x), ψ(y)) = Rμ p (ba −1 x, ba −1 y) = μ A ((ba −1 x)−1 (ba −1 x)) p

= μ A (x −1 y) = Rμ p (x, y). p

Rμn (ψ(x), ψ(y)) = Rμn (ba −1 x, ba −1 y) = μnA ((ba −1 x)−1 (ba −1 x)) = μnA (x −1 y) = Rμn (x, y). Therefore, R(ψ(x), ψ(y)) = R(x, y). Hence ψ is an automorphism on G. Also ψ(a) = b. Hence G is vertex transitive.  Theorem 3.9 Every vertex-transitive bipolar fuzzy graph is regular. Proof Let G = (V, R) be any vertex-transitive bipolar fuzzy graph. Let u,v ∈ V . Then there is an automorphism f on G such that f (u) = v. Note that

150

3 Special Types of Bipolar Fuzzy Graphs

ind(u) =



R(x, u) =

x∈V

 (Rμ p (x, u), Rμn (x, u)) x∈V

 (Rμ p ( f (x), f (u)), Rμn ( f (x), f (u)) = x∈V

 (Rμ p ( f (x), v), Rμn ( f (x), v)) = x∈V

 (Rμ p (y, v), Rμn (y, v) = x∈V

= ind(v), outd(u) =



R(x, u) =

x∈V

=



 (Rμ p (u, x), Rμn (u, x)) x∈V

(Rμ p ( f (u), f (x)), Rμn ( f (u), f (x))

x∈V

=



(Rμ p (v, f (x)), Rμn (v, f (x))

x∈V

=



(Rμ p (v, y), Rμn (v, y)

x∈V

= outd(v). 

Hence G is regular. Theorem 3.10 Cayley bipolar fuzzy graphs are regular. The proof of Theorem 3.10 follows from the Theorems 3.8 and 3.9.

Theorem 3.11 Let G = (V, R) be a bipolar fuzzy graph then bipolar fuzzy relation p R is reflexive if and only if μ A (1) = 1 and μnA (1) = −1. Proof R is reflexive if and only if R(x, x) = (1, −1), for all x ∈ V . R(x, x) = (μ A (x −1 x), μnA (x −1 x)) p p

= (μ A (1), μnA (1)) for all x ∈ V. p

Hence R is reflexive if and only if μ A (1) = 1 and μnA (1) = −1.



Theorem 3.12 Let G = (V, R) denotes bipolar fuzzy graph. Then bipolar fuzzy p p relation R is symmetric if and only if (μ A (x), μnA (x)) = (μ A (x −1 ), μnA (x −1 )) for all x ∈ V. Proof Suppose that R is symmetric. Then for any x ∈ V ,

3.4 Cayley Bipolar Fuzzy Graphs

151

(μ A (x), μnA (x)) = (μ A (x −1 x 2 ), μnA (x −1 x 2 )) p

p

= R(x, x 2 ) = R(x 2 , x), since R is symmetric = (μ A ((x 2 )−1 x), μnA (x 2 )−1 x) p

= (μ A ((x −2 x), μnA (x −2 x) p

= (μ A ((x −1 ), μnA (x −1 ). p

Conversely suppose that (μ A (x), μnA (x)) = (μ A (x −1 ), μnA (x −1 )) for all x ∈ V . Then for all x, y ∈ V , p

p

R(x, y) = (μ A (x −1 y), μnA (x −1 y)) p

= (μ A (y −1 x), μnA (y −1 x)) = R(y, x). p



Hence R is symmetric. p

Theorem 3.13 A bipolar fuzzy relation R is anti-symmetric if and only if (μ A (x), p μnA (x)) = (μ A (x −1 ), μnA (x −1 )) = (1, −1), for all x ∈ V . p

Proof Suppose that R is anti-symmetric and x ∈ V , then (μ A (x), μnA (x)) = p (μ A (x −1 ), μnA (x −1 )), which implies R(1, x) = R(x, 1). Hence x = 1, since R is anti-symmetric. p p Conversely suppose that {x | (μ A (x), μnA (x)) = (μ A (x −1 ), μnA (x −1 )) = {1, 1}}. p −1 Then for any x, y ∈ V , R(x, y) = R(y, x) ⇔ (μ A (x y), μnA (y −1 x)). This implies p p that (μ A (x −1 y), μnA (y −1 x)) = (μ A ((x −1 y)−1 ), μnA ((x −1 y)−1 )). That is x −1 y = 1. Equivalently, x = y. Hence R is anti-symmetric.  p

Definition 3.26 Let (S, ∗) be a semigroup. Let A = (μ A , μnA ) be a bipolar fuzzy subset of S. Then A is said to be a bipolar fuzzy subsemigroup of S if for all x, y ∈ S, p p p p μ A (x y) ≥ μ A (x) ∧ μ A (y) and μnA (x y) ≤ μ A (x) ∨ μnA (y). p

Theorem 3.14 A bipolar fuzzy relation R is transitive if and only if A = (μ A , μnA ) is a bipolar fuzzy subsemigroup of (V, ∗). Proof Suppose that R is transitive and let x, y ∈ V . Then R 2 ≤ R. Now for any x ∈ p V , we have R(1, x) = (μ A (x), μnA (x)). This implies that {R(1, z) ∧ R(z, x y) | z ∈ p p p 2 V } = R (1, x y) ≤ R(1, x y). That is ∨{μ A (z) ∧ μ A (z −1 x y) | z ∈ V } ≤ μ A (x y) and p p p n n −1 n ∧{μ A (z) ∨ μ A (z x y) | z ∈ V } ≥ μ A (x y). Hence μ A (x y) ≥ μ A (x) ∧ μ A (y) and p p n n n μ A (x y) ≤ μ A (x) ∨ μ A (y). Hence A = (μ A , μ A ) is a bipolar fuzzy subsemigroup of (V, ∗). p Conversely, suppose that A = (μ A , μnA ) is a bipolar fuzzy subsemigroup of (V, ∗). p p p p That is, for all x, y ∈ V μ A (x y) ≥ μ A (x) ∧ μ A (y) and μnA (x y) ≤ μ A (x) ∨ μnA (y). Then for any x, y ∈ V ,

152

3 Special Types of Bipolar Fuzzy Graphs

R 2 (x, y) = (Rμ2 p (x, y), Rμ2 n (x, y)) Rμ2 p (x, y) = ∨{Rμ p (x, z) ∧ Rμ p (z, y) | z ∈ V } = ∨{μ A (x −1 z) ∧ μ A (z −1 y) | z ∈ V } p

p

≤ μ A (x −1 y) = Rμ p (x, y). p

Rμ2 n (x, y) = ∧{Rμn (x, z) ∨ Rμn (z, y) | z ∈ V } = ∧{μnA (x −1 z) ∨ μnA (z −1 y) | z ∈ V } ≥ μnA (x −1 y) = Rμn (x, y). Hence Rμ2 p (x, y) ≤ Rμ p (x, y) and Rμ2 n (x, y) ≥ Rμn (x, y). Hence R is transitive.  Theorem 3.15 A bipolar fuzzy relation R is a partial order if and only if A = p (μ A , μnA ) is a bipolar fuzzy subsemigroup of (V, ∗) satisfying the following properties: p

1. μ A (1) = 1 and μnA (1) = −1, p p 2. (μ A (x), μnA (x)) = (μ A (x −1 ), μnA (x −1 )) = (1, −1), for all x ∈ V . p

Theorem 3.16 A bipolar fuzzy relation R is a linear order if and only if (μ A , μnA ) is a bipolar fuzzy subsemigroup of (V, ∗) satisfying the following properties: p

1. μ A (1) = 1 and μnA (1) = −1, p p 2. {x | (μ A (x), μnA (x)) = (μ A (x −1 ), μnA (x −1 )) = (1, −1)} and p p −1 3. {x | μ A (x) ∨ μ A (x ) > 0, μnA (x) ∧ μnA (x −1 ) < 0} = V. Proof Suppose R is a linear order then clearly by Theorem 3.15 conditions 1,2, and 3 are satisfied. For any x ∈ V , (R ∨ R −1 )(1, x) > 0. This implies that R(1, x) ∨ p p R(x, 1) > 0. Hence {x | μ A (x) ∨ μ A (x −1 ) > 0, μnA (x) ∧ μnA (x −1 ) < 0}. Conversely, suppose that the conditions 1,2, and 3 hold. By Theorem 3.15, R is partial order. Now for any x, y ∈ V , we have (x −1 y), (y −1 x) ∈ V . Then, {x | p p  μ A (x) ∨ μ A (x −1 ) > 0, μnA (x) ∧ μnA (x −1 ) < 0}. Therefore R is linear order. Theorem 3.17 A bipolar fuzzy relation R is an equivalence relation if and only if p (μ A , μnA ) is a bipolar fuzzy subsemigroup of (V, ∗) satisfying p

1. μ A (1) = 1 and μnA (1) = −1, p p 2. (μ A (x), μnA (x)) = (μ A (x −1 ), μnA (x −1 )) for all x ∈ V . Theorem 3.18 G is a Hasse diagram if and only if for any collection x1 , x2 , x3 , p . . . , xn of vertices in V with n ≥ 2 and μ A (xi ) > 0, μnA (xi ) < 0, for i = 1, 2, 3, . . . , p n n, we have, μ A (x1 x2 . . . xn ) = 0 and μ A (x1 x2 . . . xn ) = 0.

3.4 Cayley Bipolar Fuzzy Graphs

153

Proof Suppose G is a Hasse diagram and let x1 , x2 , . . . , xn be vertices in V with p n ≥ 2 and μ A (xi ) > 0, μnA (xi ) < 0, for i = 1, 2, . . . , n. Then it is obvious that p R(x1 x2 . . . xi−1 , x1 x2 . . . xi ) = (μ A (xi ), μnA (xi )), for i = 1, 2, . . . , n, where x0 = 1. Therefore (1, x1 , x1 x2 , . . . , x1 x2 . . . xn ) is a path from 1 to x1 x2 . . . xn . Since G is a p Hasse diagram, we have R(1, x1 x2 . . . xn ) = 0. This implies that μ A (x1 x2 . . . xn ) = 0 n and μ A (x1 x2 . . . xn ) = 0. Conversely, suppose that for any collection x1 , x2 , . . . xn of vertices in V with p p n ≥ 2 and μ A (xi ) > 0, μnA (xi ) < 0, for i = 1, 2, . . . , n, we have, μ A (x1 x2 . . . xn ) = n 0 and μ A (x1 x2 . . . xn ) = 0. Let (x0 , x1 , x2 , . . . xn ) be a path in G from x0 to xn p −1 with n ≥ 2. Then R(xi−1 , xi ) > 0, for i = 1, 2, . . . , n. Therefore μ A (xi−1 xi ) > −1 −1 n 0, μ A (xi−1 xi ) < 0, for i = 1, 2, . . . , n. Now consider the elements x0 x1 , x1−1 x2 , p −1 −1 xn in V . Then by assumption μ A (x0−1 x1 x1−1 x2 . . . xn−1 xn ) = 0 and μnA (x0−1 . . . , xn−1 p −1 −1 −1 −1 n x1 x1 x2 . . . xn−1 xn ) = 0. That is, μ A (x0 xn ) = 0 and μ A (x0 xn ) = 0. Hence, R(x0 ,  xn ) = 0. Thus G is a Hasse diagram. Let G = (V, R) be any bipolar fuzzy graph, then G is connected (weakly connected, semiconnected, locally connected, or quasi- connected) if and only if the induced fuzzy graph (V, R0+ ) is connected (weakly connected, semiconnected, locally connected, or quasi-connected). p

Definition 3.27 Let (S, ∗) be a semigroup and A = (μ A , μnA ) be a bipolar fuzzy subset of S. Then the subsemigroup generated by A is the meet of all bipolar fuzzy subsemigroups of S which contain A. It is denoted by < A >. p

Lemma 3.1 Let (S, ∗) be a semigroup and A = (μ A , μnA ) be a bipolar fuzzy subset of S. Then bipolar fuzzy subset < A > is precisely given by p

p

p

p

p

< μ A > (x) = ∨{μ A (x1 ) ∧ μ A (x2 ) ∧ . . . ∧ μ A (xn ) | x = x1 x2 . . . xn with μ A (xi ) > 0 f or i = 1, 2, . . . , n},

< μnA > (x) = ∧{μnA (x1 ) ∨ μnA (x2 ) ∨ . . . ∨ μnA (xn ) | x = x1 x2 . . . xn with μnA (xi ) < 0 f or i = 1, 2, . . . , n},

x ∈ S. Proof Let A = (μ´ A , μ´ nA ) be a bipolar fuzzy subset of S defined by p

p

p

p

p

p

μ´ A (x) = ∨{μ A (x1 ) ∧ μ A (x2 ) ∧ . . . ∧ μ A (xn ) | x = x1 x2 . . . xn with μ A (xi ) > 0 f or i = 1, 2, . . . , n},

μ´ nA (x) = ∧{μnA (x1 ) ∨ μnA (x2 ) ∨ . . . ∨ μnA (xn ) |, x = x1 x2 . . . xn withμnA (xi ) < 0 f or i = 1, 2, . . . , n}

for any x ∈ S. p p p p Let x, y ∈ S. If μ A (x) = 0 or μ A (y) = 0, then μ A (x) ∧ μ A (y) = 0 and μnA (x) = p p p 0 or μnA (y) = 0, then μnA (x) ∨ μnA (y) = 0. Therefore, μ´ B (x y) ≥ μ A (x) ∧ μ A (y) p and μ´ nB (x y) ≤ μnA (x) ∨ μnA (y). Again , if μ A (x) = 0, μnA (x) = 0, then by definition p p p p of μ´ A (x) and μ´ nA (x), we have μ´ B (x y) ≥ μ A (x) ∧ μ A (y) and μ´ nB (x y) ≤ μnA (x) ∨ p p μnA (y). Hence (μ´ A , μ´ nA ) is a bipolar fuzzy subsemigroup of S containing (μ A , μnA ). p Now let L be any bipolar fuzzy subsemigroup of S containing (μ A , μnA ). Then for any

154

3 Special Types of Bipolar Fuzzy Graphs p

x ∈ S with x = x1 x2 . . . xn with μ A (xi ) > 0, μnA (xi ) < 0, for i = 1, 2, . . . , n, we p p p p p p p have μ L (xi ) ≥ μ L (x1 ) ∧ μ L (x2 ) ∧ . . . ∧ μ L (xn ) ≥ μ A (x1 ) ∧ μ A (x2 ) ∧ . . . ∧ μ A n n n n n n (xn ) and μ L (xi ) ≥ μ L (x1 ) ∧ μ L (x2 ) ∧ . . . ∧ μ L (xn ) ≥ μ A (x1 ) ∧ μ A (x2 ) ∧ . . . ∧ p p p p μnA (xn ). Thus μ L (x) ≥ ∨{μ A (x1 ) ∧ μ A (x2 ) ∧ . . . ∧ μ A (xn ) | x = x1 x2 . . . xn with p n n μ A (xi ) > 0 f or i = 1, 2, . . . , n}, μ L (x) ≤ ∧{μ A (x1 ) ∨ μnA (x2 ) ∨ . . . ∨ μnA (xn ) | x p = x1 x2 . . . xn with μnA (xi ) < 0 f or i = 1, 2, . . . , n}, for any x ∈ S. Hence μ L (x) ≥ p p p p n n n μ´ A (x), μ L (x) ≤ μ´ A (x), for all x ∈ S. Thus μ´ A (x) ≤ μ L (x), μ´ A (x) ≥ μ A (x). Thus p A = (μ´ A , μ´ nA ) is the meet of all bipolar fuzzy subsemigroup containing p n  (μ A , μ A ). p

Theorem 3.19 Let (S, ∗) be a semigroup and A = (μ A , μnA ) be a bipolar fuzzy p subset of S. Then for any α ∈ [0, 1], (< μα >, < μnα >) = (< μ p >α , < μn >α ) p + p + n p + n and (< (μ )α >, < (μ )α >) = (< μ >α , < μn >+ α ), where (< μα >, < μα >) p n p n denotes the subsemigroup generated by (μα , μα ) and < (μ , μ ) > denotes bipolar fuzzy subsemigroup generated by (μ p , μn ). Proof Let x ∈ S, then p

x ∈ (< μ p >α , < μn >α ) ⇔ there exists x1 , x2 , . . . , xn in (μα , μnα ) such that x = x1 x2 . . . xn ⇔ there exists x1 , x2 , . . . , xn in S such that μ p (xi ) ≥ α, μn (xi ) ≤ α, for all i = 1, 2, . . . , n and x = x1 x2 . . . xn ⇔< μ p > (x) ≥ α and < μn > (x) ≤ α ⇔ x ∈< μ p >α and x ∈< μn >α . p

Therefore (< μα >, < μnα >) = (< μ p >α , < μn >α ). p n + Similarly, (< (μ+ )α >, < (μ+ )nα >) = (< μ p >+ α , < μ >α ).



p

Remark 3.8 Let (S, ∗) be a semigroup and A = (μ A , μnA ) be a bipolar fuzzy subset of S. Then by Theorem 3.19, we have < supp(A) = A+ >= supp < A >. Let G denote the Cayley bipolar fuzzy graphs G = (V, R) induced by (V, ∗, μ p , μ ). Then we have the following results. n

Theorem 3.20 Let A be any subset of V and G = (V , R ) be the Cayley graph induced by (V , ∗, A). Then G is connected if and only if < A >⊇ V − v1 . Theorem 3.21 G is connected if and only if supp < A >⊇ V − v1 . Proof Indeed,

G is connected ⇔ (V, R0+ )is connected ⇔< A+ 0 >⊇ V − v1 ⇔< supp(A) >⊇ V − v1 ⇔ supp < A >⊇ V − v1 ,

as expected.



3.4 Cayley Bipolar Fuzzy Graphs

155

Theorem 3.22 Let A be a nonempty subset of a set V and G = (V , R ) be the Cayley graph induced by the triplet (V , ∗, A). Then G is weakly connected graph if and only if < A ∪ A−1 >⊇ V − v1 , where A−1 = {x −1 | x ∈ A}. Definition 3.28 Let (S, ∗) be a group and A be a bipolar fuzzy subset of S. Then we define A−1 as bipolar fuzzy subset of S given by A−1 (x) = A(x −1 ) for all x ∈ S. Theorem 3.23 G is weakly connected if and only if supp (< A ∪ A−1 >) ⊇ V − v1 . Proof Indeed, G is weakly connected ⇔ (V, R0+ )is weakly connected + −1 ⇔< A+ >⊇ V − v1 0 ∪ (A0 )

⇔< supp(A) ∪ supp(A)−1 >⊇ V − v1 ⇔ supp < A ∪ (A)−1 >⊇ V − v1 ⇔ supp < A ∪ A−1 >⊇ V − v1 , 

as expected.

Theorem 3.24 Let A be any subset of a set V and G = (V , R ) be the Cayley graph induced by the triplet (V , ∗, A). Then G is semiconnected if and only if < A > ∪ < A−1 >⊇ V − v1 , where A−1 = {x −1 | x ∈ A}. Theorem 3.25 G is semiconnected if and only if supp (< A > ∪ < A−1 >) ⊇ V − v1 . Proof Observe that G is semiconnected ⇔ (V, R0+ )is semiconnected + −1 ⇔< A+ >⊇ V − v1 0 > ∪ < (A0 )

⇔< supp(A) > ∪ < supp(A)−1 >⊇ V − v1 ⇔ supp < A > ∪ < (A)−1 >⊇ V − v1 ⇔ supp(< A > ∪ < A−1 >) ⊇ V − v1 , which completes the proof.



Theorem 3.26 Let G = (V , R ) be the Cayley graph induced by the triplet (V , ∗, A). Then G is locally connected if and only if < A >=< A−1 >, where A−1 = (x −1 | x ∈ A). Theorem 3.27 Let G is locally connected if and only if supp(< A >) = supp(< A−1 >).

156

3 Special Types of Bipolar Fuzzy Graphs

Proof Note that G is locally connected ⇔ (V, R0+ ) is locally connected + −1 ⇔< A+ > 0 >=< (A0 )

⇔< supp(A) >=< supp(A)−1 > ⇔ supp < A >= supp < A−1 >, which completes the proof.



Theorem 3.28 Let G = (V , R ) be the Cayley graph induced by the triplet (V , ∗, A), where V is finite. Then G is quasi-connected if and only if it is connected. Theorem 3.29 A finite Cayley bipolar fuzzy graph G is quasi-connected if and only if it is connected. Proof Indeed, G is quasi-connected ⇔ (V, R0+ ) is Quasi-connected ⇔ (V, R0+ ) is connected ⇔ G is connected.  Definition 3.29 The μ p − strength of a path P = v1 , v2 , . . . , vn is defined as p p min(μ2 (vi , v j )) for all i and j and is denoted by Sμ . The μn − strength of a path n P = v1 , v2 , . . . , vn is defined as max(μ2 (vi , v j )) for all i and j and is denoted by Sμn . Definition 3.30 Let G = (V, μ p , μn ) be a bipolar fuzzy graph, then 1. G is called α-connected if for every pair of vertices x, y ∈ G, there is a path P from x to y such that strength (P) ≥ α, 2. G is called weakly α-connected if a bipolar fuzzy graph (V, R ∨ R −1 ) is αconnected, 3. G is called semi α-connected if for every x, y ∈ V , there is a path of strength greater than or equal to α from x to y or from y to x in G, 4. G is called locally α-connected if for every pair of vertices x and y, there is a path P of strength greater than or equal to α from x to y whenever there is a path P of strength greater than or equal to α from y to x, 5. G is called quasi α-connected if for every pair x, y ∈ V , there is some z ∈ V such that there is directed path from z to x of strength greater than or equal to α and there is a directed path from z to y of strength greater than or equal to α. Remark 3.9 Let G = (V, R) be any bipolar fuzzy graph, then G is α-connected (weakly α-connected, α-semiconnected, locally α-connected, or quasi- α-connected) if and only if the induced fuzzy graph (V, R0+ ) is connected (weakly connected, semiconnected, locally connected, or quasi-connected). Theorem 3.30 G is α-connected if and only if < A >α ⊇ V − v1 .

3.4 Cayley Bipolar Fuzzy Graphs

157

Proof Indeed, G is connected ⇔ (V, Rα ) is connected ⇔< Aα >⊇ V − v1 ⇔< A >α ⊇ V − v1 . Theorem 3.31 G is weakly α-connected if and only if < A ∪ A−1 >α ⊇ V − v1 . Proof G is weakly connected ⇔ (V, Rα ) is weakly connected ⇔< Aα ∪ (Aα )−1 > ⊇ V − v1 ⇔< (A ∪ A−1 )α >⊇ V − v1 ⇔< A ∪ (A)−1 >α ⊇ V − v1 . Theorem 3.32 G is semi α connected if and only if (< A >α ∪ < A−1 >α ) ⊇ V − v1 . Proof G is α-semiconnected ⇔ (V, R0+ ) is α-semiconnected ⇔< Aα > ∪ < A−1 α >⊇ V − v1 ⇔< A >α ∪ < A−1 >α ⊇ V − v1 . Theorem 3.33 Let G is locally α-connected if and only if < A >α =< A−1 α >. Proof G is locally α-connected ⇔ (V, Rα ) is locally connected ⇔< Aα >=< −1 >α .  A−1 α >⇔< A >α =< A Theorem 3.34 A finite Cayley bipolar fuzzy graph G is quasi α-connected if and only if it is α-connected. Proof G is quasi α-connected ⇔ (V, Rα+ ) is quasi-connected ⇔ (V, Rα+ ) is connected ⇔ G is α-connected. 

3.5 Conclusions Bipolar fuzzy graphs can be categorized on the basis of the number of vertices, number of edges, connectivity, distance, degree, and their overall structure. Different classifications of bipolar fuzzy graphs play an important role in modeling graphical structures under certain conditions of regularity, irregularity, transitivity, and connectedness. In this chapter, we have discussed, in detail, the concept of irregularity in bipolar fuzzy graphs and extended this notion, with more strong constraints, to neighborly irregular, totally irregular, highly irregular, and neighborly totally irregular bipolar fuzzy graphs. We have discussed isomorphism properties of regular, m-totally regular, and different kinds of irregular bipolar fuzzy graphs. We have presented the conditions under which a regular bipolar fuzzy graph and a totally regular bipolar fuzzy graph are equivalent. We have described the idea of bipolar fuzzy line graph and established the characterizations under which a bipolar fuzzy graph is isomorphic to its bipolar fuzzy line graph. We have discussed the notions of regularity, vertex transitivity, and connectedness in Cayley bipolar fuzzy graphs using various algebraic structures.

158

3 Special Types of Bipolar Fuzzy Graphs

Exercises 3 1. Give an example of a regular bipolar fuzzy graph G such that G (α,β) is not a regular graph, for any (α, β) ∈ [0, 1] × [−1, 0]. 2. Let G = (A, B) be a regular bipolar fuzzy graph, then determine whether or not G c is a regular bipolar fuzzy graph. 3. Let G = (A, B) be an irregular bipolar fuzzy graph, then determine whether or not G c is an irregular bipolar fuzzy graph. 4. Let G = (A, B) be an (r p , r n )-regular bipolar fuzzy graph with k number of vertices then prove that S(G) = k2 (r p , r n ). 5. Let G = (A, B) be an (m p , m n )-totally regular bipolar fuzzy graph with k number of vertices then prove that 2S(G) + O(G) = (m p , m n )k. 6. Let G = (A, B) be an r = (r p , r n )-regular and m = (m p , m n )-totally regular bipolar fuzzy graph with k number of vertices then prove that O(G) = k(m p − r p , m n − r n ). 7. Let G = (A, B) be a bipolar fuzzy graph such that G ∗ is an odd cycle. Prove that G is regular if and only if B is a constant function. 8. Prove that every complete bipolar fuzzy graph is a totally regular bipolar fuzzy graph. 9. Give an example to show that a highly irregular and neighborly irregular bipolar fuzzy graph G need not have all the vertices of distinct neighborhood degrees. 10. Give examples of two regular bipolar fuzzy graphs G and H such that the strong product G  H is not a regular bipolar fuzzy graph. 11. Let x y be an edge of a connected bipolar fuzzy graph G. Show that G − x y is connected if and only if x y is an edge of a bipolar fuzzy cycle in G.

References 1. Akram, M.: Bipolar fuzzy graphs with applications. Knowl.-Based Syst. 39, 1–8 (2013) 2. Akram, M., Dudek, W.A.: Regular bipolar fuzzy graphs. Neural Comput. Appl. 21(1), 197–205 (2012) 3. Akram, M., Dudek, W.A., Sarwar, S.: Properties of bipolar fuzzy hypergraphs. Ital. J. Pure Appl. Math. 31, 426–458 (2013) 4. Akram, M., Karunambigai, M.G.: Metric in bipolar fuzzy graphs. World Appl. Sci. J. 14, 1920–1927 (2011) 5. Akram, M., Li, S.-G., Shum, K.P.: Antipodal bipolar fuzzy graphs. Ital. J. Pure Appl. Math. 31, 97–110 (2013) 6. Akram, M., Yousaf, M.M., Dudek, W.A.: Self centered interval-valued fuzzy graphs. Afr. Math. 26(5–6), 887–898 (2015) 7. Alavi, Y., Boals, A.J., Chartand, G., Oellermann, O.R., Erdos, P.: k−path irregular graphs. Congr. Numer. 65, 201–202 (1988) 8. Alavi, Y., Chung, F.R.K., Erdos, P., Graham, R.L., Oellermann, O.R.: Highly irregular graphs. J. Graph Theory 11(2), 235–249 (1987) 9. Alshehro, N.O., Akram, M.: Cayley bipolar fuzzy graphs. Sci. World J. 2013, Article ID 156786 (2013)

References

159

10. Aravamudhan, R., Rajendran, B.: On antipodal graphs. Discret. Math. 49, 193–195 (1984) 11. Bhutani, K.R., Battou, A.: On M-strong fuzzy graphs. Inf. Sci. 155(1–2), 103–109 (2003) 12. Nagoorgani, A., Latha, S.R.: Isomorphism on irregular fuzzy graphs. Int. J. Math. Sci. Eng. Appl. 6(3), 193–208 (2012) 13. Nagoorgani, A., Latha, S.R.: On irregular fuzzy graphs. Appl. Math. Sci. 6(11), 517–523 (2012) 14. Nagoorgani, A., Malarvizhi, J.: On antipodal fuzzy graph. Appl. Math. Sci. 4(43), 2145–2155 (2010) 15. Nagoorgani, A., Radha, K.: On regular fuzzy graphs. J. Phys. Sci. 12, 33–44 (2008) 16. Rosenfeld, A.: Fuzzy graphs. In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds.) Fuzzy Sets and their Applications, pp. 77–95. Academic Press, New York, USA (1975) 17. Samanta, S., Pal, M.: Irregular bipolar fuzzy graphs. Int. J. Appl. Fuzzy Sets 2, 91–102 (2012) 18. Samanta, S., Pal, M.: Some more results on bipolar fuzzy sets and bipolar fuzzy intersection graphs. J. Fuzzy Math. 22(2), 253–62 (2014) 19. Smith, D.H.: Primitive and Imprimitive graphs. Q. J. Math. 22(4), 551–557 (1971) 20. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference. pp. 305–309 (1994) 21. Zhang, W.-R., Zhang, L.: YinYang bipolar logic and bipolar fuzzy logic. Inf. Sci. 165(3–4), 265–287 (2004) 22. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998) 23. Zhang, W.-R.: Equilibrium relations and bipolar cognitive mapping for online analytical processing with applications in international relations and strategic decision support. IEEE Trans. Syst., Man, Cybern., Part B (Cybern.) 33(2), 295–307 (2003)

Chapter 4

Bipolar Fuzzy Competition Graphs

In this chapter, we discuss the concept of bipolar fuzzy competition graphs and present several notions concerning bipolar fuzzy out neighborhoods, bipolar fuzzy in neighborhoods, bipolar fuzzy open neighborhood graphs, bipolar fuzzy closed neighborhood graphs, bipolar fuzzy k−competition graphs, and underlying bipolar fuzzy graphs. We describe various methods for the construction of bipolar fuzzy competition graphs of certain products of bipolar fuzzy digraphs. Using various constraints, we study the relations of bipolar fuzzy [k]−competition graphs, bipolar fuzzy (k)−competition graphs, and underlying bipolar fuzzy graphs. We elaborate certain algorithms to compute the strength of competition with a number of real-world applications in different fields including food webs, business marketing, politics, wireless communication networks, and social networking. We also study different types of competition graphs under complex bipolar environment. We illustrate the concepts of bipolar fuzzy common enemy graph, bipolar fuzzy competition common enemy graph, bipolar fuzzy niche graph, bipolar influence graph, bipolar fuzzy conflict graph, and bipolar fuzzy confusion graph using these applications. The work in this chapter is from [4, 15, 16].

4.1 Introduction Digraphs are the important mathematical structures to represent point-to- point interconnections for distributed and parallel systems. A digraph is a generalized case of a graph, whose edges have directions. In a digraph, a directed edge from x to y shows that the object can travel from point x to point y but not from y to x. Competition graphs arose in the work of Cohen [7] in connection with an application in ecosystems. Based on this analog, Cohen defined the notion of a competition graph © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_4

161

162

4 Bipolar Fuzzy Competition Graphs

to represent the competition between species in a food web. Competition graphs have many applications including modeling of economic systems, communication over a noisy channel, phylogenetic tree reconstruction, and channel assignment. After the primary motivation of ecological application of competition between species, a lot of work has been done on competition graphs. These representations are crisp graphs, which do not describe all the competitions of real-world problems. In a competition graph, it is assumed that vertices and edges are defined clearly. But in some cases, it is observed that vertices and edges are not defined precisely. For instance, over a noisy channel, a communication channel may have a different network range, radio frequency, bandwidth, and latency. To describe the competition in politics, a politician may have different positions according to leadership quality, public voting, education, and political power. In a social network, the influence rate of different people may be different with respect to socialism, proactiveness, and trading relationship. In a food web, species may be of different types including strong, weak, vegetarian, nonvegetarian, and preys may be energetic, harmful, and digestive. The predator–prey relations can not only be represented by competition graphs but also by the dependence of a specie onto a prey as compared to other species. The terms tasty, digestive, harmful, etc., have no precise meanings and are fuzzy in nature. Fuzzy competition graphs, fuzzy k−competition, and fuzzy p−competition graphs were introduced by Samanta and Pal [13]. Samanta et al. [12] studied various properties of m−step fuzzy competition graphs. Sahoo and Pal [14] studied intuitionistic fuzzy competition graphs, intuitionistic fuzzy p−competition graphs, intuitionistic fuzzy closed neighborhood graphs and applied this idea to ecosystems. Pramanik et al. [10] investigated interesting properties of fuzzy tolerance graphs and fuzzy φ−tolerance competition graphs for particular cases of φ (minimum, maximum, and sum). But all the predator–prey relations can not only be represented by fuzzy competition graphs. For example, a specie is at the same time strong and weak, a prey may be energetic and harmful. Similarly harmony and noise, eligibility and legal disqualification, friend and enemy are bipolar information, which are fuzzy in nature. This idea motivates the necessity of bipolar fuzzy competition graphs. Akram and Dudek [1] discussed certain properties of regular bipolar fuzzy graphs and bipolar fuzzy line graphs. Akram and Dudek, in collaboration with other researchers discussed self-centered interval-valued fuzzy graphs [3]. Alshehri and Akram [6] initiated the concept of bipolar fuzzy competition graphs, studied their application in economic markets, and identified the subscribers or facilitators from a company. Akram and Sarwar studied decision support systems in m−polar fuzzy competition graphs [4], bipolar fuzzy competition graphs of certain products of bipolar fuzzy digraphs, bipolar fuzzy k−competition graphs, underlying bipolar fuzzy graphs, bipolar fuzzy open neighborhood graphs, and bipolar fuzzy closed neighborhood graphs in [16], designed and implemented algorithms for computing the strength of competition in bipolar fuzzy graphs [15]. For further terminologies and studies on complex fuzzy graphs, readers are referred to [2, 5, 8, 9, 11, 17–19].

4.2 Bipolar Fuzzy Competition Graphs

163

4.2 Bipolar Fuzzy Competition Graphs  = (X, E)  be a digraph corresponding to a food web in which vertices represent Let G the species. There is a directed edge yz  between two species y any z if y preys on  = (X, E) of a digraph G  = (X, E)  is an undirected z. A competition graph C(G) graph with the same vertex set X and there is an edge between two vertices y and t  i.e., y and t has a common prey. if there exists z ∈ X such that yz,  tz ∈ E,  = (A, B)  be a bipolar fuzzy graph on a nonempty set X . Definition 4.1 Let G A bipolar fuzzy out neighborhood of a vertex x ∈ X in a bipolar fuzzy digraph  is a bipolar fuzzy set N + (x) = (X x+ , μ p + , μn + ), where X x+ = {y ∈ X | G N (x) N (x)  y) = (0, 0)} and μ p + : X x+ → [0, 1] and μn + : X x+ → [−1, 0] are defined B(x N (x) N (x) p p as μN + (x) (y) = μ B (x y) and μnN + (x) (y) = μnB (x y). A bipolar fuzzy in neighborhood of a vertex x ∈ X in a bipolar fuzzy digraph  is a bipolar fuzzy set N − (x) = (X x− , μ p − , μn − ), where X x− = {y ∈ X | G N (x) N (x) p  B(yx) = (0, 0)} and μN − (x) : X x− → [0, 1] and μnN − (x) : X x− → [−1, 0] are defined p p as μN − (x) (y) = μ B (yx) and μnN − (x) (y) = μnB (yx). p

Definition 4.2 Let B = (μ B , μnB ) be a bipolar fuzzy set on a nonempty set X . The p p height of B is defined as h(B) = max{μ B (x) | x ∈ X }. The depth of B = (μ B , μnB ) n is defined as d(B) = min{μ B (x) | x ∈ X }.  = (A, B)  be a bipolar fuzzy digraph on a nonempty set X . Definition 4.3 Let G  is an undirected A bipolar fuzzy competition graph of a bipolar fuzzy digraph G   and there bipolar fuzzy graph C(G) = (A, R), which has the same vertex set as in G + + is an edge between two vertices x and y, x, y ∈ X , if N (x) ∩ N (y) is nonempty, i.e., x and y prey on at least one common specie. The positive membership and negative membership values of the edge x y are defined as μ R (x y) = (μ A (x) ∧ μ A (y))h(N + (x) ∩ N + (y)), p

p

p

μnR (x y) = (μnA (x) ∨ μnA (y))h(N + (x) ∩ N + (y)).  = (A, B)  be a bipolar fuzzy digraph on X = {a, b, c, d, e, f, g} Example 4.1 Let G as shown in Fig. 4.1. The bipolar fuzzy out neighborhoods of the vertices are given in Table 4.1.  from G  is given as The construction of bipolar fuzzy competition graph C(G) follows: Consider the vertices a1 and a7 as N + (a1 ) ∩ N + (a7 ) = {(a4 , 0.2, −0.3)}, μ R (a1 a7 ) = (μ A (a1 ) ∧ μ A (a7 ))h(N + (a1 ) ∩ N + (a7 )) = 0.04, p

p

p

μnR (a1 a7 ) = (μnA (a1 ) ∨ μnA (a7 ))h(N + (a1 ) ∩ N + (a7 )) = −0.08.

164

4 Bipolar Fuzzy Competition Graphs a2 (0.4, −0.4) a6 (0.7, −0.5)

− 5,

) 0.3

) 0. 6 (0

.6 ,−

(0.

a5

3) 0.

(0.3, − 0.3)

3) 0.

(

(0.4, −0. 5)

a4 (0.5, −0.3)

−

0.5)

0.4 ) 4, − (0.

0.5) 0.4, −

,− .4 (0

a3 (

3, 0.

(0.2, −

)

0. 4

) 0.4

a1 (0.5, −0.4)

4, −

4, − (0.

(0.

a7 (0.2, −0.5)

(0.2, −0.3)

Fig. 4.1 Bipolar fuzzy digraph Table 4.1 Bipolar fuzzy out neighborhoods of vertices x N + (x) x {(a2 , 0.4, −0.4), (a4 , 0.4, −0.3)} {(a3 , 0.4, −0.4), (a5 , 0.4, −0.4)} {(a5 , 0.5, −0.3), (a3 , 0.4, −0.5)} {(a4 , 0.2, −0.3)}

{(a5 , 0.4, −0.5)} {(a6 , 0.3, −0.3)} {(a7 , 0.2, −0.5)}

a3 a5 a6

a4 (0.5, −0.3)

Fig. 4.2 Bipolar fuzzy competition graph

(0.16

a2 (0.4, −0.4)

, −0.1

(0.0

4, −

, −0.5

a5

a3 (0.4

(0 .6 ,−

)

(0 .16 ,−

0.12

0.1 6

6, −

)

(0.1

a1 (0.5, −0.4)

a6 (0.7, −0.5)

2)

0. 6)

a1 a2 a4 a7

N + (x)

)

0.08

)

a7 (0.2, −0.5)

Therefore, R(a1 a7 ) = (0.04, −0.08). Similarly, the membership values of other  is given in Fig. 4.2. edges can be calculated. The bipolar fuzzy competition graph of G  = (A, B)  be a bipolar fuzzy digraph on X = {a, b, c, d, s, f, g} Example 4.2 Let G with A = {(a, 0.3, −0.4), (b, 0.5, −0.6), (c, 0.4, −0.7), (d, 0.6, −0.5), (e, 0.8, −0.4), ( f, 0.7, −0.6), (g, 0.7, −0.8)}. The membership values of the directed edges are represented by the adjacency matrix in Table 4.2. The bipolar fuzzy digraph is shown in Fig. 4.3.

4.2 Bipolar Fuzzy Competition Graphs

165

Table 4.2 Adjacency matrix of bipolar fuzzy relation B x∈X

a

b

c

d

e

f

g

a

(0, 0)

(0.3, −0.4)

(0.3, −0.4)

(0, 0)

(0.3, −0.4)

(0, 0)

(0, 0)

b

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0.5, −0.5)

(0, 0)

c

(0, 0)

(0, 0)

(0, 0)

(0.4, −0.5)

(0, 0)

(0.4, −0.6)

(0.4, −0.6)

d

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0.5, −0.4)

(0, 0)

(0, 0)

e

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0.7, −0.4)

(0.7, −0.4)

f

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

g

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

g(0.7, −0.8)

Fig. 4.3 Bipolar fuzzy digraph )

0.6

(0.3, −0.4)

(0.4, −0.5)

c(0.4, −0.7) (0.4 ,

(0. 3, −

0.4 )

(0.5, −0.4)

d(0.6, −0.5)

(0.7, −0.4)

e(0.8, −0.4)

− 0. 5) .5 , (0

−0

f (0.7, −0.5)

a(0.3, −0.4)

(0.7,

− 4,

−0.4 )

(0.

(0.3, −0.4)

.5)

b(0.5, −0.6)

Table 4.3 Bipolar fuzzy out neighborhoods of vertices u∈X N + (u) a b c d e f g

{(b, 0.3, −0.4), (c, 0.3, −0.4), (e, 0.3, −0.4)} {( f, 0.5, −0.5)} {(d, 0.4, −0.5), ( f, 0.4, −0.5), (g, 0.4, −0.6)} {(e, 0.5, −0.4)} {( f, 0.7, −0.4), (g, 0.7, −0.4)} ∅ ∅

The bipolar fuzzy out neighborhoods of the vertices are given in Table 4.3. The bipolar fuzzy competition graph of bipolar fuzzy digraph, in Fig. 4.3 is shown in Fig. 4.4. The method for the construction of a bipolar fuzzy competition graph is given in Algorithm 4.2.1.

166

4 Bipolar Fuzzy Competition Graphs

Algorithm 4.2.1 Construction of bipolar fuzzy competition graph 1. Competition(X, C, M = [xi j ]n×n ) 2. Begin 3. do i from 1 → n 4. Take N + (xi ) = ∅ 5. do j from 1 → n  i x j ) = (0, 0))then 6. if( B(x p  i x j )) ∈ N + (xi ) 7. (x j , μ B (xi x j ), μnB D(x 8. else p  i x j )) ∈ / N + (xi ) 9. (x j , μ B (xi x j ), μnB D(x 10. end if 11. end do 12. end do 13. do i from 1 → n 14. do j from i + 1 → n 15. if(N + (xi ) ∩ N + (x j ) = ∅)then 16. R(xi x j ) = inf{A(xi ), A(x j )} × h(N + (xi ) ∩ N + (x j )) 17. else 18. R(xi x j ) = (0, 0) 19. end if 20. print*, R(xi x j ) 21. end do 22. end do 23. End Description: Line 1 passes the input of bipolar fuzzy set C and adjacency matrix  = (A, B)  on n vertices x1 , x2 , . . . , xn . M = [xi j ]n×n of a bipolar fuzzy digraph G Lines 3–12 calculate bipolar fuzzy out neighborhood of each vertex. Lines 13–22  calculate and print the adjacency matrix R of bipolar fuzzy competition graph of G. Time Complexity: Line 1 corresponds to the input of the algorithm, its time complexity is O(1). The do loop on line 3 runs n times, therefore its complexity is O(n).

a(0.3, −0.4)

d(0.6, −0.5)

(0.09, −0.12) c(0.4, −0.7)

e(0.8, −0.4)

(0.16, −0.16)

.24 −0

−0 .20

)

16 , (0. )

(0. 25,

Fig. 4.4 Bipolar fuzzy competition graph

b(0.5, −0.6) f (0.7, −0.5)

g(0.7, −0.8)

4.2 Bipolar Fuzzy Competition Graphs

167

The nested loop on line 5 runs n 2 times, its complexity is O(n 2 ). Ignoring the constant, the if conditional on line 6 has complexity O(n 2 ). The complexity of lines 10, 11, and 12 is O(n 2 ), O(n 2 ) and O(n), respectively. Similarly, the time complexity of line 13 is O(n), lines 14 − −21 is O(n(n − 1)), line 22 is O(n). The net time complexity of the algorithm is O(n 2 ). Proof of Correctness: The loop initializes by taking the value i = 1 of do loop which is always true, i.e., the loop runs for the first iteration. As the elements in the adjacency matrix are always zero or nonzero-ordered pairs, therefore for any ith iteration of do loop on line 3, the do loop on line 4 runs n times. If the membership value of xi j is positive, then line 7 is executed otherwise line 9 is executed. For every ith iteration of the loop on line 3, this process continues n times and then increments i for the next iteration maintaining the loop throughout the algorithm. For i = n, the loop calculates the bipolar fuzzy out neighborhood of every vertex and terminates successfully at line 12. Similarly, the loops on lines 13 and 14 maintain and terminate successfully. Theorem 4.1 Let G be a bipolar fuzzy graph then adding a sufficient number of isolated vertices to G produces a bipolar fuzzy competition graph of some bipolar fuzzy digraph. Proof Let G = (A, R) be a bipolar fuzzy graph on nonempty set X , where A = p p (μ A , μnA ) is a bipolar fuzzy set on the set of vertices X and R = (μ R , μnR ) is a bipolar  = (A, B)  as follows: fuzzy relation in X . Construct the bipolar fuzzy digraph G Let x, y ∈ X be any two vertices of G such that R(x y) = (0, 0). Add a vertex βx y , remove the edge x y, and draw directed edges from x and y to βx y such that p

p

p

p

p

μ A (βx y ) = μ A (x) ∧ μ A (y), μ B (xβx y ) = μ B (yβx y ) =

p μ R (x y) , p p μ A (x) ∧ μ A (y)

μnA (βx y ) = μnA (x) ∨ μnA (y), μnB (xβx y ) = μnB (yβx y ) = −

μnR (x y) . μnA (x) ∨ μnA (y)

 such that C(G)  = Continuing this process, we obtain a bipolar fuzzy digraph G G ∪ I , where I is the bipolar fuzzy set of isolated vertices added to G.  The method for the construction of bipolar fuzzy competition graph of the Cartesian product of bipolar fuzzy digraphs is elaborated in Theorem 4.2.  2 ) = (A2 , R2 ) be two bipolar fuzzy  1 ) = (A1 , R1 ) and C(G Theorem 4.2 Let C(G  2 = (A2 , B2 ),  1 = (A1 , B1 ) and G competition graphs of bipolar fuzzy digraphs G respectively. Then  2 ) = G C(G )∗ C(G )∗ ∪ G  ,  1 G C(G 1 2 where G C(G 1 )∗ C(G 2 )∗ is a bipolar fuzzy graph on the crisp graph (X 1 × X 2 , E C(G 1 )∗  1 )∗ and C(G  2 )∗ are the crisp competition graphs of G  1 and G  2, E C(G 2 )∗ ), C(G   respectively. G is a bipolar fuzzy graph on (X 1 × X 2 , E ) such that

168

4 Bipolar Fuzzy Competition Graphs

(1) E  = {(a1 , a2 )(b1 , b2 ) | b1 ∈ N − (a1 )∗ , b2 ∈ N + (a2 )∗ }, E C(G 1 )∗ E C(G 2 )∗ = (2)

{(a1 , a2 )(a1 , b2 ) | a1 ∈ X 1 , a2 b2 ∈ E C (G 2 )∗ } ∪ {(a1 , a2 )(b1 , a2 ) | a2 ∈ X 2 , a1 b1 ∈ E C (G 1 )∗ }. p

μA

p

1 A2

p

(a1 , a2 ) = μ A1 (a1 ) ∧ μ A2 (a2 ), μnA

p

p

1 A2

p

(a1 , a2 ) = μnA1 (a1 ) ∨ μnA2 (a2 ), (a1 , a2 ) ∈ X 1 × X 2 .

p

p

p

p

p

p

p

p

p

p

p

p

p

(3)

μ R ((a1 , a2 )(a1 , b2 )) = [μ A1 (a1 ) ∧ μ A2 (a2 ) ∧ μ A2 (b2 )] × ∨x2 [μ A1 (a1 ) ∧ μ  (a2 x2 ) ∧ μ  (b2 x2 )], B2 B2 (a1 , a2 )(b1 , a2 ) ∈ E C (G )∗ E C (G )∗ , x2 ∈ (N + (a2 ) ∩ N + (b2 ))∗ , 1 2

(4)

μnR ((a1 , a2 )(a1 , b2 )) = [μnA1 (a1 ) ∨ μnA2 (a2 ) ∨ μnA2 (b2 )] × ∨x2 [μ A1 (a1 ) ∧ μ  (a2 x2 ) ∧ μ  (b2 x2 )], B2 B2 (a1 , a2 )(b1 , a2 ) ∈ E C (G )∗ E C (G )∗ , x2 ∈ (N + (a2 ) ∩ N + (b2 ))∗ , 1 2

(5)

μ R ((a1 , a2 )(b1 , a2 )) = [μ A1 (a1 ) ∧ μ A1 (b1 ) ∧ μ A2 (a2 )] × ∨x1 [μ A2 (a2 ) ∧ μ  (a1 x1 ) ∧ μ  (b1 x1 )], B1 B1 (a1 , a2 )(b1 , a2 ) ∈ E C (G )∗ E C (G )∗ , x1 ∈ (N + (a1 ) ∩ N + (b1 ))∗ , 1 2

(6)

μnR ((a1 , a2 )(b1 , a2 )) = [μnA1 (a1 ) ∨ μ A1 (b1 ) ∨ μnA2 (a2 )] × ∨x1 [μ A2 (a2 ) ∧ μ  (a1 x1 ) ∧ μ  (b1 x1 )], B1 B1 (a1 , a2 )(b1 , a2 ) ∈ E C (G )∗ E C (G )∗ , x1 ∈ (N + (a1 ) ∩ N + (b1 ))∗ , 1 2

p

p

p

p

p

p

(7) μ R ((a1 , a2 )(b1 , b2 )) = p

p

p

p

p

p B1

p

p B2

p

p B1

p

p B2

[μ A1 (a1 ) ∧ μ A1 (b1 ) ∧ μ A2 (a2 ) ∧ μ A2 (b2 )] × [μ A1 (a1 ) ∧ μ  (b1 a1 ) ∧ μ A2 (b2 ) ∧ μ  (a2 b2 )],

(a1 , b1 )(a2 , b2 ) ∈ E  ,

(8) μnR ((a1 , a2 )(b1 , b2 )) =

[μnA1 (a1 ) ∨ μnA1 (b1 ) ∨ μnA2 (a2 ) ∨ μnA2 (b2 )] × [μ A1 (a1 ) ∧ μ  (b1 a1 ) ∧ μ A2 (b2 ) ∧ μ  (a2 b2 )],

(a1 , b1 )(a2 , b2 ) ∈ E  .

Proof Consider an edge (a1 , a2 )(b1 , b2 ) of G C(G 1 )∗ C(G 2 )∗ ∪ G  . Then there are three cases, Case 1: a1 = b1 , a2 = b2 , then (a1 , a2 )(a1 , b2 ) ∈ E C(G 1 )∗ E C(G 2 )∗ . By conditions (3) and (4), p

p

p

p

μ R ((a1 , a2 )(a1 , b2 )) = [μ A (a1 ) ∧ μ A (a2 ) ∧ μ A (b2 )] 1 2 2 p

p

p

× ∨x2 [μ A (a1 ) ∧ μ  (a2 x2 ) ∧ μ  (b2 x2 )], 1 B B 2

p

p

(4.1)

2

p

p

p

μ R ((a1 , a2 )(a1 , b2 )) = [μ A (a1 ) ∧ μ A (a2 )} ∧ {μ A (a1 ) ∧ μ A (b2 )] 1 2 1 2 p

p

p

p

× ∨x2 [{μ A (a1 ) ∧ μ  (a2 x2 )}, ∧{μ A (a1 ) ∧ μ  (b2 x2 )}], 1 1 B B 2

p

p

2

p

μ R ((a1 , a2 )(a1 , b2 )) = [μ A (a1 , a2 ) ∧ μ A (a1 , b2 )] p B

p B

× ∨x2 [μ  ((a1 , a2 )(a1 , x2 )) ∧ μ  ((a1 , b2 )(a1 , x2 ))], p

p

p

p

μ R ((a1 , a2 )(a1 , b2 )) =[μ A (a1 , a2 ) ∧ μ A (a1 , b2 )] × ∨x2 [μN + (a ,a )∩N + (a ,b ) (a1 , x2 )], 1 2 1 2

μ R ((a1 , a2 )(a1 , b2 )) =[μ A (a1 , a2 ) ∧ μ A (a1 , b2 )] × h(N + (a1 , a2 ) ∩ N + (a1 , b2 )). p

p

p

(4.2)

4.2 Bipolar Fuzzy Competition Graphs

169 p

p B2

p B2

μnR ((a1 , a2 )(a1 , b2 )) =[μnA1 (a1 ) ∨ μnA2 (a2 ) ∨ μnA2 (b2 )] × ∨x2 [μ A1 (a1 ) ∧ μ  (a2 x2 ) ∧ μ  (b2 x2 )], μnR ((a1 , a2 )(a1 , b2 )) = [{μnA1 (a1 ) ∨ μnA2 (a2 )} ∨ {μnA1 (a1 ) ∨ μnA2 (b2 )}] p

p B2

p

p B2

× ∨x2 [{μ A1 (a1 ) ∧ μ  (a2 x2 )}, ∧{μ A1 (a1 ) ∧ μ  (b2 x2 )}], μnR ((a1 , a2 )(a1 , b2 )) = [μnA (a1 , a2 ) ∨ μnA (a1 , b2 )] p B

p B

× ∨x2 [μ  ((a1 , a2 )(a1 , x2 )) ∧ μ  ((a1 , b2 )(a1 , x2 ))], p

μnR ((a1 , a2 )(a1 , b2 )) =[μnA (a1 , a2 ) ∨ μnA (a1 , b2 )] × ∨x2 [μN + (a

1 ,a2 )∩N

+ (a

1 ,b2 )

(a1 , x2 )],

μnR ((a1 , a2 )(a1 , b2 )) =[μnA (a1 , a2 ) ∨ μnA (a1 , b2 )] × h(N + (a1 , a2 ) ∩ N + (a1 , b2 )).

(4.3)

From Eqs. (4.2) and (4.3), (a1 , a2 )(a1 , b2 ) is an edge of C(G 1 G 2 ). Case 2: If a1 = b1 , a2 = b2 , then (a1 , a2 )(b1 , a2 ) ∈ E C(G 1 )∗ E C(G 2 )∗ . Using conditions (5) and (6), p

p

p

p

p

p B1

p B1

μ R ((a1 , a2 )(b1 , a2 )) =[μ A1 (a1 ) ∧ μ A1 (b1 ) ∧ μ A2 (a2 )] × ∨x1 [μ A2 (a2 ) ∧ μ  (a1 x1 ) ∧ μ  (b1 x1 )], p

p

p

p

(4.4)

p

μ R ((a1 , a2 )(b1 , a2 )) = [{μ A1 (a1 ) ∧ μ A2 (a2 )} ∧ {μ A1 (b1 ) ∧ μ A2 (a2 )}] p p p p B1 B1 p p = [μ A (a1 , a2 ) ∧ μ A (b1 , a2 )] p p × ∨x1 [μ  ((a1 , a2 )(x1 , a2 )) ∧ μ  ((b1 , a2 )(x1 , a2 ))] B B p p p =[μ A (a1 , a2 ) ∧ μ A (b1 , a2 )] × ∨x1 [μN + (a ,a )∩N + (b ,a ) (x1 , a2 )], 1 2 1 2 p p + =[μ A (a1 , a2 ) ∧ μ A (b1 , a2 )] × h(N (a1 , a2 ) ∩ N + (b1 , a2 )).

× ∨x1 [{μ A2 (a2 ) ∧ μ  (a1 x1 )}, ∧{μ A2 (a2 ) ∧ μ  (b1 x1 )}],

p μ R ((a1 , a2 )(b1 , a2 ))

p

μ R ((a1 , a2 )(b1 , a2 )) p

μ R ((a1 , a2 )(b1 , a2 ))

p

p B1

(4.5)

p B1

μnR ((a1 , a2 )(b1 , a2 )) =[μnA1 (a1 ) ∨ μnA1 (b1 ) ∨ μnA2 (a2 )] × ∨x1 [μ A2 (a2 ) ∧ μ  (a1 x1 ) ∧ μ  (b1 x1 )],

(4.6)

μnR ((a1 , a2 )(b1 , a2 )) =[{μnA1 (a1 ) ∨ μnA2 (a2 )} ∨ {μnA1 (b1 ) ∨ μnA2 (a2 )}] p

p B1

p

p B1

× ∨x1 [{μ A2 (a2 ) ∧ μ  (a1 x1 )}, ∧{μ A2 (a2 ) ∧ μ  (b1 x1 )}], μnR ((a1 , a2 )(b1 , a2 )) =[μnA (a1 , a2 ) ∨ μnA (b1 , a2 )] p p B B p n n =[μ A (a1 , a2 ) ∨ μ A (b1 , a2 )] × ∨x1 [μN + (a ,a )∩N + (b ,a ) (x1 , a2 )] 1 2 1 2

× ∨x1 [μ  ((a1 , a2 )(x1 , a2 )) ∧ μ  ((b1 , a2 )(x1 , a2 ))],

μnR ((a1 , a2 )(b1 , a2 ))

μnR ((a1 , a2 )(b1 , a2 )) =[μnA (a1 , a2 ) ∨ μnA (b1 , a2 )] × h(N + (a1 , a2 ) ∩ N + (b1 , a2 )).

(4.7)

Equations (4.5) and (4.7) show that (a1 , a2 )(b1 , a2 ) is an edge of C(G 1 G 2 ). Case 3: a1 = b1 , a2 = b2 , then (a1 , a2 )(b1 , b2 ) ∈ E  . Using conditions (7) and (8), p

p

p

p

p

μ R ((a1 , a2 )(b1 , b2 )) =[μ A1 (a1 ) ∧ μ A1 (b1 ) ∧ μ A2 (a2 ) ∧ μ A2 (b2 )] p

p p p B2 B1 p p p p =[μ A (a1 , a2 ) ∧ μ A (b1 , b2 )] × [μ  ((a1 , a2 )(a1 , b2 )) ∧ μ  ((b1 , b2 )(a1 , b2 ))], B B p p p =[μ A (a1 , a2 ) ∧ μ A (b1 , b2 )] × μN + (a ,a )∩N + (b ,b ) (a1 , b2 ), 1 2 1 2

× [μ A1 (a1 ) ∧ μ  (a2 b2 ) ∧ μ A2 (b2 ) ∧ μ  (b1 a1 )],

p

μ R ((a1 , a2 )(b1 , b2 )) p

μ R ((a1 , a2 )(b1 , b2 ))

μ R ((a1 , a2 )(b1 , b2 )) =[μ A (a1 , a2 ) ∧ μ A (b1 , b2 )] × h(N + (a1 , a2 ) ∩ N + (b1 , b2 )). p

p

p

(4.8)

170

4 Bipolar Fuzzy Competition Graphs

Fig. 4.5 Bipolar fuzzy  1 and G 2 digraphs G

Table 4.4 Bipolar fuzzy out neighborhoods u ∈ X1

N + (u)

{(b1 , 0.4, −0.5)} ∅ {(b1 , 0.3, −0.4)} {(c1 , 0.3, −0.4)}

a1 b1 c1 d1

μnR ((a1 , a2 )(b1 , b2 )) = p

p B2

p

p B1

[μnA1 (a1 ) ∨ μnA1 (b1 ) ∨ μnA2 (a2 ) ∨ μnA2 (b2 )] × [μ A1 (a1 ) ∧ μ  (a2 b2 ) ∧ μ A2 (b2 ) ∧ μ  (b1 a1 )], p p B B p =[μnA (a1 , a2 ) ∨ μnA (b1 , b2 )] × μN + (a ,a )∩N + (b ,b ) (a1 , b2 ), 1 2 1 2 n n =[μ A (a1 , a2 ) ∨ μ A (b1 , b2 )] × h(N + (a1 , a2 ) ∩ N + (b1 , b2 )).

μnR ((a1 , a2 )(b1 , b2 )) =[μnA (a1 , a2 ) ∨ μnA (b1 , b2 )] × [μ  ((a1 , a2 )(a1 , b2 )) ∧ μ  ((b1 , b2 )(a1 , b2 ))], μnR ((a1 , a2 )(b1 , b2 )) μnR ((a1 , a2 )(b1 , b2 ))

(4.9)

Equations (4.8) and (4.9) imply that (a1 , a2 )(b1 , b2 ) is an edge of C(G 1 G 2 ). Hence C(G 1 G 2 ) ⊆ G C(G 1 )∗ C(G 2 )∗ ∪ G  . Conversely, it can be proved on the same lines that G C(G 1 )∗ C(G 2 )∗ ∪ G  ⊆ C(G 1 G 2 ). It completes the proof.  The idea of the construction of bipolar fuzzy competition graph of the Cartesian product is explained with the following two examples.  1 = (A1 , B1 ) and G  2 = (A2 , B2 ) be two bipolar fuzzy digraphs Example 4.3 Let G X 1 and X 2 , respectively, as shown in Fig. 4.5. The bipolar fuzzy out neighborhoods  2 are given in Tables 4.4 and 4.5.  1 and G of vertices of G  1 ) and C(G  2 ) are given in Fig. 4.6. The bipolar fuzzy competition graphs C(G Now construct the bipolar fuzzy competition graph G C(G 1 )∗ C(G 2 )∗ ∪ G  = (A, R)  2 )∗ using Theorem 4.2, where A = (μ Ap , μnA ) and R = (μ Rp , μnR ).  1 )∗ and C(G from C(G The two sets of edges according to condition (1) are computed as follows:

4.2 Bipolar Fuzzy Competition Graphs

171

Table 4.5 Bipolar fuzzy out neighborhoods v ∈ X2

N + (v)

{(c2 , 0.2, −0.4)} {(c2 , 0.5, −0.5)} ∅

a2 b2 c2

Fig. 4.6 Bipolar fuzzy competition graphs

E C (G 1 )∗ E C (G 2 )∗ = {(a1 , a2 )(a1 , b2 ), (b1 , a2 )(b1 , b2 ), (c1 , a2 )(c1 , b2 ), (d1 , a2 )(d1 , b2 ), (a1 , a2 )(c1 , a2 ), (a1 , b2 )(c1 , b2 ), (a1 , c2 )(c1 , c2 )}, E  = {(b1 , a2 )(a1 , c2 ), (b1 , a2 )(c1 , c2 ), (b1 , b2 )(a1 , c2 ), (b1 , b2 )(c1 , c2 ), (c1 , a2 )(d1 , c2 ), (c1 , b2 )(d1 , c2 )}.

The degrees of membership of the edges can be calculated according to conditions (3)–(8) as p

p

p

R((a1 , a2 )(a1 , b2 )) = [A1 (a1 ) ∧ A2 (a2 ) ∧ A2 (b2 )] × [μ A (a1 ) ∧ μ  (a2 c2 ) ∧ μ  (b2 c2 )} 1 B B 2

p

p

2

p

= (μ A (a1 ) ∧ μ A (a2 ) ∧ μ A (b2 ), μnA (a1 ) ∨ μnA (a2 ) ∨ μnA (b2 )) 1

p

2

2

p B2

1

2

2

p B2

× [μ A (a1 ) ∧ μ  (a2 c2 ) ∧ μ  (b2 c2 )} 1

= (0.2, −0.4) × 0.2 = (0.04, −0.08). p p B1 B2 p p p p = (μ A (b1 ) ∧ μ A (a1 ) ∧ μ A (a2 ) ∧ μ A (c2 ), μnA (b1 ) ∨ μnA (a1 )∨ 1 1 1 1 2 2 p p μnA (a2 ) ∨ μnA (c2 )) × [μ  (a1 b1 ) ∧ μ  (a2 c2 )] 2 2 B B

R((b1 , a2 )(a1 , c2 )) = [A1 (b1 ) ∧ A1 (a1 ) ∧ A2 (a2 ) ∧ A2 (c2 )] × [μ  (a1 b1 ) ∧ μ  (a2 c2 )]

1

2

= (0.2, −0.4) × 0.2 = (0.04, −0.08).

The membership degrees of all the adjacent vertices of G C(G 1 )∗ C(G 2 )∗ and G  are given in Table 4.6.

172

4 Bipolar Fuzzy Competition Graphs

Table 4.6 Membership values of constructed edges (x1 , x2 )(y1 , y2 ) R((x1 , x2 )(y1 , y2 )) p

(b1 , a2 )(b1 , b2 )

[A1 (b1 ) ∧ A2 (a2 ) ∧ A2 (b2 )] × [(μ A1 (b1 ) ∧ p p μ  (a2 c2 ) ∧ μ  (b2 c2 )] = (0.04, −0.08)

(c1 , a2 )(c1 , b2 )

[A1 (c1 ) ∧ A2 (a2 ) ∧ A2 (b2 )] × [μ A1 (c1 ) ∧ p p μ  (a2 c2 ) ∧ μ  (b2 c2 )] = (0.04, −0.08)

(d1 , a2 )(d1 , b2 )

[A1 (d1 ) ∧ A2 (a2 ) ∧ A2 (b2 )] × [μ A1 (d1 ) ∧ p p μ  (a2 c2 ) ∧ μ  (b2 c2 )] = (0.04, −0.08)

(a1 , a2 )(c1 , a2 )

[A1 (a1 ) ∧ A1 (c1 ) ∧ A2 (a2 )] × [μ A2 (a2 ) ∧ p p μ  (a1 b1 ) ∧ μ  (c1 b1 )] = (0.04, −0.08)

(a1 , b2 )(c1 , b2 )

[A1 (a1 ) ∧ A1 (c1 ) ∧ A2 (b2 )] × [μ A2 (b2 ) ∧ p p μ  (a1 b1 ) ∧ μ  (c1 b1 )] = (0.09, −0.12)

(a1 , c2 )(c1 , c2 )

[A1 (a1 ) ∧ A1 (c1 ) ∧ A2 (c2 )] × [μ A2 (c2 ) ∧ p p μ  (a1 b1 ) ∧ μ  (c1 b1 )] = (0.09, −0.12)

(b1 , a2 )(c1 , c2 )

[A1 (b1 ) ∧ A1 (c1 ) ∧ A2 (a2 ) ∧ A2 (c2 )] × p p p [μ A1 (b1 ) ∧ μ  (c1 b1 ) ∧ μ A2 (c2 )∧

B2

B2

B2

B2

B2

B2

B1

B1

B1

B1

B1

B1

p B2

p

p

p

p

p

B1

μ  (a2 c2 )] = (0.04, −0.08) (b1 , b2 )(a1 , c2 )

[A1 (b1 ) ∧ A1 (a1 ) ∧ A2 (b2 ) ∧ A2 (c2 )] × p p p [μ A1 (b1 ) ∧ μ  (a1 b1 ) ∧ μ A2 (c2 )∧ p B2

B1

μ  (b2 c2 )] = (0.16, −0.20) (b1 , b2 )(c1 , c2 )

[A1 (b1 ) ∧ A1 (c1 ) ∧ A2 (b2 ) ∧ A2 (c2 )] × p p p [μ A1 (b1 ) ∧ μ  (c1 b1 ) ∧ μ A2 (c2 )∧ p B2

B1

μ  (b2 c2 )] = (0.09, −0.12) (c1 , a2 )(d1 , c2 )

[A1 (c1 ) ∧ A1 (d1 ) ∧ A2 (a2 ) ∧ A2 (c2 )] × p p p [μ A1 (c1 ) ∧ μ  (d1 c1 ) ∧ μ A2 (c2 )∧ p B2

B1

μ  (a2 c2 )] = (0.04, −0.08) (c1 , b2 )(d1 , c2 )

[A1 (c1 ) ∧ A1 (d1 ) ∧ A2 (a2 ) ∧ A2 (b2 )] × p p p [μ A1 (c1 ) ∧ μ  (d1 c1 ) ∧ μ A2 (c2 )∧ p B2

B1

μ  (b2 c2 )] = (0.09, −0.12)

The bipolar fuzzy graph obtained using this method is given in Fig. 4.7, where solid lines show the part of bipolar fuzzy graph obtained from G C(G 1 )∗ G C(G 2 )∗ and dotted lines represent the part G  .  2 of bipolar fuzzy digraphs G  1 and G  2 is shown in  1 G The Cartesian product G  2 are calculated  1 G Fig. 4.8. The bipolar fuzzy out neighborhoods of vertices of G in Table 4.7.  2 is shown in Fig. 4.9. It is clear  1 G The bipolar fuzzy competition graph of G  2 ).  1 G from Figs. 4.7 and 4.9 that G C(G 1 )∗ C(G 2 )∗ ∪ G  ∼ = C(G

4.2 Bipolar Fuzzy Competition Graphs

173

a2

a1

(0.4, −0.5)

(0.2, −0.4)

(0.04, −0.08)

(0.04, −0.08)

c1

(0.2, −0.4)

(0.2, −0.4)

d1

c2

0.08) (0.04, −

(0.4, −0.5)

0) 0.2 − (0.09, −0.14) , .16 (0 (0.5, −0.5) (0.04, −0.08) (0.5, −0.6)

(0 .09 ,−

(0.3, −0.4) (0.04, −0.08)

0.1 2)

(0.09, −0.12)

b1

b2

(0.3, −0.4) (0.04, −0.08 ) (0 .0 9, (0 . 0 − 4, − 0. 0.0 12 8) (0.04, −0.08) )

(0.2, −0.4)

(0.5, −0.4)

(0.5, −0.4)

Fig. 4.7 Bipolar fuzzy graph G C (G 1 )∗ C (G 2 )∗ ∪ G  Fig. 4.8 Cartesian product  1 G 2 G

a2

(0.2, a1

(0.2, −0.4)

(0.4, −0.5) (0.4, −0.5)

(0.2, −0.4)

(0.2, b1

d1

(0.4, −0.5)

(0.3, −0.4) )

(0.2, −0.4)

(0.3, −0.4)

(0.2, −0.4)

(0.3, −0.4) −0.4)

(0.2, −0.4)

(0.4, −0.5) (0.4, −0.5)

)

−0.4 (0.2,

(0.2,

c2

−0.4

(0.5, −0.6)

(0.2, −0.4) (0.2, −0.4)

c1

b2

)

−0.4

(0.5, −0.5) (0.5, −0.6) (0.3, −0.4) (0.3, −0.4) (0.3, −0.4) (0.3, −0.4) (0.5, −0.4)

(0.5, −0.4)

(0.5, −0.4)

 1 G 2 Table 4.7 Bipolar fuzzy out neighborhoods of vertices in G (x, y)

N + ((x, y))

(x, y)

N + ((x, y))

(a1 , a2 )

{(a1 , c2 ), 0.2, −0.4), (b1 , a2 ), 0.2, −0.4)}

(b1 , a2 )

{(b1 , c2 ), 0.2, −0.4)}

(a1 , b2 )

{(a1 , c2 ), 0.4, −0.5), (b1 , b2 ), 0.4, −0.5)}

(a1 , c2 )

{(b1 , c2 ), 0.4, −0.5)}

(c1 , b2 )

{(b1 , b2 ), 0.3, −0.4), (c1 , c2 ), 0.3, −0.4)}

(b1 , b2 )

{(b1 , c2 ), 0.5, −0.5)}

(d1 , a2 )

{(d1 , c2 ), 0.2, −0.4), (c1 , a2 ), 0.2, −0.4)}

(c1 , a2 )

{(c1 , c2 ), 0.2, −0.4)}

(d1 , b2 )

{(d1 , c2 ), 0.5, −0.4), (c1 , b2 ), 0.3, −0.4)}

(b1 , a2 )

{(b1 , c2 ), 0.2, −0.4)}

(d1 , c2 )

{(c1 , c2 ), 0.3, −0.4)}

(c1 , c2 )

{(b1 , c2 ), 0.3, −0.4)}

174

4 Bipolar Fuzzy Competition Graphs

Fig. 4.9 Bipolar fuzzy competition graph  1 G  2) C (G

a2 (0.2, −0.4)

(0.04, −0.08)

(0.04, −0.08)

c2

0. (0.04, −

08)

(0.4, −0.5)

)

(0.5, −0.6)

(0.3, −0.4) (0.04, −0.08)

, .16 (0

−0

.20

(0.5, −0.5) (0 .09 ,− 0.1 2)

(0.09, −0.12)

(0.09, −0.14) (0.04, −0.08)

(0.2, −0.4)

(0.2, −0.4)

b2 (0.4, −0.5)

(0.3, −0.4) (0.04, −0.08 ) (0 .0 9, (0.0 − 4, − 0. 0.0 12 8) ) (0.04, −0.08)

(0.2, −0.4)

(0.5, −0.4)

(0.5, −0.4)

Fig. 4.10 Bipolar fuzzy  5 and G 6 digraphs G

 5 = (A5 , B5 ) Example 4.4 Consider another example of bipolar fuzzy digraphs G   and G 6 = (A6 , B6 ) on X 5 and X 6 as given in Fig. 4.10. The bipolar fuzzy competition  6 ) are given in Fig. 4.11.  5 ) and C(G graphs C(G  5 G  6) = Apply Theorem 4.2 to construct bipolar fuzzy competition graph C(G ∗ ∗   (A, R) from C(G 5 ) and C(G 6 ) . The sets of edges constructed using condition (1) are E C(G )∗ E C(G )∗ = {(a5 , a6 )(a5 , d6 ), (b5 , a6 )(b5 , d6 ), (c5 , a6 )(c5 , d6 ), (d5 , a6 )(d5 , d6 ), (a5 , a6 )(d5 , a6 ), 6 5 (a5 , b6 )(d5 , b6 ), (a5 , c6 )(d5 , c6 ), (a5 , d6 )(d5 , d6 ), (c5 , a6 )(d5 , a6 ), (c5 , b6 )(d5 , b6 ), (c5 , c6 )(d5 , c6 ), (c5 , d6 )(d5 , d6 ), (a5 , a6 )(c5 , a6 ), (a5 , b6 )(c5 , b6 ), (a5 , c6 )(c5 , c6 ), (a5 , d6 )(c5 , d6 )}. E  = {(b5 , a6 )(a5 , b6 ), (b5 , a6 )(a5 , c6 ), (b5 , d6 )(a5 , b6 ), (b5 , d6 )(a5 , c6 ), (b5 , a6 )(c5 , b6 ), (b5 , a6 )(c5 , c6 ), (b5 , d6 )(c5 , b6 ), (b5 , d6 )(c5 , c6 ), (d5 , a6 )(b5 , b6 ), (d5 , a6 )(b5 , c6 ), (d5 , d6 )(b5 , b6 ), (d5 , d6 )(b5 , c6 ), (d5 , a6 )(c5 , b6 ), (d5 , a6 )(c5 , c6 ), (d5 , d6 )(c5 , b6 ), (d5 , d6 )(c5 , c6 )}.

The degrees of membership of the edges according to conditions (3) to (8) are calculated as follows:

4.2 Bipolar Fuzzy Competition Graphs

175

Fig. 4.11 Bipolar fuzzy  5) competition graphs C (G  6) and C (G

p

p B6

p B6

R((a5 , a6 )(a5 , d6 )) = [A5 (a5 ) ∧ A6 (a6 ) ∧ A6 (d6 )] × [{μ A5 (a5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )} p

p p B6 B6 p p p = [(μ A5 (a5 ) ∧ μ A6 (a6 ) ∧ μ A6 (d6 ), μnA5 (a5 ) ∨ μnA6 (a6 ) ∨ μnA6 (d6 ))] p p p p p p × [{μ A5 (a5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )} ∨ {μ A5 (a5 ) ∧ μ  (a6 b6 ) ∧ μ  (d6 b6 )}] B6 B6 B6 B6

∨ {μ A5 (a5 ) ∧ μ  (a6 b6 ) ∧ μ  (d6 b6 )}]‘

= (0.3, −0.5) × 0.3 = (0.09, −0.15). p

p B5

R((b5 , a6 )(a5 , c6 )) = [A5 (b5 ) ∧ A5 (a5 ) ∧ A6 (a6 ) ∧ A6 (c6 )] × [μ A5 (b5 ) ∧ μ  (a5 b5 )∧ p

p B6

μ A6 (c6 ) ∧ μ  (a6 c6 )} p

p

p

p

= (μ A5 (b5 ) ∧ μ A5 (a5 ) ∧ μ A6 (a6 ) ∧ μ A6 (c6 ), μnA5 (b5 ) ∨ μnA5 (a5 ) ∨ μnA6 (a6 ) p p p p ∨ μnA6 (c6 )) × {μ A5 (b5 ) ∧ μ  (a5 b5 ) ∧ μ A6 (c6 ) ∧ μ  (a6 c6 )} B5 B6

= (0.3, −0.4) × 0.3 = (0.09, −0.12).

The way of calculating the membership degrees of the edges from E C(G 1 )∗ E C(G 2 )∗ and E  is described in Table 4.8. The bipolar fuzzy graph obtained using this method is shown in Fig. 4.12. The solid lines show the part of the bipolar fuzzy graph obtained from G C(G 5 )∗ G C(G 6 )∗ and dashed lines show the part of the bipolar fuzzy graph obtained from G  .  6 is shown in Fig. 4.13 and its bipolar fuzzy  5 and G The Cartesian product of G competition graph is given in Fig. 4.14, which clearly shows that G C(G 5 )∗ C(G 6 )∗ ∪  6 ).  5 G G ∼ = C(G The method for the construction of bipolar fuzzy competition graph of the direct product of bipolar fuzzy digraphs from respective bipolar fuzzy competition graphs of the bipolar fuzzy digraphs is discussed in Theorem 4.3.  1 = (A1 , B1 ) and G  2 = (A2 , B2 ) be two bipolar fuzzy digraphs. Theorem 4.3 Let G   C(G 1 ) = (A1 , R1 ) and C(G 2 ) = (A2 , R2 ) are the bipolar fuzzy competition graphs without isolated vertices such that neither is a bipolar fuzzy empty graph. Then

176

4 Bipolar Fuzzy Competition Graphs

Table 4.8 Adjacent vertices of G C (G 5 )∗ C (G 6 )∗ ∪ G  (x1 , x2 )(y1 , y2 )

R((x1 , x2 )(y1 , y2 )) p p p p B6 B6 5 5 p p ∧μ  (a6 b6 ) ∧ μ  (d6 b6 )}] = (0.09, −0.15) B6 B6 p p p p [ A5 (c5 ) ∧ A6 (a6 ) ∧ A6 (d6 )] × [{μ A (c5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )} ∨ {μ A (c5 ) B6 B6 5 5 p p ∧μ  (a6 b6 ) ∧ μ  (d6 b6 )}] = (0.09, −0.12) B6 B6 p p p p [ A5 (d5 ) ∧ A6 (a6 ) ∧ A6 (d6 )] × [{μ A (d5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )} ∨ {μ A (d5 ) B6 B6 5 5 p p ∧μ  (a6 b6 ) ∧ μ  (d6 b6 )}] = (0.09, −0.15) B6 B6 p p p [ A5 (a5 ) ∧ A5 (c5 ) ∧ A6 (a6 )] × [μ A (a6 ) ∧ μ  (a5 b5 ) ∧ μ  (c5 b5 )] = (0.09, −0.12) B5 B5 6 p p p [ A5 (a5 ) ∧ A5 (c5 ) ∧ A6 (b6 )] × [μ A (b6 ) ∧ μ  (a5 b5 ) ∧ μ  (c5 b5 )] = (0.20, −0.16) B5 B5 6 p p p [ A5 (a5 ) ∧ A5 (c5 ) ∧ A6 (c6 )] × [μ A (c6 ) ∧ μ  (a5 b5 ) ∧ μ  (c5 b5 )] = (0.16, −0.16) B5 B5 6 p p p [ A5 (a5 ) ∧ A5 (c5 ) ∧ A6 (d6 )] × [μ A (d6 ) ∧ μ  (a5 b5 ) ∧ μ  (c5 b5 )] = (0.20, −0.16) B5 B5 6 p p p [ A5 (a5 ) ∧ A5 (d5 ) ∧ A6 (a6 )] × [μ A (a6 ) ∧ μ  (a5 b5 ) ∧ μ  (d5 b5 )] = (0.09, −0.15) B5 B5 6 p p p [ A5 (a5 ) ∧ A5 (d5 ) ∧ A6 (b6 )] × [μ A (b6 ) ∧ μ  (a5 b5 ) ∧ μ  (d5 b5 )] = (0.20, −0.20) B5 B5 6 p p p [ A5 (a5 ) ∧ A5 (d5 ) ∧ A6 (c6 )] × [μ A (c6 ) ∧ μ  (a5 b5 ) ∧ μ  (d5 b5 )] = (0.16, −0.16) B5 B5 6 p p p [ A5 (a5 ) ∧ A5 (d5 ) ∧ A6 (d6 )] × [μ A (d6 ) ∧ μ  (a5 b5 ) ∧ μ  (d5 b5 )] = (0.20, −0.20) B5 B5 6 p p p [ A5 (c5 ) ∧ A5 (d5 ) ∧ A6 (a6 )] × [μ A (a6 ) ∧ μ  (c5 b5 ) ∧ μ  (d5 b5 )] = (0.09, −0.12) B5 B5 6 p p p [ A5 (c5 ) ∧ A5 (d5 ) ∧ A6 (b6 )] × [μ A (b6 ) ∧ μ  (c5 b5 ) ∧ μ  (d5 b5 )] = (0.24, −0.16) B5 B5 6 p p p [ A5 (c5 ) ∧ A5 (d5 ) ∧ A6 (c6 )] × [μ A (c6 ) ∧ μ  (c5 b5 ) ∧ μ  (d5 b5 )] = (0.16, −0.16) B5 B5 6 p p p [ A5 (c5 ) ∧ A5 (d5 ) ∧ A6 (d6 )] × [μ A (d6 ) ∧ μ  (c5 b5 ) ∧ μ  (d5 b5 )] = (0.20, −0.16) B5 B5 6 p p p p [ A5 (b5 ) ∧ A5 (a5 ) ∧ A6 (a6 ) ∧ A6 (b6 )] × [μ A (b5 ) ∧ μ  (a5 b5 ) ∧ μ A (b6 ) ∧ μ  (a6 b6 )] B5 B6 6 5

(b5 , a6 )(b5 , d6 ) [ A5 (b5 ) ∧ A6 (a6 ) ∧ A6 (d6 )] × [{μ A (b5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )} ∨ {μ A (b5 ) (c5 , a6 )(c5 , d6 )

(d5 , a6 )(d5 , d6 )

(a5 , a6 )(c5 , a6 ) (a5 , b6 )(c5 , b6 ) (a5 , c6 )(c5 , c6 ) (a5 , d6 )(c5 , d6 ) (a5 , a6 )(d5 , a6 ) (a5 , b6 )(d5 , b6 ) (a5 , c6 )(d5 , c6 ) (a5 , d6 )(d5 , d6 ) (c5 , a6 )(d5 , a6 ) (c5 , b6 )(d5 , b6 ) (c5 , c6 )(d5 , c6 ) (c5 , d6 )(d5 , d6 ) (b5 , a6 )(a5 , b6 )

= (0.09, −0.15)

p

p B5

p

p

p

p B6

(b5 , d6 )(a5 , b6 ) [ A5 (b5 ) ∧ A5 (a5 ) ∧ A6 (d6 ) ∧ A6 (b6 )] × [μ A (b5 ) ∧ μ  (a5 b5 ) ∧ μ A (b6 ) ∧ μ  (d6 b6 )] 5

= (0.16, −0.20) (b5 , d6 )(a5 , c6 )

p

6

p

[ A5 (b5 ) ∧ A5 (a5 ) ∧ A6 (d6 ) ∧ A6 (c6 )] × [μ A (b5 ) ∧ μ  (a5 b5 ) ∧ μ A (c6 ) ∧ μ  (d6 c6 )] B B 6 5 6

5

= (0.16, −0.16) p

p B5

p

p B5

p

p

p

p B6

(b5 , a6 )(c5 , b6 )

[ A5 (b5 ) ∧ A5 (c5 ) ∧ A6 (a6 ) ∧ A6 (b6 )] × [μ A (b5 ) ∧ μ  (c5 b5 ) ∧ μ A (b6 ) ∧ μ  (a6 b6 )]

(b5 , a6 )(c5 , c6 )

[ A5 (b5 ) ∧ A5 (c5 ) ∧ A6 (a6 ) ∧ A6 (c6 )] × [μ A (b5 ) ∧ μ  (c5 b5 ) ∧ μ A (c6 ) ∧ μ  (a6 c6 )]

(b5 , d6 )(c5 , b6 )

[ A5 (b5 ) ∧ A5 (c5 ) ∧ A6 (d6 ) ∧ A6 (b6 )] × [μ A (b5 ) ∧ μ  (c5 b5 ) ∧ μ A (b6 ) ∧ μ  (d6 b6 )] B B 6 5

5

= (0.09, −0.12) p

5

= (0.09, −0.12) p

6

p

p B6 p

6

5

= (0.16, −0.16) (b5 , d6 )(c5 , c6 )

6

p B5

p

p B5

p

p B6

[ A5 (b5 ) ∧ A5 (c5 ) ∧ A6 (d6 ) ∧ A6 (c6 )] × [μ A (b5 ) ∧ μ  (c5 b5 ) ∧ μ A (c6 ) ∧ μ  (d6 c6 )] 5

= (0.16, −0.16) p

6

p B6

(d5 , a6 )(b5 , b6 ) [ A5 (d5 ) ∧ A5 (b5 ) ∧ A6 (a6 ) ∧ A6 (b6 )] × [μ A (b5 ) ∧ μ  (b5 d5 ) ∧ μ A (b6 ) ∧ μ  (a6 b6 )] 5

= (0.09, −0.15)

6

(continued)

4.2 Bipolar Fuzzy Competition Graphs

177

Table 4.8 (continued) (x1 , x2 )(y1 , y2 )

R((x1 , x2 )(y1 , y2 ))

(d5 , a6 )(b5 , c6 )

[ A5 (d5 ) ∧ A5 (b5 ) ∧ A6 (a6 ) ∧ A6 (c6 )] × [μ A (b5 ) ∧ μ  (b5 d5 ) ∧ μ A (c6 ) ∧ μ  (a6 c6 )]

p

5

= (0.09, −0.15)

p B5

p

p

p

p

p B6

6

p

(d5 , d6 )(b5 , b6 ) [ A5 (d5 ) ∧ A5 (b5 ) ∧ A6 (d6 ) ∧ A6 (b6 )] × [μ A (b5 ) ∧ μ  (b5 d5 ) ∧ μ A (b6 ) ∧ μ  (d6 b6 )] B B 6 5 6

5

= (0.16, −0.20) p

p B5

p

p B5

p

p

p

p B6

(d5 , d6 )(b5 , c6 )

[ A5 (d5 ) ∧ A5 (b5 ) ∧ A6 (d6 ) ∧ A6 (c6 )] × [μ A (b5 ) ∧ μ  (b5 d5 ) ∧ μ A (c6 ) ∧ μ  (d6 c6 )]

(d5 , a6 )(c5 , b6 )

[ A5 (d5 ) ∧ A5 (c5 ) ∧ A6 (a6 ) ∧ A6 (b6 )] × [μ A (c5 ) ∧ μ  (c5 d5 ) ∧ μ A (b6 ) ∧ μ  (a6 b6 )]

(d5 , a6 )(c5 , c6 )

[ A5 (d5 ) ∧ A5 (c5 ) ∧ A6 (a6 ) ∧ A6 (c6 )] × [μ A (c5 ) ∧ μ  (c5 d5 ) ∧ μ A (c6 ) ∧ μ  (a6 c6 )] B B 6 5

5

= (0.16, −0.16) p

5

= (0.09, −0.12) p

6

p B6

6

p

6

5

= (0.09, −0.12) p

p B5

p

p B5

p

p B6

(d5 , d6 )(c5 , b6 )

[ A5 (d5 ) ∧ A5 (c5 ) ∧ A6 (d6 ) ∧ A6 (b6 )] × [μ A (c5 ) ∧ μ  (c5 d5 ) ∧ μ A (b6 ) ∧ μ  (d6 b6 )]

(d5 , d6 )(c5 , c6 )

[ A5 (d5 ) ∧ A5 (c5 ) ∧ A6 (d6 ) ∧ A6 (c6 )] × [μ A (c5 ) ∧ μ  (c5 d5 ) ∧ μ A (c6 ) ∧ μ  (d6 c6 )]

5

= (0.25, −0.20) p

5

= (0.16, −0.16)

b6

0.1 ,− (0. 09

) −

0.

15

0.

16 )

9, .0

−0 .16 .12) ) , −0 ) 9 −0.16 , (0.16 0 . (0 (0.2, − 0.2)(0.6, −0.5) (0.4, −0.4) (0.16, −0.16) (0.09, −0.15)

Degree of membership of vertices is shown in red colour.

Fig. 4.12 Bipolar fuzzy graph G C (G 5 )∗ C (G 6 )∗ ∪ G 

6 0.1

25,

,−

(0.

.16

(0.24, −0.16)

,−

(0

,− 09 0.

(

(0 .1 6

)

12 0.

(0.2, −0.16) )

5)

0. 4)

(0.12, −0.15)) 6 0.1 6) 6, − 0.1 1 . (0 − 6, .1 (0 0.4) (0.5, −0.4) (0.4, −

(0.2, −0.20)

)

.7 ,−

0.5)

(0

2)

0. 12

(0.5, −0.5)

(0.4, −

(0.3, −0.5)

d6

16) (0.16, −0.

−

0.1

) 0.2

9,

,−

(0.4, −0.4)

,− .16 (0

(0 .0

)

(0.3, −0.5)

(0.2, −0.16

)

−0.12

d5

(0.4, −0.5)

09

(0.09, −0.12)

(0.09, −0.12)

c5 (0.3, −0.4)

(0.

(0

(0.09,

(0.09, −0.15)

b5

c6

(0.09, −0.15) (0.5, −0.5) (0.4, −0.4) ) (0 5 .16 .1 ,− −0 , ( 0 0.1 9 . 1 0 6 . − ) 6) 0 2 , 1 ( 0.2 − 0. ) , 9 0 (0.

(0.3, −0.5) a5

p B6

6

(0.2, − 0.16)

a6

6

(0.5, −0.5)

178

4 Bipolar Fuzzy Competition Graphs a6

b6

(0.3, −0.5)

a5

(0.4, −0.6) 0.4) (0.3, −

(0.3, −0.5)

c5 (0.3, −0.4)

(0.3, −0.5)

(0.3, −0.5)

(0.4, −0.4) (0.6, −0.5)

(0.3, −0.5)

(0.4, −0.5) (0.4, −0.4)

(0.4, −0.4)

(0.6, −0.4)

(0.3, −0.4) 0.4)

(0.3, −

(0.4, −0.5) (0.4, −0.4)

(0.4, −0.5)

(0.7, −0.4)

(0.3, −0.4)

(0.3, −0.4)

(0.5, −0.5)

5) (0.5, −0.

(0.4, −0.4)

(0.4, −0.4)

0.4) (0.3, −

(0.4, −0.4)

(0.4, −0.4)

(0.4, −0.5)

(0.3, −0.5)

(0.3, −0.4)

d5

d6

(0.5, −0.5) (0.4, −0.4) (0.3, −0.5)

(0.3, −0.5) b5

c6

(0.3, −0.4)

(0.4, −0.4)

(0.4, −0.4) (0.5, −0.4)

(0.5, −0.4)

(0.4, −0.4) (0.4, −0.5)

(0.4, −0.4)

(0.5, −0.4)

(0.4, −0.4) 5) (0.5, −0.

(0.5, −0.5)

Degree of membership of vertices is shown in red colour.

 5 G 6 Fig. 4.13 Cartesian product G b6

)

5) 0. 1 − .0 9,

5) 0 .1 ,−

0 9, − (0.0 (0.2, −

(0.24, −0.16)

(0. 25,

.1 6, −

0. 1

6)

−0

.16 )

0.5)

(0.12, −0.15)) .16 −0 .16) 16, 0 (0. − 6, .1 (0 , −0.4)

.16 (0

(0. 09

)

1 0.

.12)

.16

(0.2, −0.20)

)

0.2

9 .0

(0.4

−0

(0.5, −0.4) −0.16) (0.2, 6) 0.1 ,−

,−

(0

)

(0

2)

,−

, 0.2

(0.4, −0.4)

.16 (0

(0.09, −0.12)

(0.3, −0.4)

(0.3, −0.5)

6−

(0.4, −

)

(0.4, −0.5) 09 ,− (0 0.1 (0 .0 2) .7 9, ,− − 0. 0. 12 4) ) (0.09, −0.12)

(0.

(0.1

(0.4, −0.4) (0 .16 ,

16) (0.16, −0.

)

−0.12

, ) .09 (0 0.12 9, − 0 (0.

(0.2, −0.16

. 15 −0

d6 (0.5, −0.5) )

(0.5, −0.5)

(0.09,

(0.09, −0.15)

(0.3, −0.5)

c6 (0.09, −0.15)

(0.2, − 0.16

a6 (0.3, −0.5)

(0

Fig. 4.14 Bipolar fuzzy competition graph  5 G  6) C (G

(0.16, −0.16) 0.2) (0.6, −0.5) (0.4, −0.4) (0.16, −0.16) (0.5, −0.5) (0.09, −0.15)

1 × G  2 ) = [C(G  1 ) × C(G  2 )] ∪ G × , where G × = (A, R) is a bipolar fuzzy graph C(G on the crisp graph (X 1 × X 2 , E × ) defined as p

p

(1) C((a1 , a2 )) = (μ A1 (a1 ) ∧ μ A2 (a2 ), μnA1 (a1 ) ∨ μnA2 (a2 )), (2)

a1 ∈ X 1 , a2 ∈ X 2 ,

E × = {(a1 , a2 )(a1 , b2 )|a1 ∈ X 1 , a2 , b2 ∈ X 2 , N + (a1 )  = ∅, a2 b2 ∈ E C(G )∗ } ∪ {(a1 , a2 )(b1 , a2 )|a1 , b1 ∈ 2 X 1 , a2 ∈ X 2 , a1 b1 ∈ E C(G )∗ , N + (a2 )  = ∅}, 1

4.2 Bipolar Fuzzy Competition Graphs

179

(3) μ Rp ((a1 , a2 )(a1 , b2 )) = [μ Ap 1 (a1 ) ∧ μ Ap 2 (a2 ) ∧ μ Ap 2 (b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 {μ Bp (a1 c1 ) ∧ μ Bp (a2 c2 ) ∧ μ  (b2 c2 )|c1 ∈ N + (a1 )∗ , c2 ∈ N + (a2 )∗ ∩ N + (b2 )∗ }, p B2

1

2

(a1 , a2 )(a1 , b2 ) ∈ E × ,

(4) μnR ((a1 , a2 )(a1 , b2 )) = [μnA1 (a1 ) ∨ μnA2 (a2 ) ∨ μnA2 (b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 {μ Bp (a1 c1 ) ∧ μ Bp (a2 c2 ) ∧ μ  (b2 c2 )|c1 ∈ N + (a1 )∗ , c2 ∈ N + (a2 )∗ ∩ N + (b2 )∗ }, p B2

1

2

(a1 , a2 )(a1 , b2 ) ∈ E × ,

(5) μ Rp ((a1 , a2 )(b1 , a2 )) = [μ Ap 1 (a1 ) ∧ μ Ap 1 (b1 ) ∧ μ Ap 2 (a2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 {μ Bp (a1 c1 ) ∧ μ Bp (b1 c1 ) ∧ μ  (a2 c2 )|c2 ∈ N + (a2 )∗ , c1 ∈ N + (a1 )∗ ∩ N + (b1 )∗ }, B p

2

1

1

(a1 , a2 )(a1 , b2 ) ∈ E × ,

p p μnR ((a1 , a2 )(b1 , a2 )) = [μnA (a1 ) ∨ μnA (b1 ) ∨ μnA (a2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 {μ  (a1 c1 ) ∧ μ  1 1 2 B1 B1 p (b1 c1 ) ∧ μ  (a2 c2 )|c2 ∈ N + (a2 )∗ , c1 ∈ N + (a1 )∗ ∩ N + (b1 )∗ }, (a1 , a2 )(a1 , b2 ) ∈ E × . B2

(6)

Proof Let (a1 , a2 )(b1 , b2 ) be an edge of [C(G 1 ) × C(G 2 )] ∪ G × . Then there are three cases. Case 1: a1 = b1 , a2 = b2 , then (a1 , a2 )(a1 , b2 ) ∈ E × . Using conditions (3) and (4), p

p

p

p

p B1

p B2

μ R ((a1 , a2 )(a1 , b2 )) =[μ A1 (a1 ) ∧ μ A2 (a2 ) ∧ μ A2 (b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  (a1 c1 ) ∧ μ  (a2 c2 )∧ p p B1 B2 p p p =[μC (a1 , a2 ) ∧ μC (a1 , b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  ((a1 , a2 )(c1 , c2 ))∧ B p μ  ((a1 , b2 )(c1 , c2 ))], B

μ  (a1 c1 ) ∧ μ  (b2 c2 )],

p

μ R ((a1 , a2 )(a1 , b2 )) p

p

p

p

μ R ((a1 , a2 )(a1 , b2 )) =[μ A (a1 , a2 ) ∧ μ A (a1 , b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μN + ((a p μ R ((a1 , a2 )(a1 , b2 ))

+ 1 ,a2 ))∩N ((a1 ,b2 )) p p + + =[μ A (a1 , a2 ) ∧ μ A (a1 , b2 )] × h(N ((a1 , a2 )) ∩ N ((a1 , b2 ))).

p B1

],

(4.10)

p B2

μnR ((a1 , a2 )(a1 , b2 )) =[μnA1 (a1 ) ∨ μnA2 (a2 ) ∨ μnA2 (b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  (a1 c1 ) ∧ μ  (a2 c2 )∧ p p B1 B2 p n n =[μC (a1 , a2 ) ∨ μC (a1 , b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  ((a1 , a2 )(c1 , c2 ))∧ B p μ  ((a1 , b2 )(c1 , c2 ))], B p =[μnA (a1 , a2 ) ∨ μnA (a1 , b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μN + ((a ,a ))∩N + ((a ,b )) ], 1 2 1 2

μ  (a1 c1 ) ∧ μ  (b2 c2 )],

μnR ((a1 , a2 )(a1 , b2 )) μnR ((a1 , a2 )(a1 , b2 ))

μnR ((a1 , a2 )(a1 , b2 )) =[μnA (a1 , a2 ) ∨ μnA (a1 , b2 )] × h(N + ((a1 , a2 )) ∩ N + ((a1 , b2 ))).

(4.11)

Equations (4.10) and (4.11) show that (a1 , a2 )(a1 , b2 ) is an edge of C(G 1 × G 2 ). Case 2: a1 = b1 , a2 = b2 , then (a1 , a2 )(b1 , a2 ) ∈ E × . It can be proved on the same lines as Case 1.  1 ) × C(G  2 ). By Case 3: a1 = b1 , a2 = b2 , then (a1 , a2 )(a1 , b2 ) is an edge of C(G definition of direct product of bipolar fuzzy digraphs,

180

4 Bipolar Fuzzy Competition Graphs p

p

p

μ R ((a1 , a2 )(b1 , b2 )) =μ R1 (a1 b1 ) ∧ μ R2 (a2 b2 ), p

p

p

p

p

p B1

μ R ((a1 , a2 )(b1 , b2 )) =[μ A1 (a1 ) ∧ μ A1 (b1 ) ∧ μ A2 (a2 ) ∧ μ A2 (b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  (a1 c1 ) p B1

p B2

p B2

∧ μ  (b1 c1 ) ∧ μ  (a2 c2 ) ∧ μ  (b2 c2 )], p

p

p

p

p

p B1

μ R ((a1 , a2 )(b1 , b2 )) =[μ A1 (a1 ) ∧ μ A1 (b1 ) ∧ μ A2 (a2 ) ∧ μ A2 (b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  (a1 c1 ) p B2

p B1

p B2

∧ μ  (a2 c2 ) ∧ μ  (b1 c1 ) ∧ μ  (b2 c2 )], p

p

p

μ R ((a1 , a2 )(b1 , b2 )) =[μ A (a1 , a2 ) ∧ μ A (b1 , b2 )] p p B B p p p =[μ A (a1 , a2 ) ∧ μ A (a1 , b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μN + ((a ,a ))∩N + ((b ,b )) (c1 , c2 )], 1 2 1 2 p p =[μ A (a1 , a2 ) ∧ μ A (b1 , b2 )] × h(N + ((a1 , a2 )) ∩ N + ((b1 , b2 ))).

× ∨c1 ∈X 1 ,c2 ∈X 2 [μ  ((a1 , a2 )(c1 , c2 )) ∧ μ  ((b1 , b2 )(c1 , c2 ))],

p μ R ((a1 , a2 )(b1 , b2 )) p μ R ((a1 , a2 )(a1 , b2 ))

(4.12)

μnR ((a1 , a2 )(b1 , b2 )) =μnR1 (a1 b1 ) ∨ μnR2 (a2 b2 ), p B1 p p p ∧ μ  (b1 c1 ) ∧ μ  (a2 c2 ) ∧ μ  (b2 c2 )], B1 B2 B2 p =[μnA1 (a1 ) ∨ μnA1 (b1 ) ∨ μnA2 (a2 ) ∨ μnA2 (b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  (a1 c1 ) B

μnR ((a1 , a2 )(b1 , b2 )) =[μnA1 (a1 ) ∨ μnA1 (b1 ) ∨ μnA2 (a2 ) ∨ μnA2 (b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  (a1 c1 ) μnR ((a1 , a2 )(b1 , b2 ))

1

p B2

p B1

p B2

∧ μ  (a2 c2 ) ∧ μ  (b1 c1 ) ∧ μ  (b2 c2 )], p B

μnR ((a1 , a2 )(b1 , b2 )) =[μnA (a1 , a2 ) ∨ μnA (b1 , b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μ  ((a1 , a2 )(c1 , c2 )) p B

∧ μ  ((b1 , b2 )(c1 , c2 ))], p

μnR ((a1 , a2 )(b1 , b2 )) =[μnA (a1 , a2 ) ∨ μnA (a1 , b2 )] × ∨c1 ∈X 1 ,c2 ∈X 2 [μN + ((a μnR ((a1 , a2 )(a1 , b2 ))

+ 1 ,a2 ))∩N ((b1 ,b2 )) n n + + =[μ A (a1 , a2 ) ∨ μ A (b1 , b2 )] × h(N ((a1 , a2 )) ∩ N ((b1 , b2 ))).

(c1 , c2 ),

(4.13)

Equations (4.12) and (4.13) imply that (a1 , a2 )(b1 , b2 ) is an edge of C(G 1 × G 2 ).  2 )] ∪ G × ⊆ C(G 1 × G 2 ). The converse part is similar.  1 ) × C(G  Hence [C(G The construction of bipolar fuzzy competition graphs of the direct product using Theorem 4.3 is explained with the following two examples.  3 = (A3 , B3 ) and G  4 = (A4 , B4 ) Example 4.5 Consider bipolar fuzzy digraphs G on X 3 and X 4 as shown in Fig. 4.15 and their bipolar fuzzy competition graphs are given in Fig. 4.16.  4 ) = [C(G  3 ) × C(G  4 )] ∪ G × , where G × is a 3 × G It is to be shown that C(G × bipolar fuzzy digraph on (X 3 , ×X 4 , E ). The edge set E × using condition (2) is constructed as follows: E × = {(a3 , a4 )(a3 , c4 ), (c3 , a4 )(c3 , c4 ), (d3 , a4 )(d3 , c4 ), (a3 , a4 )(c3 , a4 ), (a3 , c4 )(c3 , c4 ), (a3 , d4 )(c3 , d4 )}. The degree of membership of all the edges from E × using conditions (3) and (4) are calculated as follows:

4.2 Bipolar Fuzzy Competition Graphs

181

Fig. 4.15 Bipolar fuzzy  3 and G 4 digraphs G

Fig. 4.16 Bipolar fuzzy  3) competition graphs C (G  4) and C (G

p p p B3 B4 B4 p p p = (μ A3 (a3 ) ∧ μ A4 (a4 ) ∧ μ A4 (c4 ), μnA3 (a3 ) ∨ μvA4 (a4 ) ∨ μnA4 (c4 )) p p p × [μ  (a3 b3 ) ∧ μ  (a4 d4 ) ∧ μ  (c4 d4 )] B3 B4 B4

R((a3 , a4 )(a3 , c4 )) = [A3 (a3 ) ∧ A4 (a4 ) ∧ A4 (c4 )] × [μ  (a3 b3 ) ∧ μ  (a4 d4 ) ∧ μ  (c4 d4 )]

= (0.4, −0.5) × 0.4 = (0.16, −0.20).

Other membership values are given in Table 4.9. The bipolar fuzzy graph constructed using Theorem 4.3 is shown in Fig. 4.17.  4 ) is shown with solid  3 ) × C(G The part of bipolar fuzzy graph obtained from C(G ×  3 and G  4 is shown lines and G is shown with dashed lines. The direct product of G in Fig. 4.18 and its bipolar fuzzy competition graph in Fig. 4.19, which is the same  4 )] ∪ G × ∼  4 ). 3 × G  3 ) × C(G as Fig. 4.17. Clearly, [C(G = C(G  5 = (A5 , B5 ) and G  6 = (A6 , B6 ) be two bipolar fuzzy digraphs Example 4.6 Let G as given in Fig. 4.20 and their bipolar fuzzy competition graphs without isolated vertices are shown in Fig. 4.21.  6 )] ∪ G × = (A, R) using Theorem 4.3, where G × is a  5 ) × C(G Construct [C(G bipolar fuzzy graph on (X 5 × X 6 , E × ).

182

4 Bipolar Fuzzy Competition Graphs

 3 ) × C (G  4 )] ∪ G × Table 4.9 Adjacent vertices of [C (G (x3 , x4 )(y3 , y4 ) R((x3 , x4 )(y3 , y4 )) (a3 , a4 )(a3 , c4 )

p B3

p B4

[A3 (a3 ) ∧ A4 (a4 ) ∧ A4 (c4 )] × [μ  (a3 b3 ) ∧ μ  (a4 d4 ) ∧ p B4

μ  (c4 d4 )]

= (0.16, −0.20) p p [A3 (c3 ) ∧ A4 (a4 ) ∧ A4 (c4 )] × [μ  (c3 b3 ) ∧ μ  (a4 d4 ) ∧

(c3 , a4 )(c3 , c4 )

B3

p B4

μ  (c4 d4 )]

B4

= (0.09, −0.12) p p [A3 (d3 ) ∧ A4 (a4 ) ∧ A4 (c4 )] × [μ  (d3 c3 ) ∧ μ  (a4 d4 ) ∧

(d3 , a4 )(d3 , c4 )

B3

p B4

μ  (c4 d4 )]

B4

= (0.08, −0.08) p p [A3 (a3 ) ∧ A3 (c3 ) ∧ A4 (a4 )] × [μ  (a3 b3 ) ∧ μ  (c3 b3 ) ∧

(a3 , a4 )(c3 , a4 )

B3

p B4

μ  (a4 d4 )]

B3

= (0.09, −0.12) p p [A3 (a3 ) ∧ A3 (c3 ) ∧ A4 (c4 )] × [μ  (a3 b3 ) ∧ μ  (c3 b3 ) ∧

(a3 , c4 )(c3 , c4 )

B3

p B4

μ  (c4 d4 )]

B3

= (0.09, −0.12) p p [A3 (a3 ) ∧ A3 (c3 ) ∧ A4 (d4 )] × [μ  (a3 b3 ) ∧ μ  (c3 b3 ) ∧

(a3 , d4 )(c3 , d4 )

B3

p B4

μ  (d4 b4 )]

B3

= (0.09, −0.12)

a4

b4 (0.16, −0.20)

(0.4, −0.5) a3

(0.5, −0.6)

0. 12

)

(0

,− .09

(0.4, −0.5)

2)

0. 1

(0.4, −0.5)

(0.3, −0.6)

(0.4, −0.5)

(0.09, −0.12)

(0.09, −0.12)

(0.3, −0.5) ,−

d4

(0.4, −0.5)

(0.09, −0.12)

b3

(0 .0 9

c4

(0.3, −0.4)

c3 (0.3, −0.4)

(0.09, −0.12)

(0.3, −0.4)

(0.3, −0.4)

(0.3, −0.4) d3

(0.5, −0.4)

(0.4, −0.4) (0.08, −0.08)

 4 )] ∪ G ×  3 ) × C (G Fig. 4.17 Bipolar fuzzy graph [C (G

(0.4, −0.4)

4.2 Bipolar Fuzzy Competition Graphs

183

Fig. 4.18 Direct product 3 × G 4 G

a4

b4

(0.4, −0.5)

c4

d4 (0.4, −0.5)

(0.4, −0.5)

(0.3, −0.5)

a3

(0.4,

−0.4)

(0

(0.3, −0.4) (0.2 , −0 (0.3, −0.4) .4)

Fig. 4.19 Bipolar fuzzy competition graph 3 × G  4) C (G

(0.3, −0.4)

a4

b4

c4

(0.3, −0.5) ,−

9 0.0

,−

0.

4)

(0.4, −0.5)

2) 0.1

(

(0.4, −0.5)

(0.3, −0.6)

(0.4, −0.5)

(0.09, −0.12)

(0.5, −0.6)

0. 12 )

4) 0.

d4

(0.4, −0.5)

(0.09, −0.12)

(0.09, −0.12)

(0 .0 9

−

(0.4, −0.4)

(0.4, −0.4)

(0.16, −0.20)

(0.4, −0.5)

, .3

,− .2 (0

−0.4) (0.2,

(0.5, −0.4)

0. 5)

(0.3, −0.4)

(0.3, −0.4)

d3

.4 ,−

(0.4, −0.5)

(0.3,

c3

(0

)

.5) , −0 (0.4, −0.5) ((00.3 .3, − (0.3, −0.6) 0.4)

(0.5, −0.6)

b3

−0.5

(0.3, −0.4) (0.3, −0.4) (0.09, −0.12)

(0.3, −0.4)

(0.3, −0.4)

(0.3, −0.4) (0.5, −0.4)

(0.4, −0.4)

(0.4, −0.4)

(0.08, −0.08)

E × ={(a5 , a6 )(a5 , d6 ), (b5 , a6 )(b5 , d6 ), (c5 , a6 )(c5 , d6 ), (a5 , a6 )(c5 , a6 ), (a5 , d6 ) (c5 , d6 ), (b5 , a6 )(c5 , a6 ), (b5 , d6 )(c5 , d6 )}. The degree of membership of all the edges from E × can be calculated as p

p

p

p

R((a5 , a6 )(a5 , d6 )) =(μ A (a5 ) ∧ μ A (a6 ) ∧ μ A (d6 ), μnA (a5 ) ∨ μnA (a6 ) ∨ μnA (d6 )) × [{μ  (a5 b5 ) 6 6 B 5 6 6 5 5

p

p

p

p

p

∧ μ  (a6 b6 ) ∧ μ  (d6 b6 )} ∨ {μ  (a5 b5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )}] B B B B B 6

6

5

6

6

=(0.3, −0.5) × 0.3 = (0.09, −0.15).

Other membership values are given in Table 4.10. The bipolar fuzzy graph constructed using Theorem 4.3 is shown in Fig. 4.22.

184

4 Bipolar Fuzzy Competition Graphs

Fig. 4.20 Bipolar fuzzy  5 and G 6 digraphs G

Fig. 4.21 Bipolar fuzzy  5) competition graphs C (G  6) and C (G

 3 ) × C(G  4 ) is shown with solid The part of bipolar fuzzy graph obtained from C(G  5 and G  6 is shown lines and G × is shown with dashed lines. The direct product of G in Fig. 4.23 and its bipolar fuzzy competition graph in Fig. 4.24 which clearly shows  6) ∼  5 ) × C(G  6 )] ∪ G × . 5 × G that C(G = [C(G  1 = (A1 , B1 ) and G  2 = (A2 , B2 ) be two bipolar fuzzy Proposition 4.1 Let G   2 ) = (A2 , R2 ) as the bipodigraphs on X 1 and X 2 with C(G 1 ) = (A1 , R1 ) and C(G  2 ) = (A1 ∪ A2 , R) can be  lar fuzzy competition graphs, respectively. Then C(G 1 ∪ G   constructed from C(G 1 ) = (A1 , R1 ) and C(G 2 ) = (A2 , R2 ) in the following way. 1 ∪ G  2 ) = C(G  1 ) ∪ C(G  2 ), 1. If X 1 ∩ X 2 = ∅, then C(G 2. if X 1 ∩ X 2 = ∅, then R : E C(G 1 )∗ ∪ E C(G 2 )∗ ∪ E˘ → [0, 1] × [−1, 0] is a bipolar fuzzy set, where / E2 , a2 e ∈ E2 , a2 e ∈ / E1 }, E˘ = {a1 a2 | e ∈ X 1 ∩ X 2 and a1 e ∈ E1 , a1 e ∈

4.2 Bipolar Fuzzy Competition Graphs

185

Table 4.10 Membership values of constructed edges (x3 , x4 )(y3 , y4 ) R((x3 , x4 )(y3 , y4 )) (b5 , a6 )(b5 , d6 )

p B5

p B6

p B5

p B6

p B6

∨{μ  (b5 d5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )}] = (0.09, −0.15) (c5 , a6 )(c5 , d6 )

(a5 , a6 )(c5 , a6 ) (a5 , d6 )(c5 , d6 ) (b5 , a6 )(c5 , a6 ) (b5 , d6 )(c5 , d6 )

p p p B5 B6 B6 p p p p p ∨{μ  (c5 d5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )} ∨ {μ  (c5 b5 ) ∧ μ  (a6 b6 ) ∧ B5 B6 B6 B5 B6 p μ  (d6 b6 )} B6 p p p ∨{μ  (c5 b5 ) ∧ μ  (a6 c6 ) ∧ μ  (d6 c6 )}] = (0.09, −0.15) B5 B6 B6 p p p [A5 (a5 ) ∧ A5 (c5 ) ∧ A6 (a6 )] × [{μ  (a5 b5 ) ∧ μ  (c5 b5 ) ∧ μ  (a6 b6 )} B5 B5 B6 p p p ∨{μ  (a5 b5 ) ∧ μ  (c5 b5 ) ∧ μ  (a6 c6 )}] = (0.09, −0.12) B5 B5 B6 p p p [A5 (a5 ) ∧ A5 (c5 ) ∧ A6 (d6 )] × [{μ  (a5 b5 ) ∧ μ  (c5 b5 ) ∧ μ  (d6 b6 )} B5 B5 B6 p p p ∨{μ  (a5 b5 ) ∧ μ  (c5 b5 ) ∧ μ  (d6 c6 )}] = (0.20, −0.16) B5 B5 B6 p p p [A5 (b5 ) ∧ A5 (c5 ) ∧ A6 (a6 )] × [{μ  (b5 d5 ) ∧ μ  (c5 d5 ) ∧ μ  (a6 b6 )} B5 B5 B6 p p p ∨{μ  (b5 d5 ) ∧ μ  (c5 d5 ) ∧ μ  (a6 c6 )}] = (0.09, −0.12) B5 B5 B6 p p p [A5 (b5 ) ∧ A5 (c5 ) ∧ A6 (d6 )] × [{μ  (b5 d5 ) ∧ μ  (c5 d5 ) ∧ μ  (d6 b6 )} B5 B5 B6 p p p ∨{μ  (b5 d5 ) ∧ μ  (c5 d5 ) ∧ μ  (d6 c6 )}] = (0.16, −0.16) B5 B5 B6

[A5 (c5 ) ∧ A6 (a6 ) ∧ A6 (d6 )] × [{μ  (c5 d5 ) ∧ μ  (a6 b6 ) ∧ μ  (d6 b6 )}

a6

b6

(0.3, −0.5)

d6

(0.09, −0. 12)

.15

)

(0.4, −0.5)

09, (0.

−0

)

.15

(0.4, −0.4)

−0.12

c5 (0.3, −0.4)

(0.4, −0.5)

(0.16

)

5) , −0.1 (0.09 (0.7, −0.4)

d5

(0.3, −0.5)

6)

, −0.1

(0.09 , −0.1

5)

(0.4, −0.4)

(0.09, −0.12) (0.09

6)

(0.3, −0.5)

(0.5, −0.5) (0.4, −0.4) 0.1 (0.20, −

(0.5, −0.5) (0.0 9, − 0

(0.09,

c6

(0.09, −0.15)

a5

b5

p B6

[A5 (b5 ) ∧ A6 (a6 ) ∧ A6 (d6 )] × [{μ  (b5 d5 ) ∧ μ  (a6 b6 ) ∧ μ  (d6 b6 )}

(0.5, −0.4)

, − 0. 15)

(0.6, −0.5)

 6 )] ∪ G ×  5 ) × C (G Fig. 4.22 Bipolar fuzzy graph [C (G

(0.4, −0.4)

(0.5, −0.5)

186

4 Bipolar Fuzzy Competition Graphs

Fig. 4.23 Cartesian product 5 × G 6 G

a6

b6

(0.3, −0.5) a5

b5

(0

.3 ,−

0. 5)

(0.3 , −0 .4)

(0

(0.3, −0.4) (0.3, −0.4)

−

)

(0

)

c6

d6

09, (0.

)

−0

) .15

(0.4, −0.4)

(0.4, −0.5)

6)

, −0.1 (0.16

0.12)

(0.09

5) , −0.1 (0.09 (0.7, −0.4)

, −0.1 5)

(0.4, −0.4)

(0.09, −0.12)

(0.09 , −0.1 (0.3, −0.5)

p

p

p

p

0.16)

(0.09, −

.15

(0.4, −0.5)

(0.3, −0.5)

(0.5, −0.5) (0.4, −0.4) (0.20, −

(0.09, −0. 12

(0.5, −0.5)

(0.09, −0.15) (0.5, −0.5) (0.0 9, − 0

(0.3, −0.4)

(0.5, −0.4) 4) 0. (0.5, −0.4) ,− .4

(0.4, −0.4)

b6

(0.3, −0.5)

(0.4, −0.5) (0.4, −0.4) (0.4, −0.4)

, −0 .4) . 5) 0 4, − (0. (0.4, −0.4)

(0.6, −0.5)

a6

4 0.

(0.4

0. 4

(0.3, −0.5)

5 × G  6) Fig. 4.24 C (G

)

,− .4

(0.4, −0.4)

.4) , −0 (0.3 (0. 3, − 0.5 ) (0.7, −0.4)

(0 .3 ,

(0.5, −0.5)

.5) , −0 (0.4

(0.4, −0.5)

(0.3, −0.5)

(0.3, −0.4)

d5

d6

(0.4, −0.4)

(0.5, −0.5)

(0.3, −0.5)

c5

c6

(0.5, −0.4)

5) (0.6, −0.5)

(0.4, −0.4)

(0.5, −0.5)

p

3. μ R (ab) = [(μ A1 (a) ∨ μ A2 (a)) ∧ (μ A1 (b) ∨ μ A2 (b))]× p p p p max{(μ B (ac) ∨ μ B (ac)) ∧ (μ B (bc) ∨ μ B (bc)) | c ∈ N + (a) ∩ N + (b)}, 1 2 1 2 c ab ∈ E C(G 1 )∗ ∪ E C(G 2 )∗ , p

4. μnR (ab) = [(μnA1 (a) ∧ μnA2 (a)) ∨ (μnA1 (b) ∧ μ A2 (b))]×

p p p p max{(μ  (ac) ∨ μ  (ac)) ∧ (μ  (bc) ∨ μ  (bc)) | c ∈ N + (a) ∩ N + (b)}, c B1 B2 B1 B2

ab ∈ E C(G 1 )∗ ∪ E C(G 2 )∗ , p

p

p

p

p

p

p

5. μ R (ab) = [(μ A1 (a) ∨ μ A2 (a)) ∧ (μ A1 (b) ∨ μ A2 (b))] × max{μ B (ae) ∧ μ B c

˘ (be) | e ∈ N + (a) ∩ N + (b)}, ab ∈ E,

p

1

p

2

p

6. μnR (ab) = [(μnA1 (a) ∧ μnA2 (a)) ∨ (μnA1 (b) ∧ μ A2 (b))] × max{μ B (ae) ∧ μ B ˘ (be) | e ∈ N + (a) ∩ N + (b)}, ab ∈ E.

c

1

2

4.3 Bipolar Fuzzy k-competition Graphs

187

4.3 Bipolar Fuzzy k-competition Graphs In this section, the concepts to bipolar fuzzy open neighborhood graphs, bipolar fuzzy closed neighborhood graphs and bipolar fuzzy k-competition graphs are discussed with the underlying bipolar fuzzy graphs.  = (A, B) be a bipolar fuzzy digraph on a nonempty set X . Definition 4.4 Let G  is called strong if μ Bp (x y) ≥ 1 (μ Ap (x) ∧ μ Ap (y)) and An edge xy ∈ E = supp( B) 2 1 n n n μ B (x y) ≤ 2 (μ A (x) ∨ μ A (y)). Remark 4.1 If all the edges of a bipolar fuzzy digraph are strong, then all the edges of the corresponding bipolar fuzzy competition graph may not be strong. Definition 4.5 Let A = (μ p , μn ) be a bipolar fuzzy set on X , then the cardinality of A is denoted by |A| and defined as 

|A| = |

p μA

|, |

μnA



| =

  x∈X

μ (x), p



 μ (x) . n

x∈X

Definition 4.6 Let k = (k p , k n ) be an ordered pair such that k p ∈ [0, 1] and k n ∈  = [−1, 0]. The bipolar fuzzy k−competition graph of a bipolar fuzzy digraph G   (A, B) is an undirected bipolar fuzzy graph Ck (G) = (A, R) which has the same  and there is an edge between two vertices x and y bipolar fuzzy vertex set as in G p n p if |μN + (x)∩N + (y) | > k and |μN + (x)∩N + (y) | < k n . The membership value of the edge x y is defined as lp − kp p p (μ A (x) ∧ μ A (y)) × h(N + (x) ∩ N + (y)), lp ln − kn n (μ A (x) ∨ μnA (y)) × h(N + (x) ∩ N + (y)), μnR (x y) = ln p

μ R (x y) =

for all x, y ∈ X

where |N + (x) ∩ N + (y)| = (l p , l n ).  = (A, B)  be a bipolar fuzzy digraph. If h(N + (x) ∩ N + (y)) = Theorem 4.4 Let G p p 1, |μN + (x)∩N + (y) | > 2k , and |μnN + (x)∩N + (y) | < 2k n for some x, y ∈ X, then the edge  x y is strong in Ck (G).  = (A, R) be a bipolar fuzzy k−competition graph of the bipoProof Let Ck (G)  = (A, B).  Suppose for x, y ∈ X , |N + (x) ∩ N + (y)| = (l p , l n ), lar fuzzy digraph G then lp − kp p p (μ A (x) ∧ μ A (y))h(N + (x) ∩ N + (y)), lp lp − kp p p p (μ A (x) ∧ μ A (y)), ∵ h(N + (x) ∩ N + (y)) = 1 μ R (x y) = lp p p p p p l p μ R (x y) = (l p − k p )(μ A (x) ∧ μ A (y)) > k p (μ A (x) ∧ μ A (y)), p

μ R (x y) =

⇒

188

4 Bipolar Fuzzy Competition Graphs p

⇒

p

p

2k p μ R (x y) > k p (μ A (x) ∧ μ A (y)),  1 p p p μ A (x) ∧ μ A (y) . μ R (x y) > 2

⇒

ln − kn n (μ A (x) ∧ μnA (y))h(N + (x) ∩ N + (y)), ln ln − kn n (μ A (x) ∧ μnA (y)), ∵ h(N + (x) ∩ N + (y)) = 1 μnR (x y) = ln l n μnR (x y) = (l n − k n )(μnA (x) ∧ μnA (y)) < k n (μnA (x) ∧ μnA (y)), μnR (x y) =

⇒ ⇒

2k n μnR (x y) < k n (μnA (x) ∧ μnA (y)),  1 n μ A (x) ∧ μnA (y) . μnR (x y) < 2

⇒

 Thus the edge x y is strong in Ck (G).



Definition 4.7 Let G = (A, B) be a bipolar fuzzy graph. A bipolar fuzzy open neighborhood, N (u), of a vertex u in G is a bipolar fuzzy set N (u) = (X u , p p μN (u) , μnN (u) ) where X u = {v|B(uv) = (0, 0)} and μN (u) : X u → [0, 1], μnN (u) : p p X u → [−1, 0] are the membership functions defined by μN (u) (u) = μ B (uv) and n n μN (u) (u) = μ B (uv). The bipolar fuzzy closed neighborhood, N [u], is defined as p N [u] = N (u) ∪ {(u, μ A (u), μnA (u))}. Definition 4.8 Let G = (A, B) be a bipolar fuzzy graph. A bipolar fuzzy open  neighborhood graph of G is a bipolar fuzzy graph N (G) = (A, R ), whose set of vertices is the same as G and there is an edge between two vertices u and v if  p p N (u) ∩ N (v) = ∅ and R = (μ R  , μnR  ) is a bipolar fuzzy set, where μ R  : X × X → [0, 1] and μnR  : X × X → [−1, 0] are the membership functions defined as p

p

p

μ R  (uv) = (μ A (u) ∧ μ A (v)) × h(N (u) ∩ N (v)), μnR  (uv) = (μnA (u) ∨ μnA (v)) × h(N (u) ∩ N (v)),

for all u, v ∈ X.

The bipolar fuzzy closed neighborhood graph is defined on the same lines in Definition 4.9. Definition 4.9 Let G = (A, B) be a bipolar fuzzy graph. The bipolar fuzzy closed  neighborhood graph of G is a bipolar fuzzy graph N [G] = (A, R ) whose set of vertices is the same as G and there is an edge between two vertices u and v if  p p N [u] ∩ N [v] = ∅ and R = (μ R  , μnR  ) is a bipolar fuzzy set, where μ R  : X × X → n [0, 1] and μ R  : X × X → [−1, 0] are the membership functions defined as p

p

p

μ R  (uv) = (μ A (u) ∧ μ A (v)) × h(N [u] ∩ N [v]), μnR  (uv) = (μnA (u) ∨ μnA (v)) × h(N [u] ∩ N [v]),

for all u, v ∈ X.

4.3 Bipolar Fuzzy k-competition Graphs

189

Considering the bipolar fuzzy open and closed neighborhood of the vertices, another type of bipolar fuzzy graphs are introduced in the following definitions. Definition 4.10 Let k be an ordered pair (given in Definition 4.6) then a bipolar fuzzy (k)−competition graph of a bipolar fuzzy graph G = (A, B) is a bipolar  fuzzy graph Nk (G) = (A, R ) which has the same vertex set as in G and there is an p edge between two vertices u and v if |μN (u)∩N (v) | > k p and |μnN (u)∩N (v) | < k n . The membership value of the edge uv is defined as lp − kp p p (μ A (u) ∧ μ A (v)) × h(N (u) ∩ N (v)), lp ln − kn n (μ A (u) ∨ μnA (v)) × h(N (u) ∩ N (v)), μnR  (uv) = ln p

μ R  (uv) =

for all u, v ∈ X

where |N (u) ∩ N (v)| = (l p , l n ). Definition 4.11 Let k be an ordered pair (given in Definition 4.6) then a bipolar fuzzy [k]− competition graph of a bipolar fuzzy graph G = (A, B) is a bipolar  fuzzy graph Nk [G] = (A, R ) which has the same vertex set as in G and there is an p edge between two vertices u and v if |μN [u]∩N [v] | > k p and |μnN [u]∩N [v] | < k n . The membership value of the edge uv is defined as lp − kp p p (μ A (u) ∧ μ A (v)) × h(N [u] ∩ N [v]), lp ln − kn n (μ A (u) ∨ μnA (v)) × h(N [u] ∩ N [v]), μnR  (uv) = ln p

μ R  (uv) =

for all u, v ∈ X

where |N [u] ∩ N [v]| = (l p , l n ). Theorem 4.5 For every edge of a bipolar fuzzy graph G, there exists one edge in N [G]. Proof Let N [G] = (A, R  ) be a bipolar fuzzy closed neighborhood graph corresponding to the bipolar fuzzy graph G = (A, B). Let uv be an edge in G, then u, v ∈ N [u] and u, v ∈ N [v]. So, u, v ∈ N [u] ∩ N [v] and therefore h(N [u] ∩ N [v]) = 0. Hence, μ R  (uv) =

lp − kp p p (μ A (u) ∧ μ A (v)) × h(N [u] ∩ N [v]) > 0, lp

μnR  (uv) =

ln − kn n (μ A (u) ∨ μnA (v)) × h(N [u] ∩ N [v]) < 0. ln

p

That is, uv is an edge of N [G] which completes the proof.



 = (A, B)  be a bipolar fuzzy digraph on X such that E = Definition 4.12 Let G  The underlying bipolar fuzzy graph of G  is a bipolar fuzzy graph U(G)  = supp( B). p (A, B) where B = (μ B , μnB ) is defined as

190

4 Bipolar Fuzzy Competition Graphs ⎧ p n  vu if uv  ∈ E,  ∈ / ⎪ ⎨ (μ B (uv), μ B (uv)), p p n n  if vu  ∈ E, uv  ∈ / B(uv) = (μ B (uv), μ B (uv)) = (μ  (vu), μ  (vu)), Bp ⎪ ⎩ Bp  (μ  (uv) ∧ μ  (vu), μn (uv) ∨ μn (vu)), if uv,  vu  ∈ E. B

B

B

 E,  E,

B

The relations between bipolar fuzzy neighborhood graphs and bipolar fuzzy competition graphs are established in the following theorems.  = (A, B)  be a symmetric bipolar fuzzy digraph without any Theorem 4.6 Let G    is the underlying bipolar fuzzy graph loops then Ck (G) = Nk (U(G)), where U(G)  of G.  = (A, B) be an underlying bipolar fuzzy graph corresponding to Proof Let U(G)  = (A, B).  Also, let Nk (U(G))  = (A, R  ) and Ck (G)  = a bipolar fuzzy digraph G  (A, R). The bipolar fuzzy k−competition graph Ck (G) as well as the underlying  has the same  Hence Nk (U(G)) bipolar fuzzy graph have the same vertex set as G.   vertex set as G. It remains only to show that R(uv) = R (uv), for every u, v ∈ X . There are two cases. p  then Case 1: If for all u, v ∈ X , R(uv) = (μ R (uv), μnR (uv)) = (0, 0) in Ck (G), p  is symmetric, either either |μN + (u)∩N + (v) | > k p or |μnN + (u)∩N + (v) | < k n . Since G p  Hence, R  (uv) = (0, 0) and so |μN (u)∩N (v) | > k p or |μnN (u)∩N (v) | < k n in U(G).  R(uv) = R (uv), for all u, v ∈ X . p  then Case 2: If for some u, v ∈ X , (μ R (uv), μnR (uv)) = (0, 0) in Ck (G), p n p  is a symmetric bipolar fuzzy |μN + (u)∩N + (v) | > k and |μN + (u)∩N + (v) | < k n . Since G digraph, h(N (u) ∩ N (v)) = h(N + (u) ∩ N + (v)). Then for all u, v ∈ X , lp − kp p p p (μ A (u) ∧ μ A (v))h(N + (u) ∩ N + (v)) = μ  (uv), R lp ln − kn n (μ A (u) ∧ μnA (v))h(N + (u) ∩ N + (v)) = μnR  (uv), μnR (uv) = ln p

μ R (uv) =

(l p , l n ) = |N + (u) ∩ N + (v)|.



Hence R(uv) = R (uv). Since u and v were taken to be arbitrary, the result holds  for all edges uv of Ck (G).  = (A, B)  be a symmetric bipolar fuzzy digraph having loops at Theorem 4.7 Let G  = Nk [U(G)],  where U(G)  is the underlying bipolar fuzzy every vertex, then Ck (G)  graph of G.  = (A, B) be an underlying bipolar fuzzy graph corresponding to Proof Let U(G)  = (A, B).  Also, let Nk (U(G))  = (A, R  ) and Ck (G)  = bipolar fuzzy digraph G  as well as the bipolar fuzzy (A, R). The bipolar fuzzy k−competition graph Ck (G)  has the same  Hence Nk (U[G)] underlying graph have the same vertex set as G.  It remains only to show that R(uv) = R  (uv) for every u, v ∈ X . As vertex set as G. the bipolar fuzzy digraph has a loop at every vertex therefore, the bipolar fuzzy out neighborhood contains the vertex itself. There are two cases.

4.3 Bipolar Fuzzy k-competition Graphs

191

 then either |μ p + Case 1: If for all u, v ∈ X , R(uv) = (0, 0) in Ck (G), N (u)∩N + (v) |  > p n p n  k or |μN + (u)∩N + (v) | < k . Since G is symmetric, either |μN [u]∩N [v] | > k p or  Hence, R  (uv) = (0, 0) and so R(uv) = R  (uv) for all |μnN [u]∩N [v] | < k n in U(G). u, v ∈ X . p p  then Case 2: If for some u, v ∈ X , μ R (uv) > 0 and μ R (uv) < 0 in Ck (G), p n p n  is a symmetric bipolar fuzzy |μN + (u)∩N + (v) | > k and |μN + (u)∩N + (v) | < k . Since G digraph with loops, |N [u] ∩ N [v]| = |N + (u) ∩ N + (v)| and h(N [u] ∩ N [v]) = h(N + (u) ∩ N + (v)). Thus for all u, v ∈ X , lp − kp p p p (μ A (u) ∧ μ A (v))h(N + (u) ∩ N + (v)) = μ  (uv), R lp lp − kp n (μ A (u) ∧ μnA (v))h(N + (u) ∩ N + (v)) = μnR  (uv), μnR (uv) = lp p

μ R (uv) =

(l p , l n ) = |N + (u) ∩ N + (v)|.



Hence R(uv) = R (uv). Since u and v were taken to be arbitrary, the result holds   for all edges uv of Ck (G).

4.4 Complex Bipolar Fuzzy Competition Graphs In this section, the concept of complex bipolar fuzzy competition graphs is discussed with several notions concerning complex bipolar fuzzy out neighborhoods, complex bipolar fuzzy in neighborhoods, complex bipolar fuzzy open neighborhood graphs, complex bipolar fuzzy closed neighborhood graphs, complex bipolar fuzzy k−competition graphs, and underlying complex bipolar fuzzy graphs. Definition 4.13 A complex bipolar fuzzy digraph on a nonempty set X is a pair p  where A = (r Ap ew A , r An ewnA ) : X → {z | z ∈ C, |z| ≤ 1}2 is a complex G = (A, D), p n  = (r p ew D , r n ew D ) : X × X → {z | z ∈ C, |z| ≤ 1}2 bipolar fuzzy set on X and D   D D is a complex bipolar polar fuzzy relation in X such that, for all x, y ∈ X , p

p

p

r D (x y) ≤ r A (x) ∧ r A (y) and r Dn (x y) ≥ r An (x) ∨ r An (y), (for amplitude terms) p

p

p

w D (x y) ≤ w A (x) ∧ w A (y) and w nD (x y) ≥ w nA (x) ∨ w nA (y), (for phase terms) √ p p where i = −1, r A : X → [0, 1], r An : X → [−1, 0], r D : X × X → [0, 1] and r Dn : p p n X × X → [−1, 0] are mappings, w A , w D ∈ [0, π], w A , w nD ∈ [π, 2π] or w nA , w nD ∈ [−π, 0].  is not a symmetric complex bipolar fuzzy relation. Note that D Definition 4.14 A complex bipolar fuzzy graph on a nonempty set X is a pair p n p G = (A, D), where A = (r A ew A , r An ew A ) : X → {z | z ∈ C, |z| ≤ 1}2 is a complex n p wp bipolar fuzzy set on X and D = (r D e D , r Dn ew D ) : X × X → {z | z ∈ C, |z| ≤ 1}2 is a complex bipolar polar fuzzy relation in X such that, for all x, y ∈ X ,

192

4 Bipolar Fuzzy Competition Graphs p

p

p

r D (x y) ≤ r A (x) ∧ r A (y) and r Dn (x y) ≥ r An (x) ∨ r An (y), (for amplitude terms) p

p

p

w D (x y) ≤ w A (x) ∧ w A (y) and w nD (x y) ≥ w nA (x) ∨ w nA (y), (for phase terms), √ p p where i = −1, r A : X → [0, 1], r An : X → [−1, 0], r D : X × X → [0, 1] and r Dn : p p n X × X → [−1, 0] are mappings, w A , w D ∈ [0, π], w A , w nD ∈ [π, 2π] or w nA , w nD ∈ [−π, 0]. Note that D(x y) = (0, 0) for all x y ∈ X × X − E , where E ⊆ X × X is the set of edges. A is called a complex bipolar fuzzy vertex set of G and D is a complex bipolar fuzzy edge set of G. A bipolar fuzzy relation D on X is symmetric if D(x y) = D(yx) for all x, y ∈ X . Notice that D(x y) = (0, 0) for x y ∈ / E.  = (A, D)  be a complex bipolar fuzzy graph on a nonempty Definition 4.15 Let G set X . A complex bipolar fuzzy out neighborhood of a vertex x ∈ X in a com is a complex bipolar fuzzy set N + (x) = (X x+ , μ p + , plex bipolar fuzzy digraph G N (x)

μnN + (x) ), where X x+ = {y ∈ X | (μ D (x y), μnD (x y)) = (0, 0)} and (μN + (x) , μnN + (x) ) : X x+ → {z | z ∈ C, |z| ≤ 1}2 is defined as p

p

p

μN + (x) (y) = μ D (x y) = r D (x y)eiw D (x y) , μnN + (x) (y) = μnD (x y) = r Dn (x y)eiw D (x y) . p

p

p

n

A complex bipolar fuzzy in neighborhood of a vertex x ∈ X in a complex bipolar  is a complex bipolar fuzzy set N − (x) = (X x− , μ p − , μn − ), fuzzy digraph G N (x) N (x) p p where X x− = {y ∈ X | (μ D (yx), μnD (yx)) = (0, 0)} and (μN − (x) , μnN − (x) ) : X x− → {z | z ∈ C, |z| ≤ 1}2 is defined as p

μN − (x) (y) = μ D (yx) = r D (yx)eiw D (yx) , μnN − (x) (y) = μnD (yx) = r Dn (yx)eiw D (yx) . p

p

p

n

 p n p Definition 4.16 Let A = r A eiw A , r An eiw A be a complex bipolar fuzzy set on a nonempty set X . The height of A is defined as h(A) = rh(A) eiwh(A) , where p

p

rh(A) = max{r A (x) | x ∈ X } and wh(A) = max{w A (x) | x ∈ X }.  = (A, D)  be a complex bipolar fuzzy digraph on a nonempty Definition 4.17 Let G set X . A complex bipolar fuzzy competition graph of a complex bipolar fuzzy  is an undirected complex bipolar fuzzy graph C(G)  = (A, R) which has digraph G  the same vertex set as in G and there is an edge between two vertices x and y, x, y ∈ X , if N = N + (x) ∩ N + (y) is nonempty. The complex bipolar fuzzy set p n p R = r R eiw R , r Rn eiw R is defined as p

p

p

r R (x y) = (r A (x) ∧ r A (y)) × rh(N ) , r Rn (x y) = (r An (x) ∨ r An (y)) × rh(N ) , (for amplitude terms)  

 p p wh(N ) w A (x) w A (y) p × , ∧ w R (x y) = π π π π

4.4 Complex Bipolar Fuzzy Competition Graphs

 w nR (x y) = π

w nA (x) w nA (y) ∨ π π

193

 ×

 wh(N ) , π

(for phase terms).

Theorem 4.8 Let G be a complex bipolar fuzzy graph then adding sufficient number of isolated vertices to G produces a complex bipolar fuzzy competition graph of some complex bipolar fuzzy digraph. Proof Let G = (A, R) be a complex bipolar fuzzy graph on nonempty set X , where p n p A = (r A eiw A , r An eiw A ) is a complex bipolar fuzzy set on the set of vertices X and p n p R = (r R eiw R , r Rn eiw R ) is a complex bipolar fuzzy relation in X . Construct the complex  = (A, D)  as follows: Let x, y ∈ X be any two vertices of G bipolar fuzzy digraph G such that R(x y) = (0, 0). Add a vertex βx y , remove the edge x y and draw directed edges from x and y to βx y such that  p p n n p p A(βx y ) = (r A (x) ∧ r A (y))ei(w A (x)∧w A (y)) , (r An (x) ∨ r An (y))ei(w A (x)∨w A (y)) p D

p D

r  (xβx y ) = r  (yβx y ) = p D

p D

w  (xβx y ) = w  (yβx y ) =

p r R (x y) , p p r A (x) ∧ r A (y)

n n rD  (xβx y ) = r D  (yβx y ) = −

p w R (x y) p p w A (x) w A (y)

π

∧

r Rn (x y) p r A (x) ∨ r An (y)

, w nD (xβx y ) = w nD (yβx y ) = −

w nR (x y) n w A (x) w nA (y) π

π

∨

.

π

 such that Continuing this process, we obtain a complex bipolar fuzzy digraph G  = G ∪ I , where I is the complex bipolar fuzzy set of isolated vertices added C(G) to G.   = (A, D) be a complex bipolar fuzzy digraph on a nonempty Definition 4.18 Let G  is called strong if set X . An edge xy in G 1 p p (r (x) ∧ r A (y)), 2 A 1 r Dn (x y) ≤ (r An (x) ∨ r An (y)), 2  1 p p p w D (x y) ≥ w A (x) ∧ w A (y) , 2  1 n n w A (x) ∧ w nA (y) . w D (x y) ≤ 2 p

r D (x y) ≥

p

n

Definition 4.19 Let A = (r p eiw , r n eiw ) be a complex bipolar fuzzy set on X then the cardinality of A is denoted by |A| and defined as        p  n p i x∈X w p (x) n i x∈X w n (x) |A| = |μ A |, |μ A | = r (x)e , r (x)e . x∈X

x∈X

194

4 Bipolar Fuzzy Competition Graphs p

n

Definition 4.20 Let k = (k p eiθ , k n eiθ ) be an ordered pair of complex numbers such that k p ∈ [0, 1], k n ∈ [−1, 0], θ p ∈ [0, π], θn ∈ [−π, 0]. The complex bipolar  = (A, D)  is fuzzy k−competition graph of a complex bipolar fuzzy digraph G  = (A, R) which has the same an undirected complex bipolar fuzzy graph Ck (G)  and there is an edge between two vertices complex bipolar fuzzy vertex set as in G x and y if l p > k p , φ p > θ p , l n < k n , φn < θ n ,  p n where |N | =| N + (x) ∩ N + (y) |= l p eiφ , l n eiφ . The membership values of the edge x y are defined as lp − kp p p (r A (x) ∧ r A (y)) × rh(N ) , lp ln − kn n (r A (x) ∨ r An (y)) × rh(N ) , (for amplitude terms) r Rn (x y) = ln

 p   p w A (x) w A (y) wh(N ) π(φ p − θ p ) p ∧ × , w R (x y) = φp π π π  

 n wh(N ) w A (x) w nA (y) π(φn − θn ) ∨ × , (for phase terms) w nR (x y) = φn π π π p

r R (x y) =

for all x, y ∈ X . Definition 4.21 Let G = (A, D) be a complex bipolar fuzzy graph on a nonempty set X . A complex bipolar fuzzy open neighborhood of a vertex x ∈ X in a complex p bipolar fuzzy graph G is a complex bipolar fuzzy set N (x) = (X x , μN (x) , μnN (x) ), p where X x = {y ∈ X | D(x y) = (0, 0)} and (μN (x) , μnN (x) ) : X x →{z | z ∈ C, |z| ≤ 1}2 is defined as p

μN (x) (y) = μ D (x y) = r D (x y)eiw D (x y) , μnN (x) (y) = μnD (x y) = r Dn (x y)eiw D (x y) . p

p

p

n

The complex bipolarp fuzzy closed neighborhood, N [x], is defined as N [x] =   n p N (x) ∪ (x, r A (x)eiw A (x) , r An (x)eiw A (x) . Definition 4.22 Let G = (A, D) be a complex bipolar fuzzy graph on a nonempty set X . A complex bipolar fuzzy open neighborhood graph of a complex bipolar fuzzy graph G is an undirected complex bipolar fuzzy graph N (G) = (A, R  ) which has the same vertex set as in G and there is an edge between two vertices x and y, x,  y ∈ X , if N = N (x) ∩ N (y) is nonempty. The complex bipolar fuzzy set p n p R  = r R  eiw R , r Rn  eiw R is defined as p

p

p

r R  (x y) = (r A (x) ∧ r A (y)) × rh(N ) , r Rn  (x y) = (r An (x) ∨ r An (y)) × rh(N ) , (for amplitude terms)  

 p p wh(N ) w A (x) w A (y) p × , w R  (x y) = π ∧ π π π

4.4 Complex Bipolar Fuzzy Competition Graphs

 w nR  (x y) = π

w nA (x) w nA (y) ∨ π π

195

 ×

 wh(N ) , π

(for phase terms)

for all x, y ∈ X . The bipolar fuzzy closed neighborhood graph is defined on the same lines in Definition 4.23. Definition 4.23 Let G = (A, D) be a complex bipolar fuzzy graph on a nonempty set X . A complex bipolar fuzzy closed neighborhood graph of a complex bipolar fuzzy graph G is an undirected complex bipolar fuzzy graph N [G] = (A, R ∗ ) which has the same vertex set as in G and there is an edge between two vertices x and y, ∗ x, y ∈ X , if N = N [x] ∩ N [y] is nonempty. The complex bipolar fuzzy set R =  n p iw p ∗ n iw r R ∗ e R , r R ∗ e R∗ is defined as p

p

p

r R ∗ (x y) = (r A (x) ∧ r A (y)) × rh(N ) , r Rn ∗ (x y) = (r An (x) ∨ r An (y)) × rh(N ) , (for amplitude terms)  

 p p wh(N ) w A (x) w A (y) p ∧ × , w R ∗ (x y) = π π π π

 n   w A (x) w nA (y) wh(N ) w nR ∗ (x y) = π ∨ × , (for phase terms) π π π for all x, y ∈ X .  = (A, D)  be a complex bipolar fuzzy digraph on X such Definition 4.24 Let G  The underlying complex bipolar fuzzy graph that E is the crisp set of edges in G.  is a complex bipolar fuzzy graph U(G)  = (A, D) such that D = (μ Dp , μnD ) = of G p n p (r D eiw D , r Dn eiw D ) is defined as, ⎧ p n  vu ⎪  ∈ E,  ∈ / E ⎨ (μ D (uv), μ D (uv)), if uv p n  D(uv) = (μ  (vu), μ  (vu)), if vu  ∈ E, uv  ∈ / E ⎪ ⎩ ( f Dp , f n ), D if uv,  vu  ∈ E where e

p

p

f p = (r D (uv) ∧ r D (vu))ei(w D (uv)∧w D (vu))

i(w nD (uv)∨w nD (vu))

p

p

and

f n = (r Dn (uv) ∨ r Dn (vu))

.

4.5 Applications of Bipolar Fuzzy Competition Graphs In this section, certain algorithms are elaborated to compute the strength of competition among objects with a number of real-world applications in different fields including food webs, business marketing, politics, wireless communication networks, and social networking.

196

4 Bipolar Fuzzy Competition Graphs

4.5.1 Variants of Bipolar Fuzzy Competition Graphs in Food Webs Food webs are the graphical representations of natural interconnection of food chains between different species. Competition graphs arose in connection with an application in food webs. In a food web, vertices represent species and there is a directed edge between two species x and y if y is a food for x. Bipolar fuzzy graphs are more realistic to represent the competition between species. There are many interesting variations of the notion of bipolar fuzzy competition graphs in ecological interpretation. For instance, two species may have a common prey, a common enemy, both common prey and common enemy, or either a common prey or a common enemy. All of these notions are discussed in two different ecological niches. 1. Bipolar fuzzy competition graph in food webs  = (A, B)  represents a bipolar fuzzy food web on the set of species X in Let G  which xy ∈ E if x preys on y. Bipolar fuzzy food webs can play an important role for investigating the flow of energy and predator−−prey relationship in ecosystem. Bipolar fuzzy competition graph can be constructed from bipolar fuzzy food web (Definition 4.25) to study the extent to which species compete for common prey.  = (A, B)  be a bipolar fuzzy food web. The bipolar fuzzy Definition 4.25 Let G  = (A, R) has the same set of vertices as G  and there is competition graph C(G) an edge between two vertices x and y if N − (x) ∩ N − (y) = ∅. The membership function R is defined as   p p p μ R (x y) = (μ A (x)(x) ∧ μ A (y))h N − (x) ∩ N − (y) ,   μnR (x y) = (μnA (x)(x) ∨ μnA (y))h N − (x) ∩ N − (y) . The method for calculating the strength of competition between species in ecosystem in given in Algorithm 4.5.1. Algorithm 4.5.1 Bipolar fuzzy competition graph in ecosystem Input the number n of species x1 , x2 , . . . , xn . Input the adjacency matrix of bipolar fuzzy food web. Construct the table of bipolar fuzzy in neighborhoods of all the species.  = (A, R) using Definition Construct the bipolar fuzzy competition graph C(G) 4.25. 5. If N − (x) ∩ N − (y) = ∅ for any two species x and y then the strength of competition between x and y for common food can be calculated using Formula (4.14) as

1. 2. 3. 4.

p

S(x, y) =

p

p

μ A (x) + μ A (y) + μ R (x y) + (3 + μnA (x) + μnA (y) + μnR (x y)) . 6 (4.14)

4.5 Applications of Bipolar Fuzzy Competition Graphs

197 (0.8, −0.1)

(0.9, −0.4)

(0.65, −0.3)

baboon

vulture

lion

( 0. 6, −0 .1)

1) 0.

(0.95, −0.1)

)

−0.1 )

)

(0.7,

(0. 6,

−0 .1)

(0.7, −0.35)

(0

Producer

leopard

fiscal shrike

.3 ,

−0 .0 5

)

(0.2, −0.7)

(0 .3 ,− 0. 09

(0.7, −0

(0.3, −0.4)

−

) 0.1

.1)

6,

− 7,

. (0

(0.

(0.55, −0.5)

giraffe

)

(0.95, −0.1)

snake

rhinoceros

,− 0 .1

african skunk

(0.55, −0.45)

(0.7, −0.3)

(0 .7 5

(0

grasshopper (0.2, −0.8)

(0.7, −0.09

(0.95, −0.1)

2) 0. ,− .4

caracal (0.4, −0.7)

(0. 7, −

impala 0. 2

)

mouse (0.5, −0.5)

Producer

Fig. 4.25 Bipolar fuzzy food web

Consider the example of bipolar fuzzy food web of 13 species giraffe, lion, vulture, rhinoceros, African skunk, fiscal shrike, grasshopper, baboon, leopard, snake, caracal, mouse, and impala. The positive degree of membership of each specie represents the extent to which a specie is strong according to its power and can defend itself to exist in the animal kingdom. The negative degree of membership of specie represents its weakness that it can be killed or dominated by other species. The bipolar fuzzy food web is shown in Fig. 4.25. The degree of membership of lion is (0.95, −0.1), which shows that lion is 95% strong in its kingdom according to hunting power and 10% weak because a group of animals together can dominate a lion. The directed edge between giraffe and lion has degree of membership (0.7, −0.1), which represents that lion obtains 70% energy from giraffe and a giraffe is 10% harmful to lion because it can kill a lion with its long legs. This is an acyclic bipolar fuzzy digraph. A bipolar fuzzy competition graph can be constructed to investigate the strength of competition between species for common food\prey. The bipolar fuzzy in neighborhoods are given in Table 4.11. The bipolar fuzzy competition graph is shown in Fig. 4.26. There is a food competition between lion and vulture, fiscal shrike and baboon, snake and caracal, leopard and caracal. The membership value of an edge between two species represents the degree of benefits and harm of common food. The strengths of competition between species x and y calculated using Formula (4.14) are given in Table 4.12. According to Table 4.12, there is a strong competition between caracal and leopard for common food with respect to hunting powers and weaknesses.

198

4 Bipolar Fuzzy Competition Graphs

Table 4.11 Bipolar fuzzy in neighborhoods of species Species N − (u) : u is a specie ∅ {(giraffe, 0.7, −0.1), (rhinoceros, 0.7, −0.1)} ∅ {(rhinoceros, 0.6, −0.1), (African skunk, 0.6, −0.1), (leopard, 0.8, −0.1)} {(fiscal shrike, 0.6, −0.1)} {grasshopper, 0.3, −0.05} ∅ {(grasshopper, 0.3, −0.09), (snake, 0.3, −0.4)} {(baboon, 0.75, −0.1), (impala, 0.7, −0.09)} {(caracal, 0.7, −0.1), (mouse, 0.5, −0.4)} {(mouse, 0.4, −0.2), (impala, 0.7, −0.2)} ∅ ∅

giraffe lion rhinoceros vulture African skunk fiscal shrike grasshopper baboon leopard snake caracal mouse impala

lion

(0.9, −0.4)

(0.65, −0.3)

(0.95, −0.1) (0.39, −0.6)

baboon

vulture

(0.95, −0.1) (0.95, −0.1)

(0.55, −0.45)

(0.55, −0.5)

giraffe

0. 7) 9, − (0 .4

rhinoceros

(0.28, −0.4)

(0 .0 6, −

0. 1

2)

(0.7, −0.3)

leopard

snake

african skunk

fiscal shrike caracal

(0.2, −0.7)

(0.7, −0.35)

(0.4, −0.7)

0.2, −0.8)

grasshopper

impala (0.5, −0.5)

mouse

Fig. 4.26 Bipolar fuzzy competition graph of bipolar fuzzy food web Table 4.12 Strength of competition between species Species Strength of Species competition lion, vulture snake, caracal

0.9983 0.9633

shrike, baboon caracal, leopard

Strength of competition 0.855 1.0483

4.5 Applications of Bipolar Fuzzy Competition Graphs

199

2. Bipolar fuzzy common enemy graph Bipolar common enemy graphs can be constructed from bipolar fuzzy food webs to study the strength of common enemies between species as given in Definition 4.26.  = (A, B)  be a bipolar fuzzy food web. The bipolar fuzzy Definition 4.26 Let G  = (A, R) has the same set of species as G  and there is common enemy graph C(G) an edge between two species x and y if N + (x) ∩ N + (y) = ∅, i.e., x and y has a common predator and the membership function R is defined as μ R (x y) = (μ A (x) ∧ μ A (y))h(N + (x) ∩ N + (y)), p

p

p

μnR (x y) = (μnA (x) ∨ μnA (y))h(N + (x) ∩ N + (y)). The method for calculating the strength of common enemies between species in ecosystem is presented in Algorithm 4.5.2 Algorithm 4.5.2 Bipolar fuzzy common enemy graph 1. 2. 3. 4. 5.

Input the number of species. Input the bipolar fuzzy set A of species. Construct the table of bipolar fuzzy out neighborhoods of all the species. Construct the bipolar fuzzy competition graph G = (A, R) using Definition 4.26. If N + (x) ∩ N + (y) = ∅ for any two species x and y then the strength of competition between x and y for common food can be calculated using Formula (4.15) as p

S(x, y) =

p

p

μ A (x) + μ A (y) + μ R (x y) + (3 + μnA (x) + μnA (y) + μnR (x y)) . 6 (4.15)

Consider the example of a bipolar fuzzy food web of 13 species as shown in Fig. 4.25.The bipolar fuzzy out neighborhoods are given in Table 4.13. The bipolar fuzzy common enemy graph is shown in Fig. 4.27. There are common enemies between giraffe and rhinoceros, rhinoceros and African skunk, rhinoceros and leopard, African skunk and leopard, grasshopper and snake, mouse and impala and baboon and impala. The positive membership value of an edge between two species represents the degree of energy both provide to common predator/enemy and the negative degree of membership shows the extent to which the species can harm their common enemy. Let x and y be two species then the strength of common enemies between x and y is calculated using Formula (4.15). The strength of having common enemies between all the species is given in Table 4.14, which shows that impala and baboon have the largest number of common enemies.

200

4 Bipolar Fuzzy Competition Graphs

Table 4.13 Bipolar fuzzy out neighbohoods of species Species N + (u) : u is a specie {(lion, 0.7, −0.1)} ∅ {(lion, 0.7, −0.1), (vulture, 0.6, −0.1)} ∅ {(vulture, 0.6, −0.1)} {(African skunk, 0.6, −0.1)} {(fiscal shrike, 0.3, −0.05), (baboon, 0.3, −0.09)} {(leopard, 0.75, −0.1)} {(vulture, 0.8, −0.1)} {(baboon, 0.3, −0.4)} {(snake, 0.7, −0.1)} {(caracal, 0.4, −0.2)} {(caracal, 0.7, −0.2), (leopard, 0.7, −0.09)}

giraffe lion rhinoceros vulture African skunk fiscal shrike grasshopper baboon leopard snake caracal mouse impala

(0.65, −0.3)

caracal

vulture

(0.7, −0.35)

(0.95, −0.1)

lion (0.33, −0.06) (0.55, −0.45)

african skunk (0.7, −0.3)

rhinoceros

(0.95, −0.1)

18) −0 .

(0.9, −0.4)

06) (0.42, −0.

)

baboon

(0.95, −0.1)

fiscal shrike

snake

(0.2, −0.7) 6, −

(0.2, −0.8)

(0.0

grasshopper

3) 0.0

(0.5, −0.5)

mouse

Fig. 4.27 Bipolar fuzzy common enemy graph

)

1 0.2 ,−

−0.28

(

85 0.3

(0.28,

(0.55, −0.5)

giraffe

3, ( 0. 3

leopard

(0.16, −

0.28)

impala (0.4, −0.7)

4.5 Applications of Bipolar Fuzzy Competition Graphs Table 4.14 Strength of common enemies between species x, y S(x, y) x, y giraffe, rhinoceros rhinoceros, African skunk African skunk, leopard rhinoceros, leopard

201

S(x, y)

0.9408 0.91

grasshopper, snake impala, baboon

0.8567 0.9933

0.9067 0.9217

mouse, impala

0.9233

3. Bipolar fuzzy competition common enemy graph Bipolar fuzzy competition common enemy graphs can be constructed from bipolar fuzzy food webs to study the relationship of common enemies as well as competition for prey between species as given in Definition 4.27.  = (A, B)  be a bipolar fuzzy food web. The bipolar fuzzy Definition 4.27 Let G  = (A, R) has the same vertices as G  competition common enemy graph C(G) + + and there is an edge between two vertices x and y if N (x) ∩ N (y) = ∅ and N − (x) ∩ N − (y) = ∅, that is, x and y has a common predator and a common prey. The membership function R is defined as μ R (x y) = (μC (x) ∧ μC (y))[h(N − (x) ∩ N − (y)) ∧ h(N + (x) ∩ N + (y))], p

p

p

μnR (x y) = (μCn (x) ∨ μCn (y))[h(N − (x) ∩ N − (y)) ∧ h(N + (x) ∩ N + (y))]. The method for calculating the strength of power of each specie according to competition for common prey and common enemies is elaborated in Algorithm 4.5.3. Algorithm 4.5.3 Bipolar fuzzy competition common enemy graph 1. Given any bipolar fuzzy food web. 2. Construct the table of bipolar fuzzy out neighborhoods and bipolar fuzzy in neighborhoods of all the species. 3. Construct the bipolar fuzzy competition graph G = (A, R) using Definition 4.27. 4. If N + (x) ∩ N + (y) = ∅ and N − (x) ∩ N − (y) = ∅, for any two species x and y then calculate the degree of each specie x using Eq. (4.16). 

(deg p (x), degn (x)) =

  p μ R (x y), μnR (x y) .

N + (x)∩N + (y)=∅, N − (x)∩N − (y)=∅

(4.16) 5. The strength of power of each specie x can be computed using Formula (4.17) as S(x) =

deg p (x) + 1 + degn (x) . 2

(4.17)

202

4 Bipolar Fuzzy Competition Graphs (0.95, −0.1)

lion 0. 1)

(0.7, −0.35)

0.3 )

(0.95, −0.1)

leopard

(0 .7,

−0 .3)

6, −

(0.6, −0.1)

(0.

(0

.6 ,

−

bobcat

)

(0.2, −0.7)

owl

(0.

hawk

racoon

(0.65, −0.3)

(0.7, −0.3)

.1 −0 6,

. (0

(0.7, 0.4, 0.1, 0.8

)

(0.2, −0.7)

7, 0. −

(0 . 8, −

(0.4, −0.

) −0.5

5) 0.

7)

,−

0.3

)

1)

.7

(0.7,

(0

)

(0

Producer grasshopper (0.2, −0.8)

snake (0.95, −0.1)

mouse

.8

,−

1 0.

fox , −0 (0.7

(0.4, −0.7)

.5)

(0.5, −0.5)

Producer

Fig. 4.28 Bipolar fuzzy food web

Consider the example of a bipolar fuzzy food web of 10 species as shown in Fig. 4.28. The description of degrees of membership of vertices and directed edges is the same as in Fig. 4.25. The bipolar fuzzy out neighborhoods and bipolar fuzzy in neighborhoods of species are given in Table 4.15. The bipolar fuzzy competition common enemy graph is shown in Fig. 4.29. The strength of power of each specie according to food competition and common enemies using Formula (4.17) is calculated in Table 4.16, which depicts that hawk is the most powerful animal in this bipolar fuzzy food web. 4. Bipolar fuzzy niche graph Niche graphs are important to study the behavior of species in ecological networks. Since all the species have different characteristics with respect to each other, bipolar fuzzy niche graphs can play a substantial role to study ecological networks more precisely. A bipolar fuzzy niche graph is defined in Definition 4.28.  = (A, B)  be a bipolar fuzzy food web. The bipolar fuzzy Definition 4.28 Let G   and there is an edge between niche graph C(G) = (C, R) has the same vertices as G + + − two vertices x and y if N (x) ∩ N (y) = ∅ or N (x) ∩ N − (y) = ∅, i.e., x and y has a common predator or a common prey, where

4.5 Applications of Bipolar Fuzzy Competition Graphs

203

Table 4.15 Bipolar fuzzy out and in neighborhoods of species x is a specie N − (x) and N + (x) grasshopper owl

N − (grasshopper) = ∅, N + (grasshopper)={(owl, 0.4, −0.7)} N − (owl)={(grasshopper, 0.4, −0.7), (mouse, 0.7, −0.5)} N + (owl) = {(bobcat, 0.7, −0.3)} N − (bobcat)={(hawk, 0.6, −0.3)}, N + (bobcat) = ∅

bobcat lion leopard racoon

N − (lion)={(racoon, 0.6, −0.1), (hawk, 0.6, −0.1)}, N + (lion) = ∅ N − (leopard)={(fox, 0.7, −0.1), (hawk, 0.6, −0.1)}, N + (leopard) = ∅ N − (racoon)={(mouse, 0.7, −0.5)} N + (racoon) = {(lion, 0.6, −0.1), (hawk, 0.7, −0.3)} N − (hawk)={(racoon, 0.7, −0.3), (mouse, 0.8, −0.3), (snake, 0.7, −0.1)}

hawk

N + (hawk) = {(lion, 0.6, −0.1), (leopard, 0.6, −0.1), (bobcat, 0.6, −0.3)}

mouse

N − (mouse) = ∅, N + (mouse)={(owl, 0.7, −0.5), (racoon, 0.7, −0.5),

fox snake

N − (fox) = {(mouse, 0.7, −0.5)}, N + (fox) = {(leopard, 0.7, −0.1)}

(hawk, 0.8, −0.3), (snake, 0.8, −0.1), (fox, 0.7, −0.5)} N − (snake) = {(mouse, 0.8, −0.1)}, N + (snake) = {(hawk, 0.7, −0.1)}

(0.95, −0.1)

(0.7, −0.35)

lion

bobcat

(0.95, −0.1) (0.12, −0.18)

leopard

owl (0.2, −0.7)

(0.2, −0.7)

(0.12, −0.18)

racoon

hawk (0.2

(0.65, −0.3)

0 4, − ) .18

(0 − 4, .1 )

07 0.

fox

grasshopper snake

(0.2, −0.8) (0.5, −0.5)

(0.95, −0.1)

mouse

Fig. 4.29 Bipolar fuzzy competition common enemy graph

(0.4, −0.7)

204

4 Bipolar Fuzzy Competition Graphs

Table 4.16 Strength of competition between species Specie Degree of each specie

Power in food web

(0.12, −0.18) (0.26, −0.25) (0.48, −0.54) (0.14, −0.07)

owl racoon hawk snake

(0.12 + 1 + 0.18)/2 = 0.65 0.755 1.01 0.605

(0.95, −0.1)

) (0.7, −0.35

(

,− 0.42

racoon

(0.30, −0.06)

(0.2, −0.7)

hawk

0.0 7

)

55, .2 −0

0.4 9

(0.4

(0. 14, − ,−

)

(0 .14

.08

1) 0.2 4, −

(0.1

−0

4 0.

(0.95, −0.1) (0.65, −0.3)

, .52 (0

,− 14 0.

.1 (0

)

leopard

(0.14, −0.21)

9)

− 4,

5)

−0 .06

(0.42, −0.06)

bobcat

3 0.

( 0. 57 ,

lion

) 0.06

)

(

1)

owl

snake

(0.14, −0.07)

(0.2, −0.7) (0.14, − 0.49)

(0

.

) 07 0. − , 35

(0.95, −0.1)

fox (0.4, −0.7)

(0.08, −0.2)

mouse (0.5, −0.5)

grasshopper (0.2, −0.8)

Fig. 4.30 Bipolar fuzzy niche graph

μ R (x y) = (μ A (x) ∧ μ A (y))[h(N − (x) ∩ N − (y)) ∨ h(N + (x) ∩ N + (y))], p

p

p

μnR (x y) = (μnA (x) ∨ μnA (y))[h(N − (x) ∩ N − (y)) ∨ h(N + (x) ∩ N + (y))]. The bipolar fuzzy niche graph of Fig. 4.28 is shown in Fig. 4.30. The edge between species which have only common prey is represented by dashed lines and dotted line represent the species relationship having only common enemies and solid lines represent the species which have both common prey and common enemies.

4.5 Applications of Bipolar Fuzzy Competition Graphs

205

4.5.2 Competitive Market In business marketing, there are rivalry competitions among companies which are trying to increase the profit and demand of their products. There are more than one companies which sell the identical products to retailers and other companies. As different companies often market the same types of products, every company tries to attract consumer’s attention to its product. There is always a competitive environment in business marketing. Graph theory is a key approach to study the competition between buyers and sellers using graphical structures. But sometimes these graphical structures cannot be defined precisely. For example, companies are different according to annual profit and loss. These are bipolar information, which are fuzzy in nature. This idea motivates the necessity of bipolar fuzzy competition graphs.The method for calculating the strength of competition between companies is explained in Algorithm 4.5.4. Algorithm 4.5.4 Algorithm for computing the strength of competition in business market 1. 2. 3. 4. 5.

Input the bipolar fuzzy set A of n companies. Input the adjacency matrix of bipolar fuzzy relation B of companies. Construct the table of bipolar fuzzy out neighborhoods of all the companies. Construct the bipolar fuzzy competition graph G = (A, R) using Definition 4.3. If N + (x) ∩ N + (y) = ∅ for any two companies x and y then calculate the degree of each specie x as, (deg p (x), degn (x)) =



p

(μ R (x y), μnR (x y)).

N + (x)∩N + (y)=∅

6. The strength of competition of each company x can be computed using formula: S(x) =

deg p (x) + 1 + degn (x) . 2

(4.18)

Consider the example of a business competition between 7 companies DEL, CB, HW, AK, LR, RP, SONY, RA, LR, 3 retailers, 1 retailer outlet, and 1 multinational brand as shown in Fig. 4.31. The vertices represent companies, retailers, Outlets, and brands. The positive degree of membership of each vertex represents the degree of annual loss of each company\retailer\industry and the negative degree of membership represents the annual profit. These characteristics can be collected in the form of a set as, {annual lose, annual profit}. The positive degree of membership of each directed edge xy represents the degree of rejectability of the product of company x by company y. The negative degree of membership represents the quantity of product purchased by the company y from x. The relationship between the companies can be written in a set as,

206

4 Bipolar Fuzzy Competition Graphs (0.1, −0.95)

Multinational Brand

(0.2, −0.9) (0.2, −0.7)

0.8 ) ,−

(0.6, −0.9)

Retailer3 ( 0.1

, −0

.8)

1, −

(0.2, −0.9)

0 .6

AK

.7)

(0.2, −0.9)

(0.

(0.2 , −0

RP

)

(0.2, −0.8)

Retailer2

LR

−0.7

)

Retailer1

.3,

(0.1, −0.6)

2, −

(0.5, −0.6) 0.6 )

(0.4, −0.8)

DEL

) −0.6

(0.

.7)

(0.1,

( 0. 3 , −0

(0.3, −0.8)

(0

CB

)

2

(0.5, −0.7)

(0.5,

−0 .8)

(0.

,−

) 0.7

TS Chemical and Plastic Industry (0.3, −0.6)

(0

.4,

(0 .3 ,− 0. 7

(0 .1

SONY (0.3, −0.8)

) .65 −0

(0.2, −0.9)

.7) , −0 ( 0 .2

0. 9)

.1,

.2 ,−

) 0.8 ,− .05

(0

(0

(0

Retail Outlet

.8) −0

(0.3, −0.8)

Fig. 4.31 Bipolar fuzzy marketing digraph

{rejectability, product quantity purchased}. The strength of competition between companies can be discussed using bipolar fuzzy competition graph. The bipolar fuzzy out neighborhoods are calculated in Table 4.17. The bipolar fuzzy competition graph of Fig. 4.31 is shown in Fig. 4.32. The degree of membership of each edge between companies shows the strength of competition between the companies for rejectability and purchase of their product. The strength of competition of each company is calculated in Table 4.18, which shows its competitive value within business market. Table 4.18 shows that AK is the most competitive company in market.

4.5.3 Political Competition Local governments are the public administration in towns, districts, and cities to govern a specific geographic area. Many countries have local governments in their cities. Local governments are important because the people of the area have much contact with local officials as compared to the federal government. The selection criteria of local government officials are different in different countries. Some countries

4.5 Applications of Bipolar Fuzzy Competition Graphs

207

Table 4.17 Bipolar fuzzy out neighborhoods of companies Company N + (u) : u is a company {(DEL, 0.4, −0.8), (AK, 0.3, −0.8), (Retailer1, 0.1, −0.6), (CB, 0.3, −0.7), (TS, 0.3, −0.6)}

chemical and plastic industries DEL AK LR Retailer1 CB TS Retailer2 SONY Retailer3 RP M. Brand R. Outlet

{(LR, 0.3, −0.7)} {(Multinational Brand, 0.05, −0.8)} {(Multinational Brand, 0.1, −0.65)} {(SONY, 0.2, −0.7), (RP, 0.1, −0.6), (Retailer2, 0.5, −0.7)} {(Retailer2, 0.2, −0.7)} {(Retailer2, 0.2, −0.6)} {(RP, 0.1, −0.8)} {(Retailer3, 0.2, −0.7), (R. Outlet, 0.2, −0.7), (M. Brand, 0.1, −0.7)} {(R.Outlet, 0.2, −0.9)} {(Retailer3, 0.2, −0.9), (R. Outlet, 0.1, −0.8)} ∅ ∅

(0.1, −0.95)

Multinational Brand

(0.2, −0.9)

Retail Outlet (0.3, −0.8)

0.16) (0.04, −

SONY )

Retailer3 RP

(0.0

0.16 4, −

( 0. 015 ,− 0. 0

4)

(0.2, −0.9)

(0.6, −0.9)

AK (0

(0.2, −0.9)

3 .0

0. 0

6)

(0.1, −0.6)

0.035)

1, −

Retailer1

0.12)

(0.025, −

(0. 0

Retailer2

(0.02, −

7)

0 0. ,−

(0.2, −0.8)

LR CB (0.3, −0.8)

(0

(0.5, −0.7)

.0

6, −

0.

)

12

)

(

2 0. 0

,−

2 0.1

DEL (0.4, −0.8)

TS (0.5, −0.6)

Chemical and Plastic Industry (0.3, −0.8)

Fig. 4.32 Bipolar fuzzy competition graph of business network

208

4 Bipolar Fuzzy Competition Graphs

Table 4.18 Strength of competition between companies Company Degree of company (0.055, −0.105) (0.04, −0.075) (0.125, −0.43) (0.04, −0.16) (0.04, −0.16) (0.01, −0.06) (0.05, −0.30) (0.08, −0.24) (0.08, −0.24)

LR AK SONY Retailer3 RP Retailer2 Retailer1 CB TS

Competition strength 0.475 0.4825 0.3475 0.44 0.44 0.475 0.375 0.42 0.42

have direct public elections and some countries elect the local government officials by the decisions of a governing council. Usually, there are five common positions in a local government, namely, mayor, city manager, city council member, county commissioner, and city attorney. A procedure for the governing council to select the politicians on the basis of leadership quality, eligibility, and incapacity of candidates is explained in Algorithm 4.5.5. Algorithm 4.5.5 Algorithm for calculating the strength of competition for political seats 1. 2. 3. 4.

Given a bipolar fuzzy digraph of candidates and political seats. Construct bipolar fuzzy out neighborhoods of all the candidates. Construct bipolar fuzzy competition graph using Definition 4.3. Calculate the strength of competition A(x, p) of each candidate x for a particular seat p. If x, a1 , a2 , . . . , an are the candidates, which are competing for political seat p, then A(x, p) = (A p (x, p), An (x, p)) =

 1 p p p μ (xa1 ) + μ R (xa2 ) + · · · + μ R (xan ), μnR (xa1 ) + μnR (xa2 ) + · · · + μnR (xan ) . n R

5. Calculate S(x, p), the strength of competition of each candidate x for seat p as, S(x, p) = A p (x, p) + 1 + An (x, p). Consider the example of 7 persons in a country named Jeorge Estregan, Ramil, Dave Miranda, Jaun Menzoda, Maria Evita, Neil Andrew, and Bresies Jackson competing for different positions. A bipolar fuzzy digraph is shown in Fig. 4.33, which represents the candidates competing for particular seats. The positive degree of membership of each candidate represents the degree of leadership quality and negative degree of membership represents the incapacity to manage public affairs. The candidate characteristics can be represented in a set as {leadership, incapacity}. The positive degree of membership of each directed edge between a candidate and a position

4.5 Applications of Bipolar Fuzzy Competition Graphs

209

(0.7, −0.45) Jeorge Estregan

(0.7, −0.25)

(0 .6,

−0

0.3 7 ,−

Mayor (0.9, −0.45)

(0.5, −0.

Ramil

6, − (0.

City manager (0.9, −0.35)

(0.6 .35 )

3)

County commissioner (0.85, 0.45)

(0.6 ,

75)

0.3

− 0.

(0.6, −0.3)

3)

5) , −0.2 Council member (0.8, −0.325)

, (0.6 (0 .6

−

,−

) 0.25

(0.5, −0.3)

Jaun Menzoda

(0.6, −0.375)

(0.6, −0.4)

) 0.3

(0.7, −0.4)

(0 .7

− .5, (0

Dave Miranda

5)

(0.7, −0.25)

Neil Andrew

0. 3

)

City attorney (0.8, −0.25)

Maria Evita

Bresies Jackson

(0.7, −0.3) (0.7, −0.25)

(0.7, −0.3)

(0.7, −0.7)

Fig. 4.33 Bipolar fuzzy digraph of political seats

indicates the degree of eligibility for the position. The negative degree of membership represents the degree of legal disqualification for that position. These membership properties can be written in a set as {eligibility, legal disqualification}. The positive and negative degrees of membership of each political position represent the average past leadership quality and incapacity of politicians for these positions. A bipolar fuzzy competition graph can be constructed to describe the competition between candidates for the political positions. The bipolar fuzzy out neighborhoods are given in Table 4.19.

Table 4.19 Bipolar fuzzy out neighborhoods of candidates Candidate N + (candidate) Bresies Jackson Neil Andrew Ramil Jeorge Estregan Dave Miranda Jaun Menzoda Maria Evita

{(Council member, 0.6, −0.6)} {(Council member, 0.6, −0.25), (City manager, 0.6, −0.3), (City attorney, 0.8, −0.25)} {(City manager, 0.6, −0.375), (Mayor, 0.5, −0.3)} {(Mayor, 0.7, −0.375), (County commissioner, 0.6, −0.4)} {(Mayor, 0.5, −0.3), (City attorney, 0.7, −0.25)} {(County commissioner, 0.6, −0.35), (City attorney, 0.7, −0.25)} {(County commissioner, 0.7, −0.3), (Council member, 0.7, −0.3)}

210

4 Bipolar Fuzzy Competition Graphs (0.7, −0.45)

(0.7, −0.25)

,− (0.35

Jeorge Estregan

) 0.125

17 5)

(0.

42,

−0

.17

5)

(0.42, −0.21)

Neil Andrew (0.6, −0.3)

0. 2

1)

Jaun Menzoda

.18)

Council member

(0.3 6

) .18

, −0

−0

(0 .

42,

49 ,−

(0.

County commissioner

.18)

City manager

Mayor

(

(0.7, −0.4)

(0.6, −0.375) Ramil

)

4 0.2 ,−

75)

0 6, −

0.

0.18

(0.3

(0 .4 9, −

0, −

(0.30, −0.125)

Dave Miranda

2 0.4

(0.3

City attorney

Maria Evita

Bresies Jackson

(0.7, −0.3)

)

0.18 (0.36, −

(0.7, −0.35)

(0.42, −0.18 )

Fig. 4.34 Bipolar fuzzy political competition graph

The bipolar fuzzy competition graph is shown in Fig. 4.34. The solid lines show the strength of competition between two candidates. The dashed lines indicate the candidates competing for particular seats. For example, Neil Andrew and Ramil both are competing for the seat of the City manager. The strength of competition between them is (0.36, −0.18). In Table 4.20, A(x, p) represents the competition of candidate x for seat p with respect to leadership quality and incapacity to manage public affairs. The strength of competition of all the candidates with respect to particular seats is calculated in Table 4.20. From Table 4.20, Bresies Jackson and Neil Andrew are equally eligible to be selected for the seat of Council member, Dave Miranda for city attorney, Maria Evita for County commissioner, Jeorge Estregan for Mayor and Neil and Ramil for the seat of the City manager.

4.5.4 Social Competition in Bipolar Fuzzy Environment Social competition is a widespread mechanism to figure out a best-suited group economically, politically, or educationally. Social competition occurs when an individual’s opinions, decisions, and behaviors are influenced by others. Graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network. Bipolar fuzzy networks can be used to study the influence and competition between objects more precisely. The strength of influence between

4.5 Applications of Bipolar Fuzzy Competition Graphs Table 4.20 Strength of competition of candidates for political seats (Candidate, seat) In competition A(candidate, seat) (Bresies, Council member) (Neil, Council member) (Maria, Council member) (Neil, City attorney) (Jaun, City attorney) (Dave, City attorney) (Maria, County commissioner) (Jaun, County commissioner) (Jeorge, County commissioner) (Neil, City manager) (Ramil, City manager) (Ramil, Mayor) (Jeorge, Mayor) (Dave, Mayor)

211

S(candidate, seat)

Neil, Maria

(0.39, −0.18)

1.21

Bresies, Maria

(0.36, −0.18)

1.18

Bresies, Neil

(0.39, −0.18)

1.21

Jaun, Dave Neil, Dave Neil, Jaun Jeorge, Jaun

(0.42, −0.1925) (0.455, −0.1925) (0.455, −0.175) (0.455, −0.195)

1.2275 0.64875 1.28 1.26

Jeorge, Maria

(0.42, −0.21)

1.21

Jaun, Maria

(0.455, −0.225)

1.23

Ramil Neil Jeorge, Dave Ramil, Dave Ramil, Jeorge

(0.36, −0.18) (0.36, −0.18) (0.30, −0.15625) (0.325, −0.15625) (0.325, −0.125)

1.18 1.18 1.14375 1.16875 1.2

different objects in a bipolar fuzzy influence graph can be calculated by the method presented in Algorithm 4.5.6. Algorithm 4.5.6 Algorithm for calculating the strength of influence in a social network  of social relationship among families. 1. Given any bipolar fuzzy digraph G 2. Construct the table of bipolar fuzzy in neighborhoods of all the families. 3. Construct the bipolar fuzzy influence graph G = (A, R) as,  and there is an edge between two vertices x G has the same set of vertices as G − − and y if N (x) ∩ N (y) = ∅. 4. If N − (x) ∩ N − (y) = ∅ for any two vertices x and y then calculate the average strength of competition of each vertex x as, (Average p (x), Averagen (x)) =

(deg p (x), degn (x)) . number of vertices adjacent to x

5. Find the total strength of competition of each vertex x as S(x) = Average p (x) + 1 + Averagen (x).

212

4 Bipolar Fuzzy Competition Graphs

5) 0.2

(

, −0 .2 5 )

−0 .3)

Gaudagni (0.5, −0.3)

Ridolfi

6)

(0.6, −0.3)

0.3 )

(0.4, −0.25)

, .4

0.

(0.4

(0 .5,

(0.5, −0.3)

(0

5, 0.

(0.6, −0.3)

(0.4, −0.25)

5)

Strozzi

Tornabuon

(0.6, −0.35)

2 0.

.25 )

Lambertes (0.4, −0.25)

−

−0

4,

.4,

. (0

(0

(0.4, −0.25)

Perozzi

(0.4, −0.25)

(0.3 , −0 .2)

Bischeri (0.4, −0.25)

(0.3, −0.2)

,− 0.4

(0 .6, −

Castellan

(0.5, −0.25)

(0.6, −0.4)

Albizzi

(0.6, −0.4)

Medici

(0.9, −0.4)

)

(0.5, −0.3)

5 −0.2

5) 0.2

Babadori

(0.5, −0.25)

(0.6, −0.25)

)

(

3 0. − 5, 0.

Acciaiuol

(0.5, −0.25)

, (0.5

6, − (0.

Salviati

Ginori

(0.5, −0.3)

(0.

4, − 0.2

(0.5, −0.25)

5)

Pazzi

(0.4, −0.25)

(0.5, −0.3)

Fig. 4.35 Bipolar fuzzy social digraph

Consider a bipolar fuzzy social network of Florentine trading families Peruzzi, Lamberts, Bischeri, Strozzi, Guadagni, Tornabuoni, Castellan, Ridolfi, Albizzi, Barbados, Medici, Acciaiuol, Salviati, Ginori, and Pazzi. The vertices in bipolar fuzzy digraph represent the name of trading families. The positive degree of membership of each family represents the strength of centrality and negative degree of membership represents the strength of weak position of each family in the network. The description of the degree of membership value families can be written in the form of a set as {centrality, weak position}. The directed edge xy indicates that the family x is influenced by y. The degree of membership of each directed edge indicates the extent to which the opinions and suggestions of one family influence the others and negative membership value represents the strength of suggestions ignored by family x. The influence factors can be written in the form of a set as {followed suggestions, ignored suggestions}. The degree of membership of Medici is (0.9, −0.4), which shows that Medici has 90% central position in trading network and 40% weak position. The degree of membership between Redolfi and Medici is (0.7, −0.3), which indicates that Redolfi follows 70% of the suggestions of Medici and ignore 30% of its suggestions. The bipolar fuzzy influence graph is shown in Fig. 4.35.

4.5 Applications of Bipolar Fuzzy Competition Graphs

213

Table 4.21 Bipolar fuzzy in neighborhoods of families in social network Family

N − (family)

Family

N − (family)

Acciaiuol

{(Babadori, 0.5, −0.25)}

Pazzi

∅

Ginori

{(Albizzi, 0.5, −0.25)}

Salviati

{(Pazzi, 0.4, −0.25)}

Babadori

{(Castellan, 0.5, −0.25)}

Castellan

{(Strozzi, 0.4, −0.25)}

Tornabuon {(Gaudagni, 0.5, −0.3)} Lambertes Medici

∅

Perozzi

{(Castellan, 0.5, −0.25)}

Strozzi

{(Perozzi, 0.4, −0.25)}

{(Babadori, 0.6, −0.25), (Acciaiuol, 0.5, −0.3), (Salviati, 0.5, −0.3), (Ridolfi, 0.6, −0.3)}

Bischeri

{(Perozzi, 0.4, −0.25), (Strozzi, 0.4, −0.25), (Redolfi, 0.4, −0.25)}

Albizzi

{(Medici, 0.6, −0.4), (Gaudagni, 0.5, −0.3)}

Redolfi

{(Strozzi, 0.4, −0.25), (Tornabuon, 0.6, −0.3)}

Gaudgani

{(Bischeri, 0.3, −0.2), (Lambertes, 0.3, −0.2)}

A bipolar fuzzy influence graph can be constructed to show that which family is the most influential in the bipolar fuzzy network. The bipolar fuzzy in neighborhoods are given in Table 4.21. The bipolar fuzzy influence graph is given in Fig. 4.36, which shows the strength of social competition between families to influence other families in the social network. The strength of competition of families is calculated in Table 4.22, where Average(x) represents the strength of influence of family x according to its suggestions followed and ignored by other families. S(x) represents the total strength of competition of each family to influence other families.

4.5.5 Interactions and Conflicts in Communication Networks Due to the increase of wireless devices, signal conflicts and confusions over communication networks are a common issue. Since the air is shared by all transmitters, transmissions by any device at an equivalent frequency will cause interference. The interference of signals is highly dependent on the strength of interference of physical objects, environment factors, and electronic devices. The phenomenon of signal conflicts and confusions in communication networks are discussed in the following two example applications. 1. Bipolar fuzzy conflict graph In wireless networking, signals are transmitted through the air. Some channels are operating signals at the same frequencies and range that can cause conflict with each other and it affects the network performance badly. Competition graphs arise in the theory of network interference but these graphical representations do not include all the competitions between signals. The crisp competition graphs cannot

214

4 Bipolar Fuzzy Competition Graphs Lambertes

(0.4, −0.25)

Bischeri (0.4, −0.25)

Tornabuon

, 16 1) 0. −

Gaudagni

Strozzi

75

Ridolfi

(0.4, −0.25)

(0.

1 0.

16 ,−

,−

(0.1 6, −

(0.5, −0.3)

0 .3 (0

0.1 )

0.1)

(0.6, −0.35)

. (0

(0.16, −0.1)

Perozzi (0.4, −0.25)

)

(0.6, −0.3)

−0.1) (0.20,

Castellan

(0.5, −0.25)

Albizzi (0.6, −0.4)

Medici (0.9, −0.4)

Babadori

Pazzi

0. 15

)

(0.6, −0.25)

5, −

(0.4, −0.25)

(0 .2

Ginori

(0.5, −0.25)

Salviati

Acciaiuol

(0.5, −0.3)

(0.5, −0.3)

Fig. 4.36 Bipolar fuzzy influence graph Table 4.22 Strength of competition in social network family

Average(family) S(family)

family

Average(family) S(family)

Acciaiuol

(0.25, −0.15)

1.1

Medici

(0.25, −0.15)

1.1

Babadori

(0.16, −0.1)

1.06

Perozzi

(0.16, −0.1)

1.06

Castellan

(0.18, −0.1)

1.08

Redolfi

(0.18, −0.1)

1.08

Strozzi

(0.16, −0.1)

1.06

Besceri

(0.16, −0.1)

1.06

represent the degree of conflict between signals. To reduce this limitation, bipolar fuzzy competition graphs can be used to study the degree of conflict of signals. The method to calculate the strength of signal conflict between wireless devices in a bipolar fuzzy conflict graph is given in Algorithm 4.5.7. Algorithm 4.5.7 Algorithm for the calculation of the strength of conflict between signals 1. Given a bipolar fuzzy network of n wireless devices. 2. Construct bipolar fuzzy conflict graph using Definition 4.3. 3. Calculate the strength of signal conflict of each sender device in wireless connection as follows: If x, a1 , a2 , . . . , an are the sender devices such that

4.5 Applications of Bipolar Fuzzy Competition Graphs Sender 3 (0.6, −0.6)

Sender 1 (0.8, −0.3)

8, −

, −0

(0.8, −0.7)

) 0.3

(0.7

Receiver 2 (0.9, −0.8)

(0.6, −0.5)

(0.

Sender 2 (0.7, −0.5)

215

.5) Receiver 3 (0.9, −0.5)

0.6 )

Sender 4 (0.9, −0.7)

)

− .7, (0

Sender 5 (0.8, −0.6)

)

0. 5

) 0.5

−0 .5)

Sender 6 (0.7, −0.5)

(0 .7,

− .7, (0

(0.6, −0.4)

Sender 7 (0.7, −0.5)

(0.9 , −0 .5)

(0 .8, −

(0 .7 ,

−

0.

0.5 (0.7, −

5)

Receiver 1 (0.8, −0.6)

Receiver 4 (0.8, −0.6)

(0.6, −0.6)

Sender 8 (0.6, −0.6)

Fig. 4.37 Communication over a noisy channel

p

(μ R (xai ), μnR (xai )) = (0, 0), for each 1 ≤ i ≤ n, then the average degree of conflict A(x) = (A p (x), An (x)) of each sender device x is, p

A(x) =

p

p

(μ R (xa1 ) + μ R (xa2 ) + · · · + μ R (xan ), μnR (xa1 ) + μnR (xa2 ) + · · · + μnR (xan )) . n

4. Calculate the strength of conflict of device x to other nearby devices as S(x) = A p (x) + 1 + An (x). Assume that there are 8 sender devices and 4 receiver channels. The bipolar fuzzy digraph is shown in Fig. 4.37 in which vertices represent senders and receivers. The positive degree of membership of each vertex represents the degree of strength of the signal. The negative degree of membership represents the degree of weakness of the signal because the signal spread apart as it gets weaker. These factors can be represented in a set as {strength, weakness}. The positive degree of membership of each directed edge between sender and receiver represents the degree of continuity of signal and negative degree of membership represents the degree of interference of physical objects, environmental interference, and electronic devices. These characteristics can be collectively written as {continuity, interference}. The bipolar fuzzy conflict graph (A, R) can be used to study the strength of signal conflict of sender devices in the wireless network. For this, the bipolar fuzzy out neighborhoods are given in Table 4.23.

216

4 Bipolar Fuzzy Competition Graphs

Table 4.23 Bipolar fuzzy out neighbohoods of noisy channel Sender N + (Sender) Sender

N + (Sender)

{(Receiver1, 0.8, −0.3)} Sender2 {(Receiver1, 0.7, −0.5)} {(Receiver3, 0.6, −0.5)} Sender8 {(Receiver4, 0.6, −0.6)} {(Receiver2, 0.8, −0.7), (Receiver3, 0.9, −0.5)} {(Receiver1, 0.7, −0.5), (Receiver3, 0.8, −0.6)} {(Receiver2, 0.7, −0.5), (Receiver4, 0.7, −0.5)} {(Receiver1, 0.7, −0.5), (Receiver3, 0.7, −0.5), (Receiver4, 0.6, −0.4)}

Sender1 Sender3 Sender4 Sender5 Sender6 Sender7

Sender 1 (0.8, −0.3)

Sender 3 (0.6, −0.6)

(0

)

.21

.36

,−

(0

.

,− 36

0. 3

6)

0)

3 0.

Sender 4 (0.9, −0.7)

9 .4 ,−

(0.4

(0. 36,

(0

(0.

49,

−0 .36 )

−0

Sender 2 (0.7, −0.5)

)

.21 )

(0.56, −0.21

(0

0 ,− .49

5)

3 0.

9, − )

0.35 (0.36, −0.30)

Sender 7 (0.7, −0.5) (0.

49,

−0

(0.42 ,−

.35 )

Sender 8 (0.6, −0.6)

0.30

) (0

− 6,

0)

0.3

.3

Sender 6 (0.7, −0.5)

Sender 5 (0.8, −0.6)

Fig. 4.38 Bipolar fuzzy conflict graph Table 4.24 Strength of conflict between wireless devices Sender A(Sender) S(Sender) Sender Sender1 Sender3 Sender5 Sender7

(0.51, −0.21) (0.36, −0.34) (0.475, −0.3175) (0.435−, 0.3017)

1.3 1.02 1.1575 1.13

Sender2 Sender4 Sender6 Sender8

A(Sender)

S(Sender)

(0.49, −0.30) (0.36, −0.36) (0.39, −0.30) (0.36, −0.3)

1.19 1.00 1.09 1.06

The bipolar fuzzy conflict graph is shown in Fig. 4.38. The degree of membership of each edge represents the degree of conflict between two devices for strength and weakness of signals due to the interference of different physical and electronic objects. The strength of conflict of all the devices is given in Table 4.24.

4.5 Applications of Bipolar Fuzzy Competition Graphs

217

2. Bipolar fuzzy confusion graph Another application of bipolar competition graphs is in the communication network over a noisy channel. Through a noisy channel, a message A may be received as A due to atmospheric noise, transit-time noise, or electronic noise. Assume that a transmitting device transmits three signals a, b, c and the receiving channel receives the signals as α, β, γ. Unfortunately, due to noise over the channel, confusion in receiving the signals is possible: a could be received as α or β, b could be received as β or γ, c could be received as γ or α. Graphical models can be used to study the confusion of signals in message strings but these models do not take into consideration the degree of noise because every path may have a different degree of noise. Therefore, bipolar fuzzy confusion graphs are more effective to study the strength of confusion in message strings. A method in Algorithm 4.5.8 is given to calculate the strength of confusion between signals in a bipolar fuzzy confusion graph. Algorithm 4.5.8 Algorithm for computing the strength of confusion between signals 1. Given a bipolar fuzzy communication network of sender and receiver communication devices. 2. Construct the bipolar fuzzy confusion graph using Definition 4.3. 3. If x y and uv are two message strings in a bipolar fuzzy confusion graph, then calculate the strength of confusion that x y could be received as uv as p

S(x y, uv) = μ R (x yuv) + 1 + μnR (x yuv). Consider the example that Bob sends a message of two letter strings, namely, aa, ab, ac, bc, bb, cc. The possible strings aa could be received as are: αα, αβ, βα, ββ. The possible receiving strings for each sent string are given in Table 4.25. The bipolar fuzzy digraph is shown in Fig. 4.39. The set of vertices is the set of strings. The positive degree of membership of each vertex represents the degree of strength of signals and negative degree of membership represents the weakness of signals. These properties of signals can be written collectively as {strength, weakness}. There is a directed edge between between two strings x y and uv if x y is sent and uv could be received. The positive degree of membership of each directed edge

Table 4.25 Communication over a noisy channel Sent string Possible received strings aa ab ca bb cc

αα, αβ, βα, ββ αβ, αγ, ββ, βγ γα, γβ, αα, αβ ββ, βγ, γβ, γγ γγ, γα, αγ, αα

218

4 Bipolar Fuzzy Competition Graphs αβ (0.7, −0.35)

(0.4, −0.2)

5) 0.1 (0. 3, −

. (0

.2)

)

−0

−0.2

(0. 4,

βα (0.7, −0.35)

, (0.4

ββ (0.7, −0.35)

− 4, (0.

) 0.2

− 5, ) 25 0.

4, − (0.

(0.5, −0.25)

5) 0.1

aa (0.7, −0.4)

0.2 (0.4, − βγ (0.6, −0.35)

ab (0.6, −0.35)

(0.3, −0.2)

(0.3 ,

−0.2

ca (0.8, −0.4)

3, − (0.

)

0.2

5)

(0 .5

)

(0 .5, −

,−

0.2 5

)

15)

γα (0.8, −0.4)

0.2

)

)

0.2

)

γβ (0.6, −0.35)

(0.4, −0.

,−

5 0.1

cc (0.7, −0.35)

.5 (0

3, − (0.

15 (0.4, −0.

(0 .3 ,−

0. 2)

bb (0.9, −0.4)

αγ (0.6, −0.35)

αα (0.7, −0.35)

)

(0.

5, −

) 0.2

γγ (0.8, −0.4)

Fig. 4.39 Bipolar fuzzy noisy channel Table 4.26 Bipolar fuzzy out neighborhoods of noisy channel xy N + (x y) aa ab ca bb cc

{(αα, 0.5, −0.25), (αβ, 0.4, −0.2), (βα, 0.4, −0.2), (ββ, 0.4, −0.2)} {(αβ, 0.3, −0.15), (αγ, 0.3, −0.2), (ββ, 0.5, −0.25), (βγ, 0.3, −0.2)} {(γα, 0.5, −0.25), (γβ, 0.5, −0.2), (αα, 0.4, −0.2), (αβ, 0.4, −0.2)} {(ββ, 0.4, −0.15), (βγ, 0.3, −0.2), (γβ, 0.4, −0.15), (γγ, 0.5, −0.2)} {(γγ, 0.5, −0.2), (γα, 0.4, −0.15), (αγ, 0.3, −0.15), (αα, 0.3, −0.25)}

represents the degree of harmony of the signal and negative degree of membership represents the degree of noise. The noise categories can be written in the form of a set as {harmony, noise}. A bipolar fuzzy confusion graph can be constructed to study the strength of confusion between message strings. For this, the bipolar fuzzy out neighborhoods given in Table 4.26. The bipolar fuzzy confusion graph (A, R) is shown in Fig. 4.40. The edge between aa and ab has a degree of membership (0.24, −0.14), which shows that the harmony of signal aa is 24% and 14% surity for aa to be received as ab due to channel noise. The strength of confusion of each pair of message string is given in Table 4.27, which shows that aa could be confused with bb or ca, ab could be received as aa or bb, bb and cc can confuse with each other, ca can confuse with bb.

4.6 Conclusions

219 aa (0.7, −0.4)

−0 .16 )

,−

(0.

28,

(

ab (0.6, −0.35)

,−

0. 1

6)

) .105 , −0

4 0 .2

(0 .28

1 (0.2

)

4 0.1

ca (0.8, −0.4)

(0.18, −0.105)

bb (0.9, −0.4)

32

,

(0. 28,

(0 .18 ,−

−0

.14

)

) 0.14 4, − (0.2 (0.

) .16 −0

0. 1

05

(0.35, −0.175)

)

cc (0.7, −0.35)

Fig. 4.40 Bipolar fuzzy confusion graph Table 4.27 Strength of confusion (x y, uv) S(x y, uv) (x y, uv) (aa, ab) (ab, bb) (bb, cc)

1.10 1.1 1.175

(aa, bb) (ab, cc) (ca, cc)

S(x y, uv) (x y, uv) 1.12 1.075 1.12

(aa, cc) (ab, ca)

S(x y, uv) (x y, uv) 1.105 1.075

(aa, ca) (bb, ca)

S(x y, uv) 1.12 1.16

4.6 Conclusions Bipolar fuzzy competition graphs are the important mathematical structures to represent point to point interconnections, conflicts, confusions, and competitions among objects in economic systems, ecological niches, noisy channels, and other conflicting networks. In this chapter, we have discussed different concepts such as bipolar fuzzy out neighborhoods, bipolar fuzzy in neighborhoods, bipolar fuzzy competition graphs, bipolar fuzzy k−competition graphs, bipolar fuzzy open neighborhood graphs, bipolar fuzzy closed neighborhood graphs, and underlying bipolar fuzzy graphs. We have described the construction of bipolar fuzzy competition graphs of certain products of bipolar fuzzy digraphs from their respective bipolar fuzzy competition graphs and illustrated these fundamental techniques with graphical examples. We have also elaborated the relation of underlying bipolar fuzzy graphs with bipolar fuzzy [k]−competition graphs and bipolar fuzzy (k)−competition graphs in the existence and nonexistence of loops. We have also studied different types competition graphs under complex bipolar fuzzy environment. We have presented algorithms to

220

4 Bipolar Fuzzy Competition Graphs

compute the strength of competition, the strength confusion and conflict, and the strength of influence using application examples of food webs, business marketing, political rivalry, and communication over noisy channels.

Exercises 4  1 and G  2 be two bipolar fuzzy digraphs as shown in Fig. 4.41. Construct 1. Let G  2 ) using Theorem 4.2.  1 G C(G  2 be two bipolar fuzzy digraphs as shown in Fig. 4.42. Construct  1 and G 2. Let G  × H ) using Theorem 4.3. C(G 1  3. Derive a method for the construction of bipolar fuzzy competition graph C(G  1 and G  2 using C(G  1 ) and  2 ) of the strong product of bipolar fuzzy digraphs G G  2 ). C(G 4. If all the edges of a complex bipolar fuzzy digraph are strong then prove using a contradiction example that all the edges of corresponding complex bipolar fuzzy competition graph may not be strong. 1 ◦ 5. Derive a method for the construction of bipolar fuzzy competition graph C(G     G 2 ) of the composition of bipolar fuzzy digraphs G 1 and G 2 using C(G 1 ) and  2 ). C(G 1 • 6. Derive a method for the construction of bipolar fuzzy competition graph C(G    G 2 ) of the lexicographic product of bipolar fuzzy digraphs G 1 and G 2 using  2 ).  1 ) and C(G C(G p n   7. Let G = (A, D) be a complex bipolar fuzzy digraph, k = (k p eiθ , k n eiθ ) be an ordered pair of complex numbers such that k p ∈ [0, 1], k n ∈ [−1, 0], θ p ∈ [0, π], θn ∈ [−π, 0]. If for some x, y ∈ X, h(N + (x) ∩ N + (y)) = (eiπ , −e−iπ ),

 p n |N + (x) ∩ N + (y)| = l p r iφ , l n eiφ

l p > 2k p , φ p > 2θ p , l n < 2k n , φn < 2θn  then prove that the edge x y is strong in Ck (G).

Fig. 4.41 Bipolar fuzzy graphs

4.6 Conclusions

221

Fig. 4.42 Bipolar fuzzy  1 and G 2 digraphs G

8. Prove or disprove that for every edge of a complex bipolar fuzzy graph G, there exists an edge in N [G]. 9. Define a complex bipolar fuzzy (k)-competition graph, Nk (G), of a complex  = (A, D)  be a symmetric complex bipolar fuzzy graph G = (A, D). Let G  = Nk (U(G)),  bipolar fuzzy digraph without any loops then prove that Ck (G)  is the underlying complex bipolar fuzzy graph of G.  where U(G) 10. Define a complex bipolar fuzzy [k]-competition graph, Nk [G], of a complex  = (A, D)  be a symmetric complex bipolar fuzzy graph G = (A, D). Let G  = bipolar fuzzy digraph having loops at every vertex, then prove that Ck (G)  where U(G)  is the underlying complex bipolar fuzzy graph of G.  Nk (U(G)), 11. Derive the methods for the construction of complex bipolar fuzzy competition  2 ), C(G 1 × G  2 ), C(G 1 • G  2 ), C(G 1 ∪ G  2 ) and C(G 1 ∩ G  2 ) of  1 G graphs C(G  2 using C(G  1 ) and C(G  2 ).  1 and G complex bipolar fuzzy digraphs G

References 1. Akram, M., Dudek, W.A.: Regular bipolar fuzzy graphs. Neural Comput. Appl. 21(supp 1), 197–205 (2012) 2. Akram, M., Sattar, A.: Competition graphs under complex Pythagorean fuzzy information. J. Appl. Math. Comput. 1–41 (2020) 3. Akram, M., Yousaf, M.M., Dudek, W.A.: Self centered interval-valued fuzzy graphs. Afrika Mathematika 26(5–6), 887–898 (2015) 4. Akram, M., Sarwar, M.: Novel applications of m−polar fuzzy competition graphs in decision support system. Neural Comput. Appl. 30, 3145–165 (2018) 5. Alkouri, A.U.M., Massa’deh, M.O., Ali, M.: On bipolar complex fuzzy sets and its application. J. Intell. Fuzzy Syst. (Preprint) (2020) 6. Alshehri, N.O., Akram, M.: Bipolar fuzzy competition graphs. Ars Combinatoria 121, 385–402 (2015) 7. Cohen, J.E.: Interval graphs and food webs: a finding and a problem. Document 17696-PR, RAND Coporation, Santa Monica, CA (1968)

222

4 Bipolar Fuzzy Competition Graphs

8. Dudek, W.A., Talebi, A.A.: Operations on level graphs of bipolar fuzzy graphs, Buletinul Academiei de Stiinte a Republici Moldova. Mathematica. 81(2), 107–126 (2016) 9. Ma, X., Zhan, J., Khan, M., Zeeshan, M., Anis, S., Awan, A.S.: Complex fuzzy sets with applications in signals. Comput. Appl. Math. 38(4), 150 (2019) 10. Pramanik, T., Samanta, S., Sarkar, B., Pal, M.: Fuzzy φ−tolerance competition graphs. Soft Comput. 21(13), 3723–3734 (2017) 11. Ramot, D., Milo, R., Friedman, M., Kandel, A.: Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 10(2), 171–186 (2002) 12. Samanta, S., Akram, M., Pal, M.: m-step fuzzy competition graphs. J. Appl. Math. Comput. 47(1–2), 461–472 (2015) 13. Samanta, S., Pal, M.: Fuzzy k−competition and p−competition graphs. Fuzzy Inf. Eng. 5(2), 191–204 (2013) 14. Sahoo, S., Pal, M.: Intuitionistic fuzzy competition graphs. J. Appl. Math. Comput. 52(1), 37–57 (2016) 15. Sarwar, M., Akram, M.: Novel concepts of bipolar fuzzy competition graphs. J. Appl. Math. Comput. 54(1–2), 511–547 (2017) 16. Sarwar, M., Akram, M.: Certain algorithms for computing strength of competition in bipolar fuzzy graphs. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 25(6), 877–96 (2017) 17. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems. pp. 835–840 (1998) 18. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 19. Zhang, G., Dillon, T.S., Cai, K.Y., Ma, J., Lu, J.: Operation properties and δ-equalities of complex fuzzy sets. Int. J. Approx. Reason. 50(8), 1227–1249 (2009)

Chapter 5

Bipolar Fuzzy Planar Graphs

In this chapter, we study the concepts of bipolar fuzzy multisets, bipolar fuzzy multigraphs, strong and complete bipolar fuzzy multigraphs, bipolar fuzzy planar graphs, and bipolar fuzzy dual graphs. We discuss different types of bipolar fuzzy edges, intersection value and planarity value of bipolar fuzzy graphs, strong and weak bipolar fuzzy faces, and the relation of planarity and duality in bipolar fuzzy graphs. We elaborate various properties of bipolar fuzzy bridges, bipolar fuzzy cut vertices, bipolar fuzzy blocks, bipolar fuzzy cycles, and bipolar fuzzy trees in terms of level graphs. We describe the importance of bipolar fuzzy planar graphs with a number of real-world applications in road networks and electrical connections. The main results of this chapter are from [6, 7].

5.1 Introduction Graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network. Graphs are used to model various practical processes in which crossing between edges is a nuisance including design problems for circuits, subways, and utility lines. The crossing of two connections normally means that the communication lines must work at different heights. This is not a big issue for electrical wires, but it creates extra expenses for some types of lines. Circuits, in particular, are easier to manufacture if their connections can be constructed in fewer layers. These applications are designed by the concept of planar graphs. Circuits where the crossing of lines is necessary cannot be represented by planar graphs. Numerous computational challenges can be solved by means of cuts of planar graphs. Planar graphs are essential in city planning, subway tunnels, pipelines, and metro lines. Due to crossing, there is a chance for an accident. Also, underground routes reduce traffic jam but the cost of the crossing of these routes is high. In a city © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_5

223

224

5 Bipolar Fuzzy Planar Graphs

planning, routes without crossing are perfect for safety. But due to lack of space, the crossing of such lines is allowed. It is easy to observe that the crossing between one congested and one non-congested route is better than the crossing between two congested routes. The term “congested” has no definite meaning. We generally use “congested”, “very congested”, “highly congested” routes, etc. These terms are called linguistic terms and they have some membership values. A congested route may be referred to as a strong route and a low congested route may be called a weak route. Thus crossing between a strong route and a weak route is more safe than the crossing between two strong routes. That is, crossing between a strong route and a weak route may be allowed in city planning with a certain amount of safety. The terms strong route and weak route lead to the strong edge and weak edge of a fuzzy graph, respectively. And the crossing between strong and weak edges leads to the concept of fuzzy planar graphs. One of the earliest questions concerning planarity was presented by Möbius during a lecture in about the year 1840, where Möbius presented the following problem: There was once a king with five sons. In his will, he stated that after his death the sons should divide the kingdom into five regions so that the boundary of each region should have a frontier line in common with each of the other four regions. Can the terms of the will be satisfied? [19]. This question can be rephrased using the geometric dual of the land regions. The geometric dual of a map G, denoted by G ∗ , can be formed by replacing each face of G with a vertex and connecting vertices if the faces of G share an edge. Then, the question of Mobius ¨ can be posed in a graph theoretic context as the problem of asking if each of the five sons has a road connecting his capital city to all his other brothers’ capital cities in such a way that no two roads intersect. The problem of the five princes can be solved if the complete graph of the five vertices, denoted by K 5 , is a planar graph. The second puzzle, sometimes called the gas-water-electricity problem requires to join three houses to three gas, water, and electricity facilities using pipes so that no two pipes cross. This problem reduces to the planarity of the complete bipartite graph K 3,3 obtained by connecting each of the three independent vertices to each of the three other independent vertices. It appears that these two puzzles are unfeasible. Rosenfeld [13] introduced the fuzzy analog of several basic graph-theoretic concepts including bridges, cut-nodes, connectedness, trees, and cycles. Sunitha and Vijayakumar [18] characterized fuzzy trees. Bhutani and Rosenfeld [11] introduced the concepts of strong arcs, fuzzy end nodes, and geodesics in fuzzy graphs, and types of arcs in a fuzzy graph are described in [8]. Various concepts of fuzzy planar graphs, special fuzzy planar graphs, fuzzy trees, fuzzy cycles, fuzzy co-cycles, fuzzy dual graphs, strongest strong cycles, θ -fuzzy graphs, and strong arcs in fuzzy graphs are studied in [1, 9, 10, 12, 15, 17]. Zhang [20] introduced the notion of a bipolar fuzzy set as a powerful technique to discuss the bipolar behavior of objects. Bipolar fuzzy graph theory [2–5] is finding an increasing number of applications in modeling real-time systems and gives more precision, flexibility, and compatibility to the system as compared to the fuzzy and crisp models. The concepts of planar graphs, fuzzy planar graphs, and fuzzy trees are extended to bipolar fuzzy planar graphs, bipolar fuzzy cycles, and bipolar fuzzy trees in [6, 7] which is the main discussion

5.1 Introduction

225

of this chapter. For further terminologies and studies on fuzzy graphs and bipolar fuzzy systems, readers are referred to [14, 16, 21–23].

5.2 Bipolar Fuzzy Planar Graphs A multiset is a modification of the concept of a set that, unlike a set, allows multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of this element in the multiset. A multigraph is a graph that can have more than one edge between a pair of vertices. That is, G = (X, E) is a multigraph if X is a set of vertices and E is a multiset of 2−element subsets of X . A fuzzy multiset A drawn from a non-empty set X can be characterized by a function C M A : X → Q, where Q is the set of all crisp multisets drawn from the unit interval. Definition 5.1 A bipolar fuzzy multiset A drawn from a non-empty set X is characterized by two functions: “count positive membership” of A (C M A ) and “count negative membership” of A (C N A ) given by C M A : X → Q 1 and C N A : X → Q 2 , where Q 1 and Q 2 are the sets of all crisp multisets drawn from the intervals [0, 1] and [−1, 0], respectively, such that for each x ∈ X , 1. The positive membership sequence is defined as a decreasingly ordered p p p sequence of elements in C M A (x) which is denoted by (μ A (x)1 , μ A (x)2 , μ A (x)3 , p p p p p . . . , μ A (x)m ), where μ A (x)1 ≥ μ A (x)2 ≥ μ A (x)3 ≥ · · · ≥ μ A (x)m , 2. The corresponding negative membership sequence is defined as an increasingly ordered sequence of elements in C N A (x) which is denoted by (μnA (x)1 , μnA (x)2 , μnA (x)3 , . . . , μnA (x)m ), where μnA (x)1 ≤ μnA (x)2 ≤ μnA (x)3 ≤ · · · ≤ μnA (x)m . A bipolar fuzzy multiset A is denoted as A=



  p p p p x : (μ A (x)1 , μ A (x)2 , μ A (x)3 , . . . , μ A (x)m ), (μnA (x)1 , μnA (x)2 , μnA (x)3 , . . . , μnA (x)m ) | x ∈ X

  p or A = { x : μ A (x)i , μnA (x)i ) , i = 1, 2, . . . , m | x ∈ X }. Definition 5.2 A bipolar fuzzy multigraph on a non-empty set X is a pair G = p p (A, B), where A = (μ A , μnA ) is a bipolar fuzzy set on X and B = {(x y, μ B (x y)i , μnB (x y)i ), i = 1, 2, . . . , m | x y ∈ X × X } is a bipolar fuzzy multiset on X × X such that for all x, y ∈ X , p

p

p

μ B (x y)i ≤ min{μ A (x), μ A (y)} and μnB (x y)i ≥ max{μnA (x), μnA (y)},

for all i = 1, 2, . . . , m.

Note that there may be more than one edge between the vertices x and y. Here, p μ B (x y)i and μnB (x y)i represent the positive membership value and negative membership value of the ith edge x y in G. m denotes the number of edges between the vertices x and y. In a bipolar fuzzy multigraph G, B is said to be a bipolar fuzzy p multiedge set. Also, μ A (x y)i = 0 = μnA (x y)i for all x y ∈ X × X − E.

226

5 Bipolar Fuzzy Planar Graphs

Table 5.1 Bipolar fuzzy vertex set a p μA μnA

0.5 −0.3

b

c

d

0.4 −0.4

0.5 −0.3

0.4 −0.4

Table 5.2 Bipolar fuzzy multiedge set ab ab p

μB μnB

0.2 −0.2

0.1 0

ab

bc

bd

0.2 −0.2

0.3 −0.3

0.1 −0.2

(0.2,-0.2) (0.1,0)

a(0.5, −0.3)

b(0.4, −0.4) (0.3,-0.3)

(0. 1, (0.2,-0.2)

c(0.5, −0.3)

-0. 2) d(0.4, −0.4)

Fig. 5.1 Bipolar fuzzy multigraph

p

Example 5.1 Let A = (μ A , μnA ) be a bipolar fuzzy set on X = {a, b, c, d}, shown p in Table 5.1, and B = (μ B , μnB ) be a bipolar fuzzy multiedge set of X × X given in Table 5.2. The bipolar fuzzy multigraph is shown in Fig. 5.1. p

Definition 5.3 Let B = {(x y, μ B (x y)i , μnB (x y)i ), i = 1, 2, . . . , m | x y ∈ X × X } be a bipolar fuzzy multiedge set in a bipolar fuzzy multigraph G. The degree of a vertex x ∈ X is denoted by deg(x) and defined as deg(x) =

 m y∈X i=1

p μ B (x y)i ,

m 

μnB (x y)i

 .

y∈X i=1

Example 5.2 In Example 5.1, the degree of the vertices a, b, c, d are calculated as deg(a) = (0.5, −0.4), deg(b) = (0.9, −0.9), deg(c) = (0.3, −0.3), and deg(d) = (0.1, −0.2).   p Definition 5.4 Let B = (x y, μ B (x y)i , μnB (x y)i ), i = 1, 2, . . . , m | x y ∈ X × X be a bipolar fuzzy multiedge set in a bipolar fuzzy multigraph G. A multiedge x y of G p p p is strong if 21 min{μ A (x), μ A (y)} ≤ μ B (x y)i and 21 max{μnA (x), μnA (y)} ≤ μnB (x y)i for each i = 1, 2, . . . , m.

5.2 Bipolar Fuzzy Planar Graphs

227

(0.4, −0.2) a(0.4, −0.2)

(0.4, −0.2)

(0.4, −0.3) b(0.5, −0.3)

c(0.4, −0.3)

(0.4, −0.2) Fig. 5.2 Bipolar fuzzy complete multigraph

p

Definition 5.5 Let B = {(x y, μ B (x y)i , μnB (x y)i ), i = 1, 2, . . . , m | x y ∈ X × X } be a bipolar fuzzy multiedge set in a bipolar fuzzy multigraph G. A bipolar fuzzy p p p multigraph G is complete if min{μ A (x), μ A (y)} = μ B (x y)i , max{μnA (x), μnA (y)} = μnB (x y)i for each i = 1, 2, . . . , m, for all x, y ∈ X . Example 5.3 Consider a bipolar fuzzy multigraph G as shown in Fig. 5.2. By routine computations, it is easy to see from Fig. 5.2 that it is a complete bipolar fuzzy multigraph. Definition 5.6 The strength of a bipolar fuzzy edge (ab)i , for some i ∈ {1, 2, . . . , m}, is denoted by the ordered pair I(ab)i = (M(ab)i , N(ab)i ) and can be measured by the value,  I(ab)i = (M(ab)i , N(ab)i ) =

 p μnB (ab)i μ B (ab)i . , p p μ A (a) ∧ μ A (b) μnA (a) ∨ μnA (b)

Definition 5.7 Let G = (A, B) be a bipolar fuzzy multigraph. A bipolar fuzzy edge (ab)i , for some i ∈ {1, 2, . . . , m}, is said to be strong if M(ab)i ≥ 0.5 or N(ab)i ≤ 0.5. Otherwise it is called a weak bipolar fuzzy edge. Definition 5.8 Let G = (A, B) be a bipolar fuzzy multigraph such that B contains p p two bipolar fuzzy edges (ab, μ B (ab)i , μnB (ab)i ) and (cd, μ B (cd) j , μnB (cd) j ) intersecting at a point P, where i and j are fixed integers. The intersecting value at point P is defined as   M(ab)i + M(cd) j N(ab)i + N(cd) j , , for some i, j. I P = (M P , N P ) = 2 2 If the number of points of intersection in a bipolar fuzzy multigraph increases, planarity decreases. Thus, for a bipolar fuzzy multigraph, I P is inversely proportional to the planarity. The concept of a bipolar fuzzy planar graph in introduced in Definition 5.9.

228

5 Bipolar Fuzzy Planar Graphs

Table 5.3 Bipolar fuzzy vertex set A A a b p μA μnA

0.5 −0.2

0.4 −0.1

Table 5.4 Bipolar fuzzy multiedge set A B ab ac ad ad bc p

μB μnB

0.2 −0.1

0.2 −0.1

0.2 −0.1

0.3 −0.1

0.2 −0.1

c

d

e

0.3 −0.1

0.6 −0.2

0.6 −0.1

bd

cd

ae

ce

de

be

0.2 −0.1

0.2 −0.1

0.2 −0.1

0.2 −0.1

0.2 −0.1

0.2 −0.1

Definition 5.9 Let G = (A, B) be a bipolar fuzzy multigraph such that P1 , P2 , . . . , Pn are the points of intersection between the edges for a certain geometrical representation, then G is said to be a bipolar fuzzy planar graph with bipolar fuzzy planarity value f = ( f p , f n ) which is defined as  f = ( f p, f n) =

1

,

1 + {M P1 + M P2 + · · · + M Pn } 1 + {N P1

 −1 . + N P2 + · · · + N Pn }

Clearly, f = ( f p , f n ) is bounded and 0 < f p ≤ 1, f n , −1 ≤ f n < 0. If there is no point of intersection for a certain geometrical representation of a bipolar fuzzy planar graph, then its planarity value is (1, −1). In this case, the underlying crisp graph of this bipolar fuzzy graph is the crisp planar graph. If f p decreases ( f n increases), then the number of points of intersection between the edges increases (decreases) and the nature of planarity decreases (increases). Thus, every bipolar fuzzy graph is a bipolar fuzzy planar graph with certain planarity value. Example 5.4 Consider a bipolar fuzzy multigraph G = (A, B) as shown in Fig. 5.3, p p where A = (μ A , μnA ) is a bipolar fuzzy set on X = {a, b, c, d, e} and B = (μ B , μnB ) is a bipolar fuzzy set on E = {ab, ac, ad, ad, bc, bd, cd, ce, ae, de, be} as given in Tables 5.3 and 5.4, respectively. The bipolar fuzzy multigraph, shown in Fig. 5.3, has two points of intersection P1 and P2 . P1 is a point of intersection between (ad, 0.2, −0.1) and (bc, 0.2, −0.1) while P2 is the point of intersection between (ad, 0.3, −0.1) and (bc, 0.2, −0.1). 1. For bipolar fuzzy edge (ad, 0.2, −0.1), Iad = (0.4, 0.5). 2. For bipolar fuzzy edge (ad, 0.3, −0.1), Iad = (0.6, 0.5). 3. For bipolar fuzzy edge (bc, 0.2, −0.1), Ibc = (0.67, 1). For the first point of intersection P1 , intersecting value I P1 is (0.53, 0.75) and that for the second point of intersection P2 , I P2 = (0.63, 0.75). Therefore, the planarity value for the bipolar fuzzy multigraph shown in Fig. 5.3 is (0.46, −0.4).

5.2 Bipolar Fuzzy Planar Graphs

a(0.5, −0.2)

(0.2, −0.1)

(0 .2 ,− 0. 1)

1) 0. ,− .2 (0

(0.2, −0.1)

1

.1) −0

P

2

(0.3 , −0 .1)

P

(0.2, −0.1)

d(0.6, −0.2)

(0 .2 ,− 0. 1)

c(0.3, −0.1)

b(0.4, −0.1)

(0.2, −0.1)

2, (0.

(0.2 , −0 .1)

Fig. 5.3 Bipolar fuzzy planar graph

229

(0 .2 , − 0. 1)

e(0.6, −0.1)

Definition 5.10 An bipolar fuzzy planar graph G = (A, B) is called strong bipolar fuzzy planar graph if the planarity value f = ( f p , f n ) of G is such that f p ≥ 0.5 and f n ≤ −0.5. Theorem 5.1 Let G be a strong bipolar fuzzy planar graph, then the number of points of intersection between strong bipolar fuzzy edges is at most one. Proof Let G = (A, B) be a strong bipolar fuzzy planar graph on X . Assume that G has at least two points of intersection P1 and P2 between two strong bipolar fuzzy p edges in G. For any strong bipolar fuzzy edge (ab, μ B (ab)i , μnB (ab)i ), p

μ B (ab)i ≥

1 1 p p min{μ A (a), μ A (b)} and μnB (ab)i ≤ max{μnA (a), μnA (b)}. 2 2

This shows that Mab ≥ 0.5 or Nab ≤ 0.5. Thus, for two intersecting strong bipolar p p fuzzy edges (ab, μ B (ab)i , μnB (ab)i ) and (cd, μ B (cd) j , μnB (cd) j ), Mab + Mcd Nab + Ncd ≥ 0.5, ≤ 0.5. 2 2 That is, M P1 ≥ 0.5, N P1 ≤ 0.5. Similarly, M P2 ≥ 0.5, N P2 ≤ 0.5. This implies that 1 + M P1 + M P2 ≥ 2, 1 + N P1 + N P2 ≤ 2. Therefore, f p = 1+M P1+M P ≤ 0.5, f n = −1 1+N P1 +N P2

1

2

≥ −0.5. It contradicts the fact that the bipolar fuzzy graph is a strong bipolar fuzzy planar graph. Thus, the number of points of intersection between strong bipolar fuzzy edges cannot be two. Obviously, if the number of points of intersection of strong bipolar fuzzy edges increases, the planarity value decreases. Similarly, if the

230

5 Bipolar Fuzzy Planar Graphs

number of points of intersection of strong bipolar fuzzy edges is one, then planarity value f p < 0.5, f n > −0.5. Any bipolar fuzzy planar graph without any crossing between edges is a strong bipolar fuzzy planar graph. Thus, we conclude that the maximum number of points of intersection between the strong bipolar fuzzy edges in G is one.  Theorem 5.2 Let G be a bipolar fuzzy planar graph with planarity value f = ( f p , f n ). If f p ≥ 0.67, f n ≤ −0.67, then G does not contain any point of intersection between two strong bipolar fuzzy edges. Definition 5.11 Let G = (A, B) be a bipolar fuzzy planar graph and p B = {(x y, μ B (x y)i , μnB (x y)i ), i = 1, 2, . . . , m | x y ∈ X × X } be a bipolar fuzzy multiedge set of G. A bipolar fuzzy face of G is a region bounded by the set of the bipolar fuzzy edges E ⊂ E of a geometric representation of G. The positive and negative membership values of the bipolar fuzzy face are defined as

p μ B (x y)i , i = 1, 2, . . . , m | x y ∈ E , min p p min{μ A (x), μ A (y)}

max

μnB (x y)i , i = 1, 2, . . . , m | x y ∈ E . max{μnA (x), μnA (y)}

Definition 5.12 A bipolar fuzzy face is called a strong bipolar fuzzy face if its positive membership value is greater than 0.5 or the negative membership value is lesser than −0.5. Otherwise, it is called a weak bipolar fuzzy face. Every bipolar fuzzy planar graph has an infinite region which is called outer bipolar fuzzy face. Other faces are called inner bipolar fuzzy faces. Example 5.5 Consider a bipolar fuzzy planar graph G = (A, B) as shown in Fig. 5.4. Then G has the following bipolar fuzzy faces. 1. The bipolar fuzzy face F1 is bounded by bipolar fuzzy edges (v1 v2 , 0.5, −0.1), (v2 v3 , 0.6, −0.1),(v1 v3 , 0.5, −0.1). 2. The outer bipolar fuzzy face F2 is surrounded by bipolar fuzzy edges (v1 v3 , 0.5, −0.1), (v1 v4 , 0.5, −0.1), (v2 v4 , 0.6, −0.1), (v2 v3 , 0.6, −0.1). 3. The bipolar fuzzy face F3 is bounded by bipolar fuzzy edges (v1 v2 , 0.5, −0.1), (v2 v4 , 0.6, −0.1), (v1 v4 , 0.5, −0.1). Clearly, the positive and negative membership values of a bipolar fuzzy face F1 are 0.833 and −0.333, respectively. The positive membership value and negative membership value of a bipolar fuzzy face F3 are also 0.833 and −0.333, respectively. Thus, F1 and F3 are strong bipolar fuzzy faces. The concept of dual of a bipolar fuzzy planar graph is given in Definition 5.13. In a bipolar fuzzy dual graph, vertices are corresponding to the strong bipolar fuzzy faces of the bipolar fuzzy planar graph and each bipolar fuzzy edge between two vertices is corresponding to each edge in the boundary between two bipolar fuzzy faces of the bipolar fuzzy planar graph as explained in Definition 5.13.

5.2 Bipolar Fuzzy Planar Graphs

231

Fig. 5.4 Bipolar fuzzy faces in G

F3 (0.5,-0.1)

v1 (0.6, −0.3)

(0.5,-0.1)

v2 (0.7, −0.3)

) .1 ,-0 6 . (0

(0.6,-0.1)

(0.5,-0.1)

F1

F2 v3 (0.8, −0.1)

v4 (0.7, −0.1)

Definition 5.13 Let G = (A, B) be a bipolar fuzzy planar graph and  p B = (x y, μ B (x y)i , μnB (x y)i ), i = 1, 2, . . . , m | x y ∈ X × X be a bipolar fuzzy multiedge set of G. Let F1 , F2 , . . . , Fk be the strong bipolar fuzzy faces of G. The bipolar fuzzy dual graph of G is a bipolar fuzzy planar graph G = (X , A , B ), where X = {xi , i = 1, 2, . . . , k}, and the vertex xi of G is considered for the face Fi of G. The positive and negative membership values of the vertices are p p given by the mapping A = (μ A , μnA ) : X → [0, 1] × [−1, 0] such that μ A (x j ) = p max{μ B (uv)i , i = 1, 2, . . . , p | uv is an edge of the boundary of the strong bipolar fuzzy face F j }, μnA (x j ) = min{μnB (uv)i , i = 1, 2, . . . , p | uv is an edge of the boundary of the strong bipolar fuzzy face F j }. There may exist more than one common edge between two bipolar fuzzy faces Fi and F j in G. Thus, there may be more than one edge between two vertices xi and p x j in a bipolar fuzzy dual graph G . Let μ B (xi x j )l denote the positive membership value of the lth edge between xi and x j , and μnB (xi x j )l denote the negative–positive membership value of the lth edge between xi and x j . The positive membership and negative membership values of the edges of the bipolar fuzzy dual graph are p p given by μ B (xi x j )l = μ B (uv)l , μnB (xi x j )l = μnB (uv)l , where (uv)l is an edge in the boundary between two strong bipolar fuzzy faces Fi and F j and l = 1, 2, . . . , s, where s is the number of common edges in the boundary between Fi and F j or the number of edges between xi and x j . If there is any strong pendant edge in the bipolar fuzzy planar graph, then there will be a self-loop in G corresponding to this pendant edge. The positive and negative membership values of the self-loop are equal to the positive and negative membership values of the pendant edge. The bipolar fuzzy dual graph of the bipolar fuzzy planar graph does not contain the point of intersection of edges for a certain representation, so it is a bipolar fuzzy planar graph with planarity value (1, −1). Thus, the bipolar fuzzy face of the bipolar fuzzy dual graph can be similarly described as in bipolar fuzzy planar graphs.

232

5 Bipolar Fuzzy Planar Graphs

Example 5.6 Consider a bipolar fuzzy planar graph G = (X, A, B) as shown in Fig. 5.5 such that X = {a, b, c, d}, A = {(a, 0.6, −0.2), (b, 0.7, −0.2), (c, 0.8, −0.2), (d, 0.9, −0.1)}, and B = {(ab, 0.5, −0.01), (ac, 0.4, −0.01), (ad, 0.55, −0.01), (bc, 0.45, −0.01), (bc, 0.6, −0.01), (cd, 0.7, −0.01)}.

The bipolar fuzzy planar graph has the following bipolar fuzzy faces. 1. The bipolar fuzzy face F1 is bounded by (ab, 0.5, −0.01), (ac, 0.4, −0.01), (bc, 0.45, −0.01). 2. The bipolar fuzzy face F2 is bounded by (ad, 0.55, −0.01), (cd, 0.7, −0.01), (ac, 0.4, −0.01). 3. The bipolar fuzzy face F3 is bounded by (bc, 0.45, −0.01), (bc, 0.6, −0.01). 4. The outer bipolar fuzzy face F4 is surrounded by (ab, 0.5, −0.01), (bc, 0.6, −0.01), (cd, 0.7, −0.01), (ad, 0.55, −0.01). Routine calculations show that all bipolar fuzzy faces are strong bipolar fuzzy faces. For each strong bipolar fuzzy face, consider a vertex for the bipolar fuzzy dual graph. So the vertex set is X = {x1 , x2 , x3 , x4 }, where the vertex xi is taken corresponding to the strong bipolar fuzzy face Fi , i = 1, 2, 3, 4. Thus, p

p

μ A (x1 ) = max{0.5, 0.4, 0.45} = 0.5, μ A (x2 ) = max{0.55, 0.7, 0.4} = 0.7, μnA (x1 ) = min{−0.01, −0.01, −0.01} = −0.01, μnA (x2 ) = min{−0.01, −0.01, −0.01} = −0.01,

p

p

μ A (x3 ) = max{0.45, 0.6} = 0.6, μ A (x4 ) = max{0.5, 0.6, 0.7, 0.55} = 0.7, (mu nA ) (x3 ) = min{−0.01, −0.01} = −0.01, (μnA (x4 ) = min{−0.01, −0.01, −0.01, −0.01} = −0.01.

There are two common edges ad and cd between the bipolar fuzzy faces F2 and F4 in G. Hence, between the vertices x2 and x4 , there exist two edges in the bipolar fuzzy dual graph of G. The positive and negative membership values of these edges are given as p

p

p

p

μ B (x2 x4 ) = μ B (cd) = 0.7, μ B (x2 x4 ) = μ B (ad) = 0.55, μnB (x2 x4 ) = μnB (cd) = −0.01, μnB (x2 x4 ) = μnB (ad) = −0.01. The positive and negative positive membership values of other edges of the bipolar fuzzy dual graph are calculated as p

p

p

p

μ B (x1 x3 ) = μ B (bc) = 0.45, μ B (x1 x2 ) = μ B (ac) = 0.4, μ B (x1 x4 ) = μ B (ab) = 0.5, μ B (x3 x4 ) = (μ B ) (bc) = 0.6, p

p

p

p

5.2 Bipolar Fuzzy Planar Graphs

233

Fig. 5.5 Bipolar fuzzy dual graph

b

a

x1

x3

x2 c d

x4

(μnB ) (x1 x3 ) = μ B (bc) = −0.01, μnB (x1 x2 ) = μnB (ac) = −0.01, p

μnB (x1 x4 ) = μnB (ab) = −0.01, (μnB ) (x3 x4 ) = μnB (bc) = −0.01. Thus, the edge set of the bipolar fuzzy dual graph is B = {(x1 x3 , 0.45, −0.01), (x1 x2 , 0.4, −0.01), (x1 x4 , 0.5, −0.01), (x3 x4 , 0.6, −0.01), (x2 x4 , 0.7, −0.01), (x2 x4 , 0.55, −0.01)}. In Fig. 5.5, the bipolar fuzzy dual graph G = (X , A , B ) of G is drawn by dotted lines. Theorem 5.3 Let G be a bipolar fuzzy planar graph whose number of vertices, number of bipolar fuzzy edges, and number of strong faces are denoted by n, p, and m, respectively. Let G be the bipolar fuzzy dual graph of G, then the following conditions are satisfied. 1. The number of vertices of G is equal to m. 2. The number of edges of G is equal to p. 3. The number of bipolar fuzzy faces of G is equal to n. Theorem 5.4 Let G = (X, A, B) be a bipolar fuzzy planar graph without weak edges and the bipolar fuzzy dual graph of G be G = (X , A , B ). The positive and negative membership values of edges in G are equal to positive and negative membership values of the bipolar fuzzy edges of G. It is known that the isomorphism between bipolar fuzzy graphs is an equivalence relation (see Definition 1.35). If there is an isomorphism between two bipolar fuzzy graphs such that one is a bipolar fuzzy planar graph, then the other will also be a bipolar fuzzy planar graph. Theorem 5.5 Let G be a bipolar fuzzy planar graph and H be a bipolar fuzzy graph. If there exists an isomorphism f : G → H, then H can be drawn as a bipolar fuzzy planar graph with the same planarity value as G.

234

5 Bipolar Fuzzy Planar Graphs

Two bipolar fuzzy planar graphs with the same number of vertices may be isomorphic. But the relations between planarity values of two bipolar fuzzy planar graphs are discussed in the following theorems. Theorem 5.6 Let G 1 and G 2 be the isomorphic bipolar fuzzy graphs with planarity values f 1 and f 2 , respectively, then f 1 = f 2 . Theorem 5.7 Let G 1 and G 2 be weak isomorphic bipolar fuzzy graphs with planarity values f 1 and f 2 , respectively, then f 1 = f 2 if the positive and negative membership values of corresponding intersecting edges are the same. Theorem 5.8 Let G 1 and G 2 be co-weak isomorphic bipolar fuzzy graphs with planarity values f 1 and f 2 , respectively, then f 1 = f 2 if the minimum of positive membership values and maximum of negative membership values of the end vertices of corresponding intersecting edges are the same.

5.3 Bipolar Fuzzy Bridges A path in a crisp graph G ∗ is an alternating sequence of vertices and edges v0 , e1 , v1 , e2 , . . .,vn−1 , en , vn . The path with n vertices is denoted by Pn . A path is sometimes denoted by Pn : v0 v1 · · · vn (n > 0). The length of a path Pn in G ∗ is n. A path Pn : v0 v1 · · · vn in G ∗ is called a cycle if v0 = vn and n ≥ 3. Note that path graph, Pn , has n − 1 edges and can be obtained from cycle graph, Cn , by removing any edge. An undirected graph G ∗ is connected if there is a path between each pair of distinct vertices. A block is a maximal biconnected (if any one vertex is removed, then the graph remains connected) subgraph of a crisp graph G ∗ . An edge e in a connected graph G ∗ is a bridge (cut edge or cut arc) if G − e is disconnected. A vertex v in a connected graph G is a cut vertex if G − v is disconnected. The graphs with exactly n − 1 bridges are exactly the trees (connected acyclic undirected graph), and the graphs in which every edge is a bridge are exactly the forests (disjoint union of trees). A spanning tree in a connected graph G is a subgraph of G which includes all the vertices of G and is also a tree. Let G = (A, B) be a bipolar fuzzy graph. The μ p -strength of a bipolar fuzzy k−1 p μ B (ei ). path P : v1 , e1 , v2 , e2 , v3 , . . . , ek−1 , vk is defined as (μ p )k (v1 , vk ) = ∧i=1 k−1 n n n k μ B (ei ). The μ -strength of P between v1 and vn is defined as (μ ) (v1 , vk ) = ∧i=1 If x, y ∈ X , the µ p −strength of connectedness between x and y is defined as (μ B )∞ (x, y) = max{(μ B )k (x, y) | k = 1, 2, . . .}, p

p

(μ B )∞ (x, y) = max{μ B (x, v1 ) ∧ μ B (v1 , v2 ) ∧ ... ∧ μ B (vk−1 , y) | x, v1 , v2 , · · · , vk−1 , y ∈ X, k = 1, 2, . . .}. p

p

p

p

The µn −strength of connectedness between x and y is (μnB )∞ (x, y) = min{(μnB )k (x, y) | k = 1, 2, · · · , n}, (μnB )∞ (x, y) = min{μnB (x, v1 ) ∨ μnB (v1 , v2 ) ∨ ... ∨ μnB (vk−1 , y) | x, v1 , v2 , · · · , vk−1 , y ∈ X, k = 1, 2, · · · , n}.

5.3 Bipolar Fuzzy Bridges

235

Fig. 5.6 Connected bipolar fuzzy graph

z

(0.8, −0.2)

y

(0.7, −0.1)

x





Also, (μ B ) ∞ (x, y) and (μnB ) ∞ (x, y) denote (μG−(x,y) )∞ (x, y) and (μnG−(x,y) )∞ (x, y), where G − (x, y) is obtained from G by deleting the arc (x, y). p

p

Definition 5.14 A bipolar fuzzy edge graph on a non-empty set X is an ordered  = (X, B), where X is the crisp vertex set and B is defined by the pair of the form G p functions μ B : X × X → [0, 1] and μnB : X × X → [−1, 0]. Definition 5.15 An edge x y in bipolar fuzzy graph G is said to be µ p -bridge if deleting x y reduces the μ p -strength of connectedness between some pair of vertices. An edge x y is said to be µn -bridge if deleting x y increases the μn -strength of connectedness between some pair of vertices. An edge x y is said to be a bipolar fuzzy bipolar fuzzy bridge if it is μ p -bridge and μn -bridge. Definition 5.16 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X, then 1. x y is called a weak bipolar fuzzy bridge if there exists (s, t) ∈ (0, h(B)] × [d(B), 0) such that x y is a bridge of G (s,t) (see Definitions 7.10 and 4.2). 2. x y is called a partial bipolar fuzzy bridge if x y is a bridge of G (s,t) , for all (s, t) ∈ (0, h(B)] × (d(B), 0] ∪ {(h(B), d(B))}. 3. x y is called a full bipolar fuzzy bridge if x y is a bridge for G (s,t) , for all (s, t) ∈ (0, h(B)] × [d(B), 0). Example 5.7 Consider a connected bipolar fuzzy graph as shown in Fig. 5.6. By routine computations, d(B) = −0.2, h(B) = 0.8. Thus, (s, t) ∈ (0, 0.8] × [−0.2, 0). For 0 < s ≤ 0.7, −0.2 ≤ t < 0, G (s,t) = (X, {x y, yz}). For 0.7 < s ≤ 0.8, −0.2 ≤ t < 0, G (s,t) = (X, {yz}). Hence, yz is a full bipolar fuzzy bridge and x y is a weak bipolar fuzzy bridge but not a partial bipolar fuzzy bridge. Both x y and yz are bridges and bipolar fuzzy bridges. Example 5.8 Consider a connected bipolar fuzzy graph as shown in Fig. 5.7. By routine calculations, d(B) = −0.4, h(B) = 0.9. For 0 < s ≤ 0.1, −0.4 ≤ t < 0,

236

5 Bipolar Fuzzy Planar Graphs

Fig. 5.7 Bipolar fuzzy bridges

z

(0.9, −0.1)

x

4) 0.

(0.8, −0.1)

,− .1 (0

y

G (s,t) = (X, {x y, x z, yz}). For 0.1 < s ≤ 0.8, −0.1 ≤ t < 0, G (s,t) = (X, {x y, x z}). For 0.8 < s ≤ 0.9,−0.1 ≤ t < 0, G (s,t) = (X, {x z}). Thus, x z is a bipolar fuzzy bridge and a partial bipolar fuzzy bridge but not a bridge. The edge yz is not any type of bipolar fuzzy bridge. Example 5.9 Let A be a bipolar fuzzy set on X = {x, y, z} and B be a bipolar fuzzy relation in X defined by p

p

p

μ A (x) = μ A (y) = μ A (z) = 1, μnA (x) = μnA (y) = μnA (z) = −1, p

p

p

μ B (x y) = μ B (yz) = μ B (x z) = 0.9, μnB (x y) = μnB (yz) = μnB (x z) = −0.1. Clearly, G = (A, B) has no bipolar fuzzy bridges of any type. Example 5.10 Let A be a bipolar fuzzy set on X = {x, y, z, w} and let B be a bipolar fuzzy relation in X defined by p

p

p

p

μ A (x) = μ A (y) = μ A (z) = μ A (w) = 1, μnA (x) = μnA (y) = μnA (z) = μnA (w) = −1, p

p

p

p

μ B (x y) = μ B (yz) = 0.1, μ B (x z) = μ B (wz) = 0.9, μnB (x y) = μnB (yz) = −0.5, μnB (x z) = μnB (wz) = −0.1.

Here, d(B) = −0.5, h(B) = 0.9. For 0 < s ≤ 0.1, −0.5 ≤ t < 0, G (s,t) = (X, {x y, yz, x z, zw}). For 0.1 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x z, zw}). Thus, zw is a full bipolar fuzzy bridge and x z is a partial bipolar fuzzy bridge but not a full bipolar fuzzy bridge. Proposition 5.1 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X such that G ∗ is a crisp graph corresponding to G. If x y is a bridge in G ∗ , then x y is p p a bipolar fuzzy bridge in G if and only if μ B (x, y) > (μ B ) ∞ (x, y) and μnB (x, y) < n ∞ (μ B ) (x, y).

5.3 Bipolar Fuzzy Bridges

237

Proposition 5.2 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X, then x y is a bipolar fuzzy bridge in G if and only if x y is not a weakest bipolar fuzzy edge of any bipolar fuzzy cycle. Proposition 5.3 An edge x y in a bipolar fuzzy graph is a bipolar fuzzy bridge if and p only if x y is a bridge of G ∗ and μ B (x, y) = h(B), μnB (x, y) = d(B). Proof Suppose x y is a full bipolar fuzzy bridge of a bipolar fuzzy graph G. Then x y is a bridge for G (s,t) , for all (s, t) ∈ (0, h(B)] × (0, d(B)]. Hence, x y ∈ B(h(B),d(B)) p and so μ B (x, y) = h(B), μnB (x y) = d(B). Since x y is a bridge for G (s,t) for all (s, t) ∈ (0, h(B)] × (0, d(B)], it follows that x y is a bridge for G ∗ . p Conversely, suppose that x y is a bridge for G ∗ and μ B (x y) = h(B), μnB (x y) = d(B). Then x y ∈ B(s,t) , for all (s, t) ∈ (0, h(B)] × [d(B), 0). Since x y is a bridge for G ∗ , therefore, x y is a bridge for G (s,t) ∀(s, t) ∈ (0, h(B)] × [d(B), 0) since each  G (s,t) is a subgraph of G ∗ . Hence, x y is a full bipolar fuzzy bridge. Proposition 5.4 Let G = (A, B) be a bipolar fuzzy graph on X and x y is not contained in a cycle of G ∗ , then the following conditions are equivalent. p

1. μ B (x y) = h(B), μnB (x y) = d(B). 2. x y is a partial bipolar fuzzy bridge. 3. x y is a full bipolar fuzzy bridge. Proof Since x y is not contained in a cycle of G ∗ , x y is a bridge of G ∗ . Hence by Proposition 5.3, 1⇔ 3. Clearly, 3⇔ 2. Assume that condition 2 holds, then x y is a bridge for G (s,t) , for all (s, t) ∈ (d(B), 0] × (0, h(B)]. So, x y ∈ B(h(B),d(B)) . Thus p  μ B (x y) = h(B), μnB (x y) = d(B), i.e., condition 1 holds. Proposition 5.5 Let G = (A, B) be a bipolar fuzzy graph on X and x y is a bridge of G ∗ , then x y is a weak bipolar fuzzy bridge and a bipolar fuzzy bridge. Proposition 5.6 Let G = (A, B) be a bipolar fuzzy graph on X, then x y is a bipolar fuzzy bridge if and only if x y is a weak bipolar fuzzy bridge. Proof Suppose x y is a weak bipolar fuzzy bridge, then there exists (s, t) ∈ (0, h(B)] × [d(B), 0) such that x y is a bridge for G (s,t) . Hence, removal of x y disconp nects G (s,t) . Thus, any path from x to y in G has an edge uv with μ B (uv) < s, p p μnB (uv) > t. Thus, the removal of x y results in (μ B ) ∞ (x, y) < s  (μ B )∞ (x, y), (μnB ) ∞ (x, y) > t  (μnB )∞ (x, y). Hence, x y is a bipolar fuzzy bridge. Conversely, suppose x y is a bipolar fuzzy bridge. Then there exists uv such p p that removal of x y results in (μ B ) ∞ (u, v) < (μ B )∞ (u, v), (μnB ) ∞ (u, v) > (μnB )∞ p (u, v). Hence, x y is on every strongest path connecting u and v and in fact, μ B (uv)  p μ B (x y) and μnB (uv)  μnB (x y). Thus, there does not exist a bipolar fuzzy path (other than x y) connecting x and y in G (μ Bp (x y),μnB (x y)) , else the other bipolar fuzzy path p without x y would be of strength greater or equal to μ B (x y), less or equal to μnB (x, y) and would be a part of a bipolar fuzzy path connecting u and v of strongest length, contrary to the fact x y is on every such bipolar fuzzy path. Hence, x y is a bridge of p p G (μ Bp (x y),μnB (x y)) and 0 < μ B (x y) ≤ h(B), d(B) ≤ μnB (x y) < 0. Thus, μ B (x y) = s n  and μ B (x y) = t.

238

5 Bipolar Fuzzy Planar Graphs

Fig. 5.8 Bipolar fuzzy cut vertices in G = (A, B)

z

x

(0.8, −0.2)

(0.7, −0.1)

,− .6 (0 2) 0.

y

5.4 Bipolar Fuzzy Cut Vertices and Bipolar Fuzzy Blocks Definition 5.17 A vertex x ∈ X in a bipolar fuzzy graph G = (A, B) is called µ p cut vertex if deleting x from G reduces the μ p -strength of connectedness between some pair of vertices. A vertex x ∈ X in called a µn -cut vertex if deleting x increases the μn -strength of connectedness between some pair of vertices. A vertex x ∈ X is a bipolar fuzzy cut vertex if it is μ p -cut vertex and μn -cut vertex. Definition 5.18 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X with crisp graph G ∗ = (X, E), then

1. a vertex x ∈ X is called a bipolar fuzzy cut vertex if (μ B ) ∞ (u, v) < (μ B )∞ (u, v) and (μnB ) ∞ (u, v) > (μnB )∞ (u, v), for some u, v ∈ X , 2. x is called a weak bipolar fuzzy cut vertex if there exists (s, t) ∈ (0, h(B)] × [d(B), 0) such that x is a cut vertex of G (s,t) , 3. x is called a partial bipolar fuzzy cut vertex if x is a cut vertex of G (s,t) , for all (s, t) ∈ (0, h(B)] × (d(B), 0] ∪ {(h(B), d(B))}, 4. x is called a full bipolar fuzzy cut vertex if x is a cut vertex of G (s,t) , for all (s, t) ∈ (0, h(B)] × [d(B), 0). p

p

Example 5.11 Consider a connected bipolar fuzzy graph G = (A, B) as shown in Fig. 5.8. Here d(B) = −0.2, h(B) = 0.8. Thus, (s, t) ∈ (0, 0.8] × [−0.1, 0). For 0 < s ≤ 0.6, −0.2 ≤ t < 0, G (s,t) = (X, {x y, yz, x z}). For 0.6 < s ≤ 0.7, −0.2 ≤ t < 0, G (s,t) = (X, {x y, x z}). For 0.6 < s ≤ 0.8,−0.1 ≤ t < 0, G (s,t) = (X, {x z}). Thus, x is a bipolar fuzzy cut vertex and a weak bipolar fuzzy cut vertex, but not a partial bipolar fuzzy cut vertex. Example 5.12 Let A be a bipolar fuzzy set on X = {x, y, z} and B be a bipolar fuzzy relation in X defined by p

p

p

μ A (x) = μ A (y) = μ A (z) = 1, μnA (x) = μnA (y) = μnA (z) = −1,

5.4 Bipolar Fuzzy Cut Vertices and Bipolar Fuzzy Blocks p

p

239

p

μ B (x y) = μ B (x z) = 0.9, μ B (yz) = 0.5 μnB (x y) = μnB (x z) = −0.1, μnB (yz) = −0.4.

Here d(B) = −0.4, h(B) = 0.9. For 0 < s ≤ 0.5, −0.4 ≤ t < 0, G (s,t) = (X, {x y, yz, x z}). For 0.5 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y, x z}). Thus, x is a bipolar fuzzy cut vertex and a partial bipolar fuzzy cut vertex but neither a cut vertex of G ∗ nor a full bipolar fuzzy cut vertex. Example 5.13 Let A be a bipolar fuzzy set on X = {x, y, z} and B be a bipolar fuzzy relation in X defined by p

p

p

μ A (x) = μ A (y) = μ A (z) = 1, μnA (x) = μnA (y) = μnA (z) = −1, p

p

μ B (x y) = μ B (x z) = 0.9, μnB (x y) = μnB (x z) = −0.1. Clearly, d(B) = −0.1, h(B) = 0.9. For 0 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y, x z}). Thus, x is a full bipolar fuzzy cut vertex of G, bipolar fuzzy cut vertex of G, and a cut vertex of G ∗ . Proposition 5.7 Let G be a bipolar fuzzy graph such that G ∗ is a cycle. Then a vertex is a bipolar fuzzy cut vertex of G if and only if it is a common vertex of two bipolar fuzzy bridges. Proposition 5.8 If z is a common vertex of at least two bipolar fuzzy bridges, then z is a bipolar fuzzy cut vertex. Proposition 5.9 If G is a complete bipolar fuzzy graph, then (μ B )∞ (u, v) = p μ B (uv) and (μnB )∞ (u, v) = μnB (uv). p

Proposition 5.10 A complete bipolar fuzzy graph has no bipolar fuzzy cut vertices. A bipolar fuzzy graph G is biconnected if a vertex is removed from G, then G remains connected (see Definition 1.23). Definition 5.19 Let G = (A, B) be a bipolar fuzzy graph on X, then 1. G is called a bipolar fuzzy block if it has no bipolar fuzzy cut vertices. 2. G is called a weak bipolar fuzzy block if there exists (s, t) ∈ (0, h(B)] × [d(B), 0) such that G (s,t) is a block. 3. G is called a partial bipolar fuzzy block if G (s,t) is a block, for all (s, t) ∈ (0, h(B)] × (d(B), 0] ∪ {(h(B), d(B))}. 4. G is called a full bipolar fuzzy block if G (s,t) is a block, for all (s, t) ∈ (0, h(B)] × [d(B), 0). Example 5.14 Consider a connected bipolar fuzzy graph G = (A, B) as shown in Fig. 5.9. By routine calculations, d(B) = −0.3, h(B) = 0.7. Thus, (s, t) ∈ (0, 0.7] × [−0.2, 0). For 0 < s ≤ 0.5, −0.3 ≤ t < 0, G (s,t) = (X, {x y, yz, x z}). For 0.5 < s ≤ 0.7, −0.2 ≤ t < 0, G (s,t) = (X, {x z}). Thus, G ∗ is a block, G is a bipolar fuzzy block and a weak bipolar fuzzy block. G is not a partial bipolar fuzzy block since G (s,t) is not a block for 0.5 < s ≤ 0.7, −0.2 ≤ t < 0.

240

5 Bipolar Fuzzy Planar Graphs

Fig. 5.9 Bipolar fuzzy block

z

(0.7, −0.3)

x

2) 0.

(0.5, −0.2)

,− .5 (0

y

Example 5.15 Consider a bipolar fuzzy graph G = (A, B) on X = {x, y, z}, where A is a bipolar fuzzy set of X and let B is a bipolar fuzzy relation in X defined by p

p

p

μ A (x) = μ A (y) = μ A (z) = 1, μnA (x) = μnA (y) = μnA (z) = −1, p

p

p

μ B (x y) = μ B (x, z) = 0.9, μ B (y, z) = 0.5 μnB (x, y) = μnB (x, z) = −0.1, μnB (y, z) = −0.4.

Clearly, d(B) = −0.4, h(B) = 0.9. For 0 < s ≤ 0.5, −0.4 ≤ t < 0, G (s,t) = (X, {x y, yz, x z}). For 0.5 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y, x z}). Thus, G ∗ is a block and G is a weak bipolar fuzzy block. However, G is not a bipolar fuzzy block since x is a bipolar fuzzy cut vertex of G. Also, G is not a partial bipolar fuzzy block since x is a cut vertex of G (s,t) for 0.5 < s ≤ 0.9, −0.1 ≤ t < 0. Example 5.16 Consider a bipolar fuzzy graph G = (A, B) on X = {x, y, z}, where A is a bipolar fuzzy set of X and B is a bipolar fuzzy relation in X defined by p

p

p

μ A (x) = μ A (y) = μ A (z) = 1, μnA (x) = μnA (y) = μnA (z) = −1, p

p

p

μ B (x y) = μ B (x, z) = μ B (y, z) = 0.9 μnB (x, y) = μnB (x, z) = μnB (y, z) = −0.1. By routine computations, d(B) = −0.1, h(B) = 0.9. For 0 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y, yz, x z}). Thus, G ∗ is a block, G is a bipolar fuzzy block and a full bipolar fuzzy block. Definition 5.20 A connected bipolar fuzzy graph G = (A, B) is said to be a firm if p

p

min{μ A (x) | x ∈ X } ≥ max{μ B (x y) | x y ∈ E}, max{μnA (x) | x ∈ X } ≤ min{μnB (x y) | x y ∈ E}. The bipolar fuzzy graph shown in Fig. 5.7 is a firm.

5.4 Bipolar Fuzzy Cut Vertices and Bipolar Fuzzy Blocks Fig. 5.10 Bipolar fuzzy block which is not a firm

241 x

z (1, −0)

(0.8, −0.2)

(0.5, −0.5)

,− .5 2) 0.

(0.5, −0.4)

(0

(1, −0)

y

Example 5.17 Consider a connected bipolar fuzzy graph as shown in Fig. 5.10. Here d(B) = −0.4, h(B) = 0.8. Thus, (s, t) ∈ (0, −0.8] × (0, −0.2]. For 0 < s ≤ 0.5, −0.4 ≤ t < 0, G (s,t) = (X, {x y, yz, x z}). For 0.5 < s ≤ 0.8, −0.2 ≤ t < 0, G (s,t) = (X, {x z}). Thus, G is a bipolar fuzzy block and full bipolar fuzzy block but not a firm.

5.5 Bipolar Fuzzy Cycles and Bipolar Fuzzy Trees The concept of a bipolar fuzzy cycle is already elaborated in Definition 1.23. In this section, the notions of bipolar fuzzy cycles and bipolar fuzzy trees are discussed with level graphs. Definition 5.21 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X, then 1. G is called a bipolar fuzzy cycle if G ∗ is a cycle and there does not exist unique p p x y ∈ E such that μ B (x y) = min{μ B (uv) | uv ∈ E = supp(B)}, μnB (x y) = n max{μ B (uv) | uv ∈ E}. 2. G is called a weak bipolar fuzzy cycle if there exists (s, t) ∈ (0, h(B)] × [d(B), 0) such that G (s,t) is a cycle. 3. G is called a partial bipolar fuzzy cycle if G (s,t) is a cycle, for all (s, t) ∈ (0, h(B)] × (d(B), 0] ∪ {(h(B), d(B))}. 4. G is called a full bipolar fuzzy cycle if G (s,t) is a cycle, for all (s, t) ∈ (0, h(B)] × [d(B), 0). Example 5.18 Consider a connected bipolar fuzzy graph G = (A, B) as shown in Fig. 5.11. By routine computations, d(B) = −0.2, h(B) = 0.9. Thus (s, t) ∈ (0, −0.9] × (0, −0.1]. For 0 < s ≤ 0.5, 0 ≤ t < −0.2, G (s,t) = (X, {x y, xw, yz, wz}). For 0.5

242

5 Bipolar Fuzzy Planar Graphs

Fig. 5.11 Bipolar fuzzy cycle

y

(0.9, −0.1)

x

(0.5, −0.2)

(0.5, −0.2)

z

Fig. 5.12 Partial bipolar fuzzy graph cycle

y

(0.9, −0.1)

(0.9, −0.1)

w x

)

,− .1 (0

(0.9, −0.1)

z

4 0.

(0.9, −0.1)

(0.9, −0.1)

w

< s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y, zw}). Thus, G is a bipolar fuzzy cycle and weak bipolar fuzzy cycle but G is not partial bipolar fuzzy cycle. Example 5.19 Consider a bipolar fuzzy graph G = (A, B) as shown in Fig. 5.12. Here d(B) = −0.4, h(B) = 0.9. Thus, (s, t) ∈ (0, h(B)] × [d(B), 0) means (s, t) ∈ G (s,t) = (X, (0, −0.9] × [−0.1, 0). For 0 < s ≤ 0.1, −0.4 ≤ t < 0, {x y, yz, wz, wx, xw}) which is not a cycle. For 0.1 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y, yz, zw, wx}) which is a cycle. Thus, G is not a bipolar fuzzy cycle. G is a partial bipolar fuzzy cycle but not a full bipolar fuzzy cycle. The proofs of the following propositions are trivial. Proposition 5.11 Suppose G is a bipolar fuzzy cycle, then G is a partial bipolar fuzzy cycle if and only if G is a full bipolar fuzzy cycle. Proposition 5.12 A bipolar fuzzy graph G = (A, B) is a full bipolar fuzzy cycle if and only if G is a bipolar fuzzy cycle and B is a constant function. Definition 5.22 A connected bipolar fuzzy graph G = (A, B) is a bipolar fuzzy tree if it has a bipolar fuzzy spanning subgraph H = (A, C) such that H ∗ is a tree p p and for all edges x y not in H , μ B (x y) < (μC )∞ (x, y), μnB (x y) > (μCn )∞ (x, y). Definition 5.23 Let G = (A, B) be a bipolar fuzzy graph on X, then

5.5 Bipolar Fuzzy Cycles and Bipolar Fuzzy Trees

243

1. G is called a bipolar fuzzy forest if G has a bipolar fuzzy spanning subgraph p H = (A, C) such that H ∗ is a forest and for all edges uv not in F, μ B (uv) < p ∞ n n ∞ (μC ) (u, v) and μ B (uv) > (μC ) (u, v). In this case, the bipolar fuzzy graph G need not be connected. 2. G is called a weak bipolar fuzzy forest if for all (s, t) ∈ (0, h(B)] × [d(B), 0), G (s,t) is a forest. 3. G is called a partial bipolar fuzzy forest if G (s,t) is a forest, for all (s, t) ∈ (0, h(B)] × (d(B), 0] ∪ {(h(B), d(B))}. 4. G is called a full bipolar fuzzy forest if G (s,t) is a forest, for all (s, t) ∈ (0, h(B)] × [d(B), 0). Example 5.20 Consider a connected bipolar fuzzy graph G = (A, B) on X = {x, y, z, w}, where A is a bipolar fuzzy set on X and B is a bipolar fuzzy relation in X defined by p

p

p

p

μ A (x) = μ A (y) = μ A (z) = μ A (w) = 1, μnA (x) = μnA (y) = μnA (z) = μnA (w) = −1, p

p

p

p

μ B (x y) = μ B (w, z) = 0.9, μ B (xw) = μ B (yz) = 0.5, μnB (x y) = μnB (wz) = −0.1, μnB (xw) = μnB (yz) = −0.4.

By routine computations, d(B) = −0.4, h(B) = 0.9. For 0 < s ≤ 0.5, −0.4 ≤ t < 0, G (s,t) = (X, {xw, yz, x y, wz}) and for 0.5 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y, wz}). Thus, G is a partial bipolar fuzzy forest but neither a bipolar fuzzy forest nor a full bipolar fuzzy forest. Proposition 5.13 A bipolar fuzzy graph G = (A, B) is a full bipolar fuzzy forest if and only if G ∗ is a forest. Proof Suppose that G is a full bipolar fuzzy forest, then G ∗ is a forest. Conversely, let G ∗ be a forest, then G (s,t) is a forest for all (s, t) ∈ (0, h(B)] ×  [d(B), 0) and each such G (s,t) is a subgraph of G ∗ . It completes the proof. Example 5.21 Consider a connected bipolar fuzzy graph G = (A, B) on X = {x, y, z} such that p

p

p

μ A (x) = μ A (y) = μ A (z) = 1, μnA (x) = μnA (y) = μnA (z) = −1, p

p

μ B (x y) = 0.9, μ B (yz) = 0.5, μnB (x y) = −0.1, μnB (yz) = −0.4. Clearly, d(B) = (0.5, −0.4), h(B) = (0.9, −0.1). For 0 < s ≤ 0.5, −0.4 ≤ t < 0, G (s,t) = (X, {x y, yz}). For 0.5 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y}). Thus, G is a full bipolar fuzzy forest without B being a constant function. Note that G ( h(B), d(B)) has more connected components than G ∗ . Proposition 5.14 A bipolar fuzzy graph G = (A, B) is a weak bipolar fuzzy forest if and only if G does not contain a bipolar fuzzy cycle whose edges are of strength (h(B), d(B)).

244

5 Bipolar Fuzzy Planar Graphs

Proof Suppose G contains a bipolar fuzzy cycle whose edges are of strength (h(B), d(B)). Then G (s,t) , (s, t) ∈ (0, h(B)] × [d(B), 0) contains this bipolar fuzzy cycle and so G ∗ is not a forest. Thus, G is not a weak bipolar fuzzy forest. Conversely, suppose G does not contain a bipolar fuzzy cycle all of whose edges are of membership value (h(B), d(B)). Then G (h(B),d(B)) does not contain a cycle and so is a forest.  Corollary 5.1 If G is a bipolar fuzzy forest, then G is also a weak bipolar fuzzy forest. Theorem 5.9 Let G = (A, B) be a bipolar fuzzy graph, then G ∗ is a forest and B is a constant function if and only if G is a full bipolar fuzzy forest, G ∗ and G (h(B),d(B)) have the same number of connected components and G is a firm. Proof Suppose that G ∗ is a forest and B is a constant function. Then for all (s, t) ∈ (0, h(B)] × [d(B), 0), G (s,t) = G ∗ and so G is a full bipolar fuzzy forest and G ∗ and G (h(B),d(B)) have the same number of connected components. Clearly, G is a firm since B is a constant function. Conversely, suppose G is a full bipolar fuzzy forest, G ∗ and G (h(B),d(B)) have the same number of connected components and G is a firm. Suppose there exists p (s1 , t1 ), (s2 , t2 ) ∈ [0, 1] × [−1, 0] such that μ B (x y) = s1 , μnB (x y) = t1 , and x y ∈ / B(s2 ,t2 ) . Hence, G (s2 ,t2 ) has more connected components than G (s1 ,t1 ) B(s1 ,t1 ) , x y ∈ since G is a firm, i.e., no vertices were lost. Thus, G (h(B),d(B)) has more connected  components than G ∗ , a contradiction. Corollary 5.2 Let G = (A, B) be a bipolar fuzzy graph, then G ∗ is a forest and B is constant function if and only if G is a full bipolar fuzzy tree and G is a firm. Definition 5.24 Let G = (A, B) be a bipolar fuzzy graph on X, then 1. G is called a bipolar fuzzy tree if G has a bipolar fuzzy spanning subgraph H = p p (A, C) which is a tree such that for all edges uv not in H , μ B (uv) < (μC )∞ (u, v) and μnB (uv) > (μCn )∞ (u, v). 2. G is called a weak bipolar fuzzy tree if for all (s, t) ∈ (0, h(B)] × [d(B), 0) G (s,t) is a tree. 3. G is called a partial bipolar fuzzy tree if G (s,t) is a tree, for all (s, t) ∈ (d(B), 0] × (0, h(B)] ∪ {(h(B), d(B))}. 4. G is called a full bipolar fuzzy tree if G (s,t) is a tree, for all (s, t) ∈ (0, h(B)] × [d(B), 0). Example 5.22 Consider a bipolar fuzzy graph G = (A, B) on X = {x, y, z}, where A and B are defined as p

p

p

μ A (x) = μ A (y) = 1, μ A (z) = 0.5, μnA (x) = μnA (y) = 0, μnA (z) = −0.2, p

p

μ B (x y) = 0.9, μ B (yz) = 0.5, μnB (x y) = 0.1, μnB (yz) = −0.4.

5.5 Bipolar Fuzzy Cycles and Bipolar Fuzzy Trees

245

By usual calculations, d(B) = −0.4, h(B) = 0.9. For 0 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y, yz}) and for 0.5 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = (X, {x y}). Thus, G ∗ is a tree, G is a full bipolar fuzzy tree, and G ∗ and G (h(B),d(B)) have the p same number of connected components. However, G is not a firm and B = (μ B , μnB ) is not a function. Example 5.23 Consider a bipolar fuzzy graph G = (A, B) on X = {x, y, z}, where A and B are defined as p

p

p

μ A (x) = μ A (y) = 1, μ A (z) = 0.5, μnA (x) = μnA (y) = −1, μnA (z) = −0.2, p

p

p

μ B (x y) = 0.9, μ B (x z) = μ B (yz) = 0.5, μnB (x y) = −0.1, μnB (x z) = μnB (yz) = −0.4.

Here d(B) = −0.4, h(B) = 0.9. For 0 < s ≤ 0.5, −0.4 ≤ t < 0, G (s,t) = (X, {x y, x z, yz}), and for 0.5 < s ≤ 0.9, −0.1 ≤ t < 0, G (s,t) = ({x, y}, {x y}). Thus, G is a partial bipolar fuzzy tree, but not a full bipolar fuzzy tree. G is also not a bipolar fuzzy tree. Proposition 5.15 If G is a bipolar fuzzy tree, then G is not a complete bipolar fuzzy graph. Proposition 5.16 If G is a bipolar fuzzy tree, then edges of spanning bipolar fuzzy subgraph H are the bipolar fuzzy bridges of G. Proposition 5.17 Let G = (A, B) be a bipolar fuzzy graph, then G is a bipolar fuzzy tree if and only if the following are equivalent. 1. x y is a bipolar fuzzy bridge. p p 2. (μ B )∞ (x, y) = μ B (x y) and (μnB )∞ (x, y) = μnB (x y). Proposition 5.18 A bipolar fuzzy graph is a bipolar fuzzy tree if and only if it has a unique maximum bipolar fuzzy spanning tree. Proposition 5.19 Let G = (A, B) be a firm. If G is a weak bipolar fuzzy tree, then G is a bipolar fuzzy tree. Proof There exist (s, t) ∈ (0, h(B)] × [d(B), 0) such that G (s,t) is a tree. Since G is a firm, G (s,t) is a spanning subgraph of G ∗ which is a tree. If uv ∈ E B(s,t) , then p  μ B (uv) < s, μnB (uv) > t and so it follows that G is a bipolar fuzzy tree. Definition 5.25 Let G = (A, B) be a bipolar fuzzy graph on X, then 1. G is called weak bipolar fuzzy connected if there exists (s, t) ∈ (0, h(B)] × [d(B), 0) such that G (s,t) is connected. 2. G is called partial bipolar fuzzy connected if G (s,t) is connected, for all (s, t) ∈ (d(B), h(B)] ∪ {h(B)}. 3. G is called full bipolar fuzzy connected if G (s,t) is connected, for all (s, t) ∈ (0, h(B)] × [d(B), 0).

246

5 Bipolar Fuzzy Planar Graphs

Proposition 5.20 If G is a connected bipolar fuzzy graph, then G is weakly connected. Proof Let G = (A, B) be a connected bipolar fuzzy graph, then G ∗ is connected, then there exists some (s, t) ∈ (0, h(B)] × [d(B), 0) such that G (s,t) = G ∗ and so G is weakly connected.  Proposition 5.21 Let G = (A, B) be a bipolar fuzzy graph, then the following conditions are satisfied. 1. If G is a weak bipolar fuzzy tree, then G is weakly connected and G is a weak bipolar fuzzy forest. Conversely, if ∃(s1 , t1 ), (s2 , t2 ) ∈ (0, h(B)] × [d(B), 0) with s1 < s2 , t1 > t2 such that G (s1 ,t1 ) is a forest and G (s2 ,t2 ) is connected, then G is a weak bipolar fuzzy tree. 2. G is a bipolar fuzzy tree if and only if G is a bipolar fuzzy forest and G is connected. 3. G is a partial bipolar fuzzy tree if and only if G is a partially connected partial bipolar fuzzy forest. 4. G is a full bipolar fuzzy tree if and only if G is a full bipolar fuzzy forest and G is fully connected. Proof 1. If G (s,t) is a tree for some (s, t) ∈ (, h(B)] × [d(B), 0), then G (s,t) is connected and is a forest. For the converse, if G (s1 ,t1 ) is a forest and G (s2 ,t2 ) is connected, then G (s2 ,t2 ) is a tree. The proofs of 2, 3, and 4 are immediate. 

5.6 Applications of Bipolar Fuzzy Planar Graphs Graph theory is considered an important part of Mathematics for solving countless real-world problems in information technology, psychology, engineering, combinatorics, and medical sciences. Everything in this World is connected, for instance, cities and countries are connected by roads, railways are linked by railway lines, flight networks are connected by air, electrical devices are connected by wires, web pages on the Internet by hyperlinks, components of electric circuits by various paths, and many more. Scientists, analysts, and engineers are trying to optimize these networks to find a way to save millions of lives by reducing traffic accidents, plane crashes, and circuit shots. Planar graphs are used to find such graphical representations of networks without any crossing or the minimum number of crossings. But there is always a bipolar uncertainty in data which can be dealt with using bipolar graphs. The applications of bipolar fuzzy planar graphs in road networks and electrical connections are discussed in this section.

5.6 Applications of Bipolar Fuzzy Planar Graphs

247

(0. 5,

−0 .2)

.4) −0

6(

−0 .1)

L

(0. 5,

0.4 ,

−0 .5)

(0 .5,

(0. 5,

−0 .2)

4, (0.

L2 (0.5, −0.3)

L5 (0.5, −0.4)

) 0.2 − , 4 .2) (0. −0 , 4 (0.

−0 .3)

L1 (0.7, −0.2)

L3 (0.8, −0.2)

L4 (0.6, −0.1) Fig. 5.13 Bipolar fuzzy road model

5.6.1 Road Network Model to Monitor Traffic Roads are a means of a frequent and unacceptable number of fatalities every year. Road accidents are increasing due to dense traffic, negligence of drivers, and speed of vehicles. Traffic accidents can be minimized by modeling road networks to monitor the traffic, apply quick emergency services, and by taking action against overspeeding. The practical approach of bipolar fuzzy planar graphs can be applied to construct road networks. Consider the problem of road networks between 6 locations L 1 , L 2 , L 3 , L 4 , L 5 , L 6 as shown in Fig. 5.13. The positive degree of membership μ p (x) of each vertex x represents the percentage that vehicles traveling to or from this city are dense. The negative degree of membership μn (x) represents the percentage that traffic is not dense. The positive degree of memberships of each edge x y indicates the percentage of road accidents through this road and the negative degree of memberships of x y shows the percentage that the road is safer. This bipolar fuzzy model can be used to check and monitor the percentage of annual accidents. Also, by monitoring and taking special security actions, the total number of accidents can be minimized.

248

5 Bipolar Fuzzy Planar Graphs

−0 .2)

E3 (0.5, −0.4)

(0.

(0. 5,

) 0.2 − , 4

.4) −0 , 4 (0.

2) 0. ,− .1) .4 −0 (0 .6, (0

(0.5, −0.2)

.1) (0.6, −0

E1 (0.7, −0.2)

E2 (0.8, −0.2)

0.5) .4, − 0 ( E4

E5 (0.6, −0.1) Fig. 5.14 Electrical connections

5.6.2 Modeling of Electrical Connections Graph theory is extensively used in designing circuit connections and installation of wires in order to prevent crossing which can cause dangerous electrical hazards. The twisted and crossing wires are a serious safety risk to human life. There is a need to install electrical wires to reduce crossing. Bipolar fuzzy planar graphs can be used to model electrical connections and to study the degree of damage that can cause due to the connection. Consider the problem of setting electrical wires between 5 electrical utilities and power plugs E 1 , E 2 , E 3 , E 4 , E 5 in a factory as shown in Fig. 5.14. The positive degree of membership μ p (E i ) of each vertex E i represents the percentage of faults and electrical sparks of utility or power plug E i and the negative degree of membership μn (E i ) represents the percentage that E i is updated and safer. The positive degree of membership of each edge E i E j indicates the percentage of electrical hazards through this connection. The negative degree of membership of E i E j shows the percentage that the connection is safer. The crossing of wires can be reduced if we change the geometrical representation of Fig. 5.14. The other representation is shown in Fig. 5.15 which has only one crossing, at point P1 , between the edges E 1 E 4 and E 2 E 5 . The electrical damage at the crossing point P1 can be reduced by using better electrical wires between E 1 and E 4 , E 2 and E 5 .

5.7 Conclusions

249

(0.5, −0.2)

(0.

(0. 5,

) 0.2 − , 4

−0 .2)

E3 (0.5, −0.4)

E1 (0.7, −0.2)

E2 (0.8, −0.2)

.4) −0 , 4 (0.

P

2) 0. ,− .1) .4 −0 (0 .6, (0

.1) (0.6, −0

1

0.5) .4, − 0 ( E4

E5 (0.6, −0.1) Fig. 5.15 Bipolar fuzzy planar graph

5.7 Conclusions Bipolar fuzzy planar graphs are used to model various practical problems in which the crossings between any two lines are prohibited with certain bipolar uncertainty. In this chapter, we have studied the concepts of bipolar fuzzy multisets, bipolar fuzzy multigraphs, strong and complete bipolar fuzzy multigraphs, bipolar fuzzy planar graphs, and bipolar fuzzy dual graphs. We have discussed the strong and weak properties of bipolar fuzzy edges, bipolar fuzzy faces, and bipolar fuzzy planar graphs. We have described the relation of planarity and duality in bipolar fuzzy graphs using different characterizations of intersection value and planarity value of bipolar fuzzy graphs. We have elaborated various properties of bipolar fuzzy bridges, bipolar fuzzy cut vertices, bipolar fuzzy blocks, bipolar fuzzy cycles, and bipolar fuzzy trees in terms of level graphs. We have presented the importance of bipolar fuzzy planar graphs with a number of real-world applications in road networks and electrical circuits.

250

5 Bipolar Fuzzy Planar Graphs

Exercises 5 1. Let G and H be connected bipolar fuzzy fuzzy graphs different from K 1 and K 2 , then prove or disprove that GH is a bipolar fuzzy planar graph if and only if both factors are bipolar fuzzy paths, or one is a bipolar fuzzy path and other is a bipolar fuzzy cycle. 2. If G = (A, B) is a bipolar fuzzy graph with p vertices, then show that G has at most p − 1 bipolar fuzzy bridges. 3. Prove that a nontrivial bipolar fuzzy tree H has at least two bipolar fuzzy end nodes. 4. If H is a bipolar fuzzy tree, then show that every vertex of H is either a bipolar fuzzy cut-node or a bipolar fuzzy end node. Also, show that the converse is not true. 5. If G is a bipolar fuzzy tree, then prove or disprove that the arcs of G are bipolar fuzzy bridges. 6. Show that a bipolar fuzzy cut vertex of a bipolar fuzzy tree is the common vertex of at least two bipolar fuzzy bridges. 7. If G is a bipolar fuzzy tree, then show that every vertex of G is either a bipolar fuzzy cut vertex or a bipolar fuzzy end node. 8. If G is a bipolar fuzzy tree, then show that it has at most n − 1 number of bipolar fuzzy bridges where n is the number of vertices in G. 9. Show that every bipolar fuzzy bridge is strong, but a strong edge need not be a bipolar fuzzy bridge.

References 1. Abdul-jabbar, N., Naoom, J.H., Ouda, E.H.: Fuzzy dual graph. J. Al. Nahrain Univ. 12(4), 168–171 (2009) 2. Akram, M.: Bipolar fuzzy graphs. Inf. Sci. 181(24), 5548–5564 (2011) 3. Akram, M.: Bipolar fuzzy graphs with applications. Knowl.-Based Syst. 39, 1–8 (2013) 4. Akram, M., Dudek, W.A.: Regular bipolar fuzzy graphs. Neural Comput. Appl. 21(supp 1), 197–205 (2012) 5. Akram, M., Alshehri, N., Davvaz, B., Ashraf, A.: Bipolar fuzzy digraphs in decision support systems. J. Multiple-Valued Logic Soft. Comput. 27(5–6), 531–551 (2016) 6. Akram, M., Farooq, A.: Bipolar fuzzy trees. New Trends Math. Sci. 4(3), 58–72 (2016) 7. Akram, M., Samanta, S., Pal, M.: Application of bipolar fuzzy sets in planar graphs. Int. J. Appl. Comput. Math. 3(2), 773–785 (2017) 8. Mathew, S., Sunitha, M.S.: Types of arcs in a fuzzy graph. Inf. Sci. 179(11), 1760–1768 (2009) 9. Mathew, S., Sunitha, M.S.: Strongest strong cycles and θ-fuzzy graphs. IEEE Trans. Fuzzy Syst. 21(6), 1096–1104 (2013) 10. Mordeson, J.N., Nair, P.S.: Cycles and cocycles of fuzzy graphs. Inf. Sci. 90(1–4), 39–49 (1996) 11. Nagoorgani, A., Radha, K.: Degree of fuzzy graph. Int. J. Algorithms Comput. Math. 2(3), 1–9 (2009) 12. Pal, A., Samanta, S., Pal, M.: Concept of fuzzy planar graphs. In: Proceedings of Science and Information Conference, pp. 557–563. IEEE (2013)

References

251

13. Rosenfeld, A.: Fuzzy graphs. In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds.) Fuzzy Sets and Their Applications, pp. 77–95. Academic, New York (1975) 14. Samanta, S., Pal, M.: Fuzzy planar graphs. IEEE Trans. Fuzzy Syst. 23(6), 1936–1942 (2015) 15. Samanta, S., Pal, M., Pal, A.: New concepts of fuzzy planar graph. Int. J. Adv. Res. Artif. Intell. 3(1), 52–59 (2014) 16. Shinoj, T.K., John, S.J.: Intuitionistic fuzzy multisets and its application in medical diagnosis. World Acad. Sci. Eng. Technol. 6(1), 1418–1421 (2012) 17. Sunitha, M.S., Vijayakumar, A.: A characterization of fuzzy trees. Inf. Sci. 113, 293–300 (1999) 18. Sunitha, M.S., Vijayakumar, A.: Complement of a fuzzy graph. Indian J. Pure Appl. Math. 33(9), 1451–1464 (2002) 19. Wilson, R.J.: Graph theory. History of Topology. Elsevier Science, Amsterdam (1999) 20. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 21. Zhang, W.-R., Zhang, L.: YinYang bipolar logic and bipolar fuzzy logic. Inf. Sci. 165(3–4), 265–287 (2004) 22. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998) 23. Zhang, W.-R.: The road from fuzzy sets to definable causality and bipolar quantum intelligenceTo the memory of Lotfi A. Zadeh. J. Intell. Fuzzy Syst. 36(4), 3019–3032 (2019)

Chapter 6

Domination in Bipolar Fuzzy Graphs

In this chapter, we discuss different concepts of dominating, total dominating, equitable dominating, total equitable dominating, and independent and equitable independent sets in bipolar fuzzy graphs. We present certain properties concerning the relationship of domination, total domination and independence numbers in complete bipolar fuzzy graphs, complete bipartite bipolar fuzzy graphs, and products of bipolar fuzzy graphs. We study the notions of private neighbor, lower and upper irredundant sets, and irredundance number in relation to independence and dominating numbers of bipolar fuzzy graphs. We describe the importance of domination in bipolar fuzzy graphs with a number of real-world applications in facility location problem, finding the set of representatives, and transmission tower location problem. The main discussion of this chapter is from [5, 14].

6.1 Introduction Domination in graph theory has a wide range of applications in different fields including, school bus routing, facility location problem, finding the set of representatives, land surveying, coding theory and, monitoring communication and electrical networks. The idea of domination first arose in the chessboard problem. The classical problems of covering chessboards with the minimum number of chess pieces were important in motivating the revival of the study of dominating sets in graphs, which commenced in the early 1970s. These problems certainly date back to de Jaenisch [11] and have been mentioned in the literature frequently since that time (cf. [9, 12]). In 1962, Ore [17] was the first to use the term “domination” for undirected graphs, and he introduced the concepts of minimal and minimum dominating sets of vertices in a graph. Somasundaram and Somasundaram [19] introduced domination and independent domination in fuzzy graphs. Gani and Chandrasekaran [15] studied the notion of fuzzy domination and independent domination using strong © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_6

253

254

6 Domination in Bipolar Fuzzy Graphs

arcs. The independent domination number and irredundance number in graphs are introduced by Cockayne [10] and Hedetniemi [13]. Nagoorgani and Vadivel [16] discussed domination, independent domination, and irredundance in fuzzy graphs using strong edges. The concept of domination in intuitionistic fuzzy graphs was investigated by Parvathi and Thamizhendhi [18]. Zhang [20] introduced the theory of bipolar fuzzy sets (also known as YinYang bipolar fuzzy sets, Yang depicts the positive side while Yin represents the negative side in a system). The concept of bipolar fuzzy sets was applied to graphs in [1–4]. The concept of certain types of dominating and independent sets in bipolar fuzzy graphs and m-polar fuzzy graphs is discussed in [5–7, 14] which is the main discussion of this chapter.

6.2 Types of Domination in Bipolar Fuzzy Graphs A dominating set of a crisp graph G ∗ = (X, E) is a subset D of X such that every vertex not in D is adjacent to at least one member of D. The domination number is the number of vertices in a smallest dominating set for G ∗ . An independent set of G ∗ is a subset I of vertices such that no two vertices of I are adjacent in G ∗ . That is, for every two vertices x, y ∈ I , x y ∈ / E. A maximal independent set is an independent set of a graph containing the largest possible number of vertices for the given graph, and the cardinality of this set is called the independence number of the graph. Definition 6.1 Let G = (A, B) be a bipolar fuzzy graph on X, then the vertex cardinality of G is denoted by |A| and defined as |A| =



p

(μ A (vi ), μnA (vi )).

vi ∈X

Definition 6.2 Let G = (A, B) be a bipolar fuzzy graph on X, then the edge cardinality of G is denoted by |B| and defined as |B| =



p

(μ B (vi v j ), μnB (vi v j )).

vi ,v j ∈X

Example 6.1 Consider a bipolar fuzzy graph G = (A, B) on X = {v1 , v2 , v3 , v4 } as shown in Fig. 6.1. The vertex cardinality of G is (0.8, −2.3) and the edge cardinality is (0.7, −1.8) cardinality. Definition 6.3 The degree of a vertex v in a bipolar fuzzy graph G = (A, B) is defined as the sum of weights of strong edges incident to v. It is denoted by dG (v). The minimum degree of G is δ(G) = min {dG (v) | v ∈ X }. The maximum degree of G is Δ(G) = max {dG (v) | v ∈ X }.

6.2 Types of Domination in Bipolar Fuzzy Graphs

v4 (0.1,

(0.2, −0.3)

.2) −0 , .0 (0

(0 .2 ,− 0. 2)

0.6)

v2 (0.2, −0.4)

(0.1,

0.2)

(0.1, −0.4)

v1 (0.2, −0.5)

(0.1, −0.5)

Fig. 6.1 Bipolar fuzzy graph G

255

v3 (0.3,

0.4)

Definition 6.4 Two vertices vi and v j in a bipolar fuzzy graph G = (A, B) are said to be neighbors if either one of the following conditions is satisfied. p

1. μ B (vi v j ) > 0 and μnB (vi v j ) ≤ 0, p 2. μ B (vi v j ) ≥ 0 and μnB (vi v j ) < 0, vi , v j ∈ X . The set of all neighbors of u is denoted by N (u). Definition 6.5 An edge uv in a bipolar fuzzy graph G = (A, B) is said to be a p p strong edge if μ B (uv) ≥ (μ B )∞ (u, v) and μnB (uv) ≤ (μnB )∞ (u, v). Definition 6.6 A vertex u ∈ X in a bipolar fuzzy graph G = (A, B) is said to be p an isolated vertex if μ B (uv) = 0 and μnB (uv) = 0, for all v ∈ X, u = v. That is, N (u) = ∅. Definition 6.7 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . Let u, v ∈ X then u dominates v in G if there exists a strong edge between u and v. Remark 6.1 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X, then the following statements are satisfied. 1. For any u, v ∈ X, if u dominates v, then v dominates u and hence domination is a symmetric relation on X . 2. For any v ∈ X, N (v) is precisely the set of all vertices in X which are dominated by v. p p 3. If μ B (uv) < (μ B )∞ (u, v) and μnB (uv) > (μnB )∞ (u, v), for all u, v ∈ X , then the dominating set of G is X itself. 4. An isolated vertex does not dominate any other vertex of G. Definition 6.8 A subset S of X is called a dominating set in G if for every v ∈ X − S there exists u ∈ S such that u dominates v. Definition 6.9 A dominating set S of a bipolar fuzzy graph G is said to be a minimal dominating set if no proper subset of S is a dominating set.

256

6 Domination in Bipolar Fuzzy Graphs (0.2, −0.3)

v2 (0.4, −0.3)

v5 (0.5, −0.5)

(0 .0,

−0 .3)

(0.1, −0.4)

(0.1, −0.2)

(0.2, −0.4)

(0. 2,

−0 .2)

.4) −0 , .1 (0

v3 (0.3, −0.4)

v1 (0.2, −0.6)

v4 (0.1, −0.7)

Fig. 6.2 Domination numbers of bipolar fuzzy graph

Definition 6.10 The minimum cardinality among all minimal dominating sets is called lower domination number of G and is denoted by d B (G) or d(G). Definition 6.11 The maximum cardinality among all minimal dominating sets is called upper domination number of G and is denoted by D B (G). Example 6.2 Consider a bipolar fuzzy graph G = (A, B) on X = {v1 , v2 , v3 , v4 , v5 } as shown in Fig. 6.2. In Fig. 6.2, {v1 , v2 , v4 , v5 }, {v2 , v4 }, {v2 , v3 , v4 } are dominating sets but {v3 , v4 } is not a dominating set. Also, {v1 , v3 }, {v1 , v4 }, {v2 , v4 }, {v2 , v3 , v5 } are minimal dominating sets, and {v1 , v4 } has minimum cardinality among all minimal dominating sets, therefore, d B (G) = 2. As {v2 , v3 , v5 } has maximum cardinality among all minimal dominating sets, D B (G) = 3. Definition 6.12 Two vertices in a bipolar fuzzy graph G = (A, B) are said to be an independent if there is no strong edge between them. Definition 6.13 Let G = (A, B) be a bipolar fuzzy graph on X . A subset S of p p X is said to be an independent set if μ B (uv) < (μ B )∞ (u, v) and μnB (uv) > n ∞ (μ B ) (u, v), for all u, v ∈ S. Definition 6.14 An independent set S of bipolar fuzzy graph G = (A, B) is said to be maximal independent set if for every vertex v ∈ X − S, the set S ∪ {v} is not an independent set. Definition 6.15 The minimum cardinality among all maximal independent sets is called lower independence number of G and it is denoted by i B (G). Definition 6.16 The maximum cardinality among all maximal independent sets is called upper independence number of G and it is denoted by I B (G) or I (G).

6.2 Types of Domination in Bipolar Fuzzy Graphs

257

Example 6.3 In Fig. 6.2, {v1 }, {v2 , v5 }, {v2 , v3 , v5 } are independent sets, {{v1 , v3 }, {v1 , v4 }, {v2 , v4 }, {v2 , v3 , v5 }} are maximal independent sets; among all maximal independent sets, {v1 , v4 } has minimum cardinality and so i B (G) = 2, {v2 , v3 , v5 } has maximum cardinality among all maximal independent sets and I B (G) = 3. Definition 6.17 Let G = (A, B) be a bipolar fuzzy graph on X without isolated vertices. A subset S ⊆ X is a total dominating set if for every vertex v ∈ X , there exists a vertex u ∈ S, u = v such that u dominates v. Definition 6.18 A total dominating set S of a bipolar fuzzy graph G = (A, B) is said to be a minimal total dominating set if no proper subset of S is a total dominating set. Definition 6.19 The minimum cardinality of a minimal total dominating set is called the lower total dominating number of G and it is denoted by t B (G). Definition 6.20 The maximum cardinality of a minimal total dominating set is called the upper total dominating number of G and it is denoted by TB (G). Example 6.4 In Fig. 6.2, {v1 , v4 , v5 }, {v1 , v2 , v3 , v4 } are minimal total dominating sets; {v1 , v4 , v5 } has minimum cardinality among all minimal total dominating sets and t B (G) = 3; {v1 , v2 , v3 , v4 } has maximum cardinality among all minimal total dominating sets and TB (G) = 4. Theorem 6.1 A dominating set D of a bipolar fuzzy graph G = (A, B) is a minimal dominating set if and only if for each d ∈ D, one of the following conditions holds. 1. d is not a strong neighbor of any vertex in D, 2. there is a vertex v ∈ X − D such that N (v) ∩ D = {d}. Proof Assume that D is a minimal dominating set of G, then for every vertex d ∈ D, D − d is not a dominating set. Hence, there exists v ∈ X − (D − {d}) which is not dominated by any vertex in D − {d}. If v = d, v is not a strong neighbor of any vertex in D. If v = d, v is not dominated by D − {v} but is dominated by D. Then the vertex v is a strong neighbor only to d in D, that is, N (v) ∩ D = {d}. Conversely, assume that D is a dominating set and for each vertex d ∈ D, one of the two conditions holds. Suppose D is not a minimal dominating set, then there exists a vertex d ∈ D such that D − {d} is a dominating set. Hence, d is a strong neighbor to at least one vertex in D − {d}, that is, condition 1 does not hold. If D − {d} is a dominating set, then every vertex in X − D is a strong neighbor to at least one vertex in D − {d}, condition 2 does not hold which contradicts our assumption that at least one of the conditions hold. So D is a minimal dominating set.

Theorem 6.2 Let G = (A, B) be a bipolar fuzzy graph without isolated vertices and D is a minimal dominating set of G, then X − D is a dominating set of G.

258

6 Domination in Bipolar Fuzzy Graphs

Proof Let D be a minimal dominating set and d ∈ D. Since G has no isolated vertices, there is a vertex v ∈ N (d) such that v must be dominated by at least one vertex in D − {d}, that is, D − {d} is a dominating set. Theorem 6.1 follows that v ∈ X − D. Thus, every vertex in D is dominated by at least one vertex in X − D and so, X − D is a dominating set.

Theorem 6.3 Let G ∗ be a crisp graph corresponding to a bipolar fuzzy graph G = (A, B), then d B (G)+ d B (G c ) < 2O(G ∗ ), where d B (G c ) is the lower domination p p number of G c and equality holds if and only if 0 < μ B (vi v j ) < (μ B )∞ (vi , v j ) and n n ∞ 0 > μ B (vi v j ) > (μ B ) (vi , v j ), for all vi , v j ∈ X . Proof The result is trivial when d B (G) + d B (G c ) < 2O(G ∗ ). Also d B (G) = O(G ∗ ) p p if and only if μ B (vi v j ) < (μ B )∞ (vi , v j ) and μnB (vi v j ) > (μnB )∞ (vi , v j ), for all p p p vi , v j ∈ X. Clearly, d B (G c ) = O(G) if and only if μ B (vi v j ) − μ B (vi v j ) < (μ B )∞ (vi , v j ) and μnB (vi v j ) − μnB (vi v j ) > (μnB )∞ (vi , v j ), for all vi , v j ∈ X which gives p

μ B (vi v j ) > 0 and μnB (vi v j ) < 0. Hence, d B (G) + d B (G c ) ≤ 2O(G ∗ ). Corollary 6.1 For any bipolar fuzzy graph G = (A, B) without isolated vertices O(G ∗ ) d B (G) = , where O(G ∗ ) is the order of crisp graph G ∗ corresponding to G. 2 Theorem 6.4 Let G = (A, B) be a bipolar fuzzy graph, then d B (G) ≤ O(G ∗ ) − Δ(G ∗ ), where Δ(G ∗ ) = max{degG ∗ (x) | x ∈ X }. Proof Let v be a vertex in a bipolar fuzzy graph G = (A, B). Assume that |N (v)| = Δ(G ∗ ), then X − N (v) is a dominating set of G, so that d B (G) ≤ |X − N (v)| =

O(G ∗ ) − Δ(G ∗ ). Theorem 6.5 An independent set of bipolar fuzzy graph G = (A, B) is maximal if and only if it is an independent set and a dominating set. Proof Let D be a maximal independent set of a bipolar fuzzy graph, then for every vertex v ∈ X − D, the set D ∪ {v} is not an independent set. For every vertex v ∈ X − D, there is a vertex u ∈ D such that u is a strong neighbor to v. Thus, D is a dominating set. Hence, D is both a dominating and independent set. Conversely, assume that D is both an independent and dominating set. Suppose that D is not a maximal independent set, then there exists a vertex v ∈ X − D such that the set D ∪ {v} is an independent set. If D ∪ {v} is an independent set, then no vertex in D is a strong neighbor to v. Thus, D cannot be a dominating set which is a contradiction. Hence, D is a maximal independent set.

Theorem 6.6 Every maximal independent set in a bipolar fuzzy graph G = (A, B) is a minimal dominating set. Proof Let S be a maximal independent set in a bipolar fuzzy graph. By Theorem 6.5, S is a dominating set. Suppose S is not a minimal dominating set, then there exists at least one vertex v ∈ S for which S − {v} is a dominating set. But if S − {v} dominates X − {S − {v}}, then at least one vertex in S − {v} must be a strong neighbor to v. This contradicts the fact that S is an independent set of G. Therefore, S must be a minimal dominating set.



6.2 Types of Domination in Bipolar Fuzzy Graphs Table 6.1 Bipolar fuzzy set A on set X A u v p μA μnA

0.3 −0.7

0.2 −0.8

259

w

x

y

z

0.2 −0.7

0.1 −0.9

0.3 −0.6

0.3 −0.5

Theorem 6.7 Let G = (A, B) be a bipolar fuzzy graph, then t B (G) = O(G ∗ ) if and only if every vertex of G has a unique neighbor. Proof If every vertex of G has a unique neighbor, then X is the only total dominating set of G and t B (G) = O(G ∗ ). Conversely, assume that t B (G) = O(G ∗ ). If there exists a vertex u with neighbors v and w, then X − {u} is a total dominating set of G. So t B (G) ≤ O(G ∗ ) which leads to a contradiction, hence every vertex of G has a unique neighbor.

Definition 6.21 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . The equitable neighborhood (EN, in short) of a vertex v ∈ X , denoted by E N (v), is defined as E N (v) = (E N P (v), E N n (v)), where E N p (v) = {u ∈ X | p p p u ∈ N (v), |deg p (u) − deg p (v)| ≤ 1, μ B (uv) = min{μ A (u), μ A (v)} and E N n (v) = n n n {u ∈ X | u ∈ N (v), |deg (u) − deg (v)| ≥ 1, μ B (uv) = max{μnA (u), μnA (v)}. Definition 6.22 The EN degree of a vertex v ∈ X , denoted bydeg E N (v), is defined p p p n as deg E N (v) = (deg E Nn(v), deg E N (v)), where deg E N (v) = u∈E N (v) μ B (vu) and n deg E N (v)= u∈E N (v) μ B (vu). p The minimum EN degree, denoted by δ E N (G), is defined as δ E N (G) = (δ E N (G), p p p n n δ E N (G)), where δ E N (G) = min{deg E N (v) | v ∈ X } and δ E N (G) = min{deg E N (v) | v ∈ X }. The maximum EN degree, denoted by Δ E N (G), is defined as Δ E N (G) = p p p (Δ E N (G), ΔnE N (G)), where Δ E N (G) = max{deg E N (v) | v ∈ X } and ΔnE N (G) = n max{deg E N (v) | v ∈ X }. Example 6.5 Let G = (A, B) be a bipolar fuzzy graph on X = {u, v, w, x, y, z} p p such that A = (μ A , μnA ) is a bipolar fuzzy subset of X , in Table 6.1, and Y = (μ B , μnB ) is a bipolar fuzzy relation in X defined in Table 6.2. The bipolar fuzzy graph is shown in Fig. 6.3. By routine verification, it is clear from Fig. 6.3 that E N (u) = {v, w}, E N (v) = {u, x, y}, E N (w) = {u, x, z}, E N (x) = {w, v}, E N (y) = {v}, and E N (z) = {w}. The EN degrees of vertices are calculated as deg E N (u) = (0.4, −1.4), deg E N (v) = (0.5, −2.1), deg E N (w) = (0.5, −1.9), deg E N (x) = (0.2, −1.5), deg E N (y) = (0.2, −0.6), and deg E N (z) = (0.2, −0.5). The minimum EN degree of a bipolar fuzzy graph G is δ E N (G) = (0.2, −2.1), and the maximum EN degree of a bipolar fuzzy graph G is Δ E N (G) = (0.5, −0.5). Definition 6.23 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . A vertex v ∈ X is called an equitable isolated vertex in G if, for every

260

6 Domination in Bipolar Fuzzy Graphs

Table 6.2 Bipolar fuzzy relation B in X B uv uw wx p μB μnB

0.2 −0.7

0.2 −0.7

0.1 −0.7

u(0.3, −0.7)

xv

vy

wz

vw

0.1 −0.8

0.2 −0.6

0.2 −0.5

0.1 −0.6

v(0.2, −0.8)

(0.2,-0.7)

.6) -0

(0.2,-0.7)

.2, (0

(0.1,-0.8)

) 0.6 1,(0.

(0.2,-0.5) z(0.3, −0.5)

(0.1,-0.7)

w(0.2, −0.7)

x(0.1, −0.9)

y(0.3, −0.6)

Fig. 6.3 EN degree of vertices in G y(0.7, −0.3)

u(0.8, −0.2)

x(0.3, −0.2)

(0.2,-0.1)

-0. 1) (0. 2,

(0. 7,

2) -0.

-0. 2)

6, (0. (0.6,-0.2)

w(0.6, −0.2)

v(0.7, −0.2)

Fig. 6.4 Equitable isolated vertex

p

p

p

u ∈ X , |deg p (u) − deg p (v)| > 1, μ B (uv) < min{μ A (u), μ A (v)}, and |deg n (u) − deg n (v)| < 1, μnB (uv) > max{μnA (u), μnA (v)}, i.e., E N (v) = ∅. Example 6.6 Consider a bipolar fuzzy graph G = (A, B) on X = {u, v, w, x, y}, as shown in Fig. 6.4. From Fig. 6.4, it is clear that deg(u) = (1.5, −0.5), deg(y) = (0.2, −0.1), deg(w) = (1.4, −0.5), and deg(x) = (0.2, −0.1). Since |deg p (u)− deg p (y)| > 1, p p p p p μ B (uy) < min{μ A (u), μ A (y)} and |deg n (u)− deg n (y)| < 1, μ B (uy) > max{μ A (u), p p p p p μ A (y)}, i.e., E N (y) = ∅. Also, |deg (w) − deg (x)| > 1, μ B (wx) < min{μ A (w), p μ A (x)} and |deg n (w)− deg n (x)| < 1, μnB (wx) > max{μnA (w), μnA (x)}, i.e., E N (x) = ∅. Therefore, y and x are equitable isolated vertices in G. Definition 6.24 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . A subset S ⊆ X is called an equitable dominating (ED) set of G if for every vertex v ∈ X − S, there exists a vertex u ∈ X such that uv ∈ E, |deg p (u) − deg p (v)| ≤ p p p 1, μ B (uv) = min{μ A (u), μ A (v)}, and |deg n (u) − deg n (v)| ≥ 1, μnB (uv) = max{μnA

6.2 Types of Domination in Bipolar Fuzzy Graphs (0.1,-0.6)

u(0.4, −0.6)

(0 .3,

(0.2,-0.6)

x(0.1, −0.7)

261

v(0.2, −0.8)

-0 .5)

(0.1,-0.1)

w(0.3, −0.5)

Fig. 6.5 Equitable dominating set of G

-0 .6) (0 .3,

.7) -0

v(0.4, −0.6)

.2, (0

(0.2,-0.1)

(0.3,-0.5)

u(0.3, −0.7)

Fig. 6.6 Degree equitable bipolar fuzzy graph

(0.1,-0.1)

w(0.5, −0.5)

x(0.2, −0.8)

(u), μnA (v)}. The ED (equitable domination) number of G, denoted by γe (G), is defined as the minimum cardinality of an ED set S. Definition 6.25 An ED subset S ⊆ X of a bipolar fuzzy graph G is called a minimal ED set of G if for every vertex v ∈ S, the set S − {v} is not an ED set, i.e., no proper subset of S is an ED set of G. Example 6.7 Consider a bipolar fuzzy graph G = (A, B) on X = {u, v, w, x}, as shown in Fig. 6.5. By Definition 6.24, it is easy to see from Fig. 6.5 that the minimal ED set of a bipolar fuzzy graph G is S = {u}. The ED number of G is γe (G) = 0.4. Definition 6.26 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . G is called a degree equitable if for every vertex v ∈ X, there exists a vertex u ∈ X p p p such that uv ∈ supp(B), |deg p (u) − deg p (v)| ≤ 1, μ B (uv) = min{μ A (u), μ A (v)}, n n n n n and |deg (u) − deg (v)| ≥ 1, μ B (uv) = max{μ A (u), μ A (v)}. Example 6.8 Consider a bipolar fuzzy graph G = (A, B) on X = {u, v, w, x}, as shown in Fig. 6.6. Routine computations show that G is a degree equitable bipolar fuzzy graph.

262

6 Domination in Bipolar Fuzzy Graphs u(0.2, −0.5)

z(0.8, −0.2)

) 0.3 3,-

(0.3,-0.1)

8, (0.

1) -0.

(0 .8,

(0.8,-0.1)

(0.1,-0.4)

-0. 3)

y(0.5, −0.3) . (0

w(0.3, −0.4)

v(0.8, −0.1)

(0.2,-0.1) (0. 2,

-0 .1)

x(0.9, −0.1)

Fig. 6.7 EI set of G

Definition 6.27 A subset I ⊆ X is called an equitable independent (EI) set of a p bipolar fuzzy graph G = (A, B) if |deg p (u) − deg p (v)| > 1, μ B (uv) p p n n n < min{μ A (u), μ A (v)}, and |deg (u) − deg (v)| < 1, μ B (uv) > max{μnA (u), μnA (v)}, for all u, v ∈ I. The EI (equitable independence) number of G, denoted by γie (G), is defined as the minimum cardinality of an EI set of G. Definition 6.28 An EI set I is called a maximal EI set of G if for every vertex u ∈ X − I , the set I ∪ {u} is not an EI set. Example 6.9 Consider a bipolar fuzzy graph G = (A, B) on X = {u, v, w, x, y, z} as shown in Fig. 6.7. By routine verifications, it is easy to see from Fig. 6.7 that S1 = {u, x} and S2 = {v, w} are maximal EI sets of G. The EI number of G is γie (G) = 2. Definition 6.29 Let G = (A, B) be a bipolar fuzzy graph on X . For any two verp p p tices u, v ∈ X , u strongly dominates v in G if μ B (uv) = min{μ A (u), μ A (v)}, p p μnB (uv) = max{μnA (u), μnA (v)}, dG (u) ≥ dG (v), and dGn (u) ≤ dGn (v). Similarly, u p p p weakly dominates v if μ B (uv) = min{μ A (u), μ A (v)}, μnB (uv) = max{μnA (u), p p μnA (v)}, dG (v) ≥ dG (u), and dGn (v) ≤ dGn (u). Definition 6.30 An ED set S ⊆ X is called a weak (strong) ED (equitable dominating) set of G if for every vertex v ∈ X − S, there is at least one vertex u ∈ S such that u weakly (strongly) dominates v. The weak (strong) ED (equitable domination) number of G, denoted by γwe (G) (γse (G)), is defined as the minimum cardinality of a weak (strong) ED set of G. Example 6.10 Consider the example of a bipolar fuzzy graph G = (A, B) as shown in Fig. 6.8. By routine computations, it is easy to see from Fig. 6.8 that the strong ED set of G is S = {u, v}. The strong ED number of G is γse (G) = 2. Definition 6.31 Let G = (A, B) be a connected bipolar fuzzy graph on X . A connected ED (equitable dominating) set is a subset X  ⊆ X of G if it satisfies the following conditions.

6.2 Types of Domination in Bipolar Fuzzy Graphs Fig. 6.8 Strong equitable dominating set of G

263

z(0.2, −0.8) (0. 2,

-0. 7)

u(0.3, −0.7)

x(0.2, −0.8)

8) ,-0. (0.1

(0.1,-0.2)

(0.2,-0.7)

(0.2,-0.7)

(0.2,-0.5) v(0.2, −0.8)

w(0.4, −0.5)

y(0.1, −0.8)

1. For every v ∈ X − X  , there exists a vertex u ∈ X  such that vu ∈ E, |deg p (u) − p p p deg p (v)| ≤ 1, μ B (uv) = min{μ A (u), μ A (v)} and |deg n (u) − deg n (v)| ≥ 1, μnB (uv) = max{μnA (u), μnA (v)}. 2. The subgraph H = (A , B  ) of G = (A, B) induced by X  is connected. Theorem 6.8 Let G = (A, B) be a bipolar fuzzy graph on X . An ED set D of G is called a minimal ED set if and only if for each v ∈ D, one of the following conditions is satisfied. (i) v is an isolated vertex in D, (ii) E N (v) ∩ (X − D) = ∅, (iii) there exists a vertex u ∈ X − D such that E N (u) ∩ D = {v}. Proof Let G = (A, B) be a bipolar fuzzy graph with minimal ED set D, then for each vertex v ∈ D, the set D  = D − {v} is not an ED set. Thus, there is at least one vertex u in X − D  such that u is not dominated by any vertex in D  . There arise two cases. If u = v, then v is an isolated vertex in D, that is, v is not adjacent to any vertex p w ∈ D such that μ B (vw) = 0 = μnB (vw). So, E N (v) ∩ (X − D) = ∅, that is, each vertex in D has a neighbor in X − S. If u = v, i.e., u ∈ X − D, then u is dominated by some vertex of D but not dominated by any vertex in D  . Thus, u is adjacent only to one vertex v ∈ D, hence E N (u) ∩ D = {v}. Conversely, assume that D is an ED set of a bipolar fuzzy graph G and for each vertex v ∈ D, one of the given conditions is satisfied. Suppose that D is not a minimal ED set, then clearly there exists a vertex v ∈ D such that D − {v} is an ED set of G. Hence, v is adjacent to at least one vertex of set D − {v}, i.e., v is not an isolated vertex in D, so condition (i) is false. Also, if we take D  = D − {v} as an ED set of G, then every vertex of X − D  is adjacent to at least one vertex in D  . Thus, conditions (ii) and (iii) are also false which is a contradiction.



264

6 Domination in Bipolar Fuzzy Graphs

Theorem 6.9 Let G = (A, B) be a bipolar fuzzy graph with order O(G), then 1. γe (G) ≤ γse (G) ≤ O(G ∗ ) − Δ E N (G), p 2. γe (G) ≤ γwe (G) ≤ O(G ∗ ) − δ E N (G). p

Proof As, by definition, every weak (strong) ED set of a bipolar fuzzy graph G is an ED set of G, γe (G) ≤ γse (G) and γe (G) ≤ γwe (G). Let u and v be two arbitrary p p p p vertices of G. If deg E N (v) = Δ E N (G) and deg E N (u) = δ E N (G), then X − E N (v) is a strong ED set of G and X − E N (u) is a weak ED set of G. Thus, γse (G) ≤ p p |X − E N (v)| and γwe (G) ≤ |X − E N (u)|, i.e., γse (G) ≤ O(G ∗ ) − Δ E N (G) and p

γwe (G) ≤ O(G ∗ ) − δ E N (G). Theorem 6.10 Let G = (A, B) be a bipolar fuzzy graph without isolated vertices and S be a minimal ED set of G, then X − S is an ED set of G. Proof Let G be a bipolar fuzzy graph with minimal ED set S, then for every vertex v ∈ S, there is at least one vertex u ∈ X − S such that |deg p (u) − deg p (v)| ≤ p p p 1, μ B (uv) = min{μ A (u), μ A (v)} and |deg n (u) − deg n (v)| ≥ 1, μnB (uv) = max{μnA (u), μnA (v)}. Thus, X − S dominates every element of S. Hence, X − S is an ED set of G.

Theorem 6.11 If H is a spanning subgraph of a bipolar fuzzy graph G, then γe (G) ≥ γe (H ). Theorem 6.12 Let G be a bipolar fuzzy graph with EI dominating set I , then I is both a minimal ED set and a maximal EI set of G. Conversely, any maximal EI set I of a bipolar fuzzy graph G is an EI dominating set of G. Proof Let G be a bipolar fuzzy graph with EI dominating set M, then for every vertex v ∈ X − M, the set M ∪ {v} is not an EI set and the set M − {v} is not an ED set of G. Hence, M is both a minimal ED set and a maximal EI set of G. Conversely, suppose that M is a maximal EI set of G, then for every vertex v ∈ X − M, the set M ∪ {v} is not an EI set of G. Thus, the set M dominates every vertex u ∈ X − M and so M is an ED set of G. Hence, M is an EI dominating set of G.

Theorem 6.13 A subset I ⊆ X is an EI set and an ED set of a bipolar fuzzy graph G if and only if I is a maximal EI set of G. Proof Suppose that M is both an ED set and an EI set of a bipolar fuzzy graph G. Assume that M is not a maximal EI set of G, then clearly there exists a vertex u ∈ X − M such that M ∪ {u} is an EI set, that is, u is not dominated by any vertex v ∈ M showing that M is not an ED set of G, a contradiction, hence M is a maximal EI set of G. Conversely, let M be a maximal EI set of G, then for every vertex u ∈ X − M, the set M ∪ {u} is not an EI set of G. Thus, the set M dominates every vertex u ∈ X − M, that is, M is an ED set of G. Hence, M is both an ED set and an EI set and an ED set of G.



6.2 Types of Domination in Bipolar Fuzzy Graphs (0.2,-0.3)

(0 .1,

5) -0.

7) ,-0. (0.2 (0.2,-0.2)

u(0.3, −0.6)

(0.2,-0.6)

w(0.2, −0.8)

x(0.3, −0.7)

.2) (0.3,-0

2, (0.

-0 .2)

v(0.4, −0.5)

265

y(0.6, −0.2)

Fig. 6.9 Total ED set of G

Definition 6.32 A total ED set of a bipolar fuzzy graph G = (A, B) is a subset S ⊆ X if for every vertex v ∈ X , there is at least one vertex u ∈ S such that uv ∈ p p p E(G), |deg p (u) − deg p (v)| ≤ 1, μ B (uv) = min{μ A (u), μ A (v)}, and |deg n (u) − n n n n deg (v)| ≥ 1, μ B (uv) = max{μ A (u), μ A (v)}. The total ED (equitable domination) number of G, denoted by γte (G), is defined as the minimum cardinality of a total ED set S. Definition 6.33 A total ED set S of a bipolar fuzzy graph G is called a minimal total ED set if for every vertex v ∈ S, the set S − {v} is not a total ED set, i.e., no proper subset of S is a total ED set of G. Example 6.11 Consider a bipolar fuzzy graph G = (A, B) on a crisp set X = {u, v, w, x, y}, as shown in Fig. 6.9. By Definition 6.32, it is easy to see from Fig. 6.9 that S = {w, y} is a total ED set of G. Remark 6.2 If G has an equitable isolated vertex, then γte (G) → ∞. Theorem 6.14 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . A total ED set S of G is called a minimal total ED set if and only if for each v ∈ S, one of the following conditions is satisfied. 1. There exists a vertex u ∈ X − S such that N (u) ∪ S = {v}. 2. < S − {v} > contains an isolated vertex. Theorem 6.15 Let G be a bipolar fuzzy graph with no isolated vertices, then γt (G) ≤ γte (G). Proof As every total ED set of a bipolar fuzzy graph G = (A, B) is also a total dominating set, γt (G) ≤ γte (G).

Definition 6.34 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X and k ≥ 1 be an integer. A subset S ⊆ X is called a k-dominating set of G if for every vertex u ∈ X − S, there exists a u − v bipolar fuzzy path which contains at least k effective edges for v ∈ S. The k-domination number of G, denoted by γk (G), is defined as the minimum cardinality among all k-dominating sets in G.

266

6 Domination in Bipolar Fuzzy Graphs

u(0.4, −0.5) (0. 4,

-0. 2)

w(0.7,

0.2)

x(0.2, −0.8)

(0.1,-0.3)

) 0.3 4,(0.

(0.2,-0.3)

(0.3,-0.1)

v(0.5, −0.3)

(0.3,-0.2)

(0. 2,0.5 )

z(0.5, −0.5)

4) ,-0. (0.3 y(0.3, 0.4)

Fig. 6.10 Total 2-dominating set of G

Definition 6.35 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X and let k ≥ 1 be an integer. A subset S ⊆ X is called a total k-dominating set of G if for every vertex u ∈ X , there exists a u − v bipolar fuzzy path which contains at least k effective edges for v ∈ S. The total k-domination number of G, denoted by γtk (G), is defined as the minimum cardinality among all total k-dominating sets in G. Example 6.12 Consider an example of a total 2-dominating set of a bipolar fuzzy graph G shown in Fig. 6.10. By Definition 6.35, it is clear from Fig. 6.10 that S = {u, z} is a minimal total 2-dominating set of a bipolar fuzzy graph G. The total 2-domination number of G is γk (G) = 2. Definition 6.36 Let G = (A, B) be a bipolar fuzzy graph on X . A subset S ⊆ X is called a k-independent dominating set of G if for every vertex u ∈ X − S, there exists a u − v bipolar fuzzy path which contains at least k + 1 effective edges for v ∈ S. The k-independent domination number of G, denoted by γik (G), is defined as the minimum cardinality among all k-independent dominating sets in G. Definition 6.37 Let G = (A, B) be a bipolar fuzzy graph on X . A subset S ⊆ X is called a restrained dominating (RD) set of G if every vertex in X − S dominates a vertex in S and also a vertex in X − S. The RD (restrained domination) number of G, denoted by γr (G), is defined as the minimum cardinality among all RD sets in G. Example 6.13 Let G = (A, B) be a bipolar fuzzy graph on X = {u, v, w, x} as shown in Fig. 6.11. From Fig. 6.11, it is clear that S = {w, x} is an RD set of G. The RD number of G is γr (G) = 2. Definition 6.38 Let G = (A, B) be a bipolar fuzzy graph on X . A subset S ⊆ X which is an RD set of both bipolar fuzzy graph G and its complement G c is called a global RD (restrained dominating) set of G. The minimum cardinality among all global RD sets in G is called a global RD (restrained domination) number of G, denoted by γgr (G).

6.2 Types of Domination in Bipolar Fuzzy Graphs u(0.3, −0.7)

267 v(0.4, −0.6)

w(0.5,

(0.3,-0.5)

(0.3,-0.4)

(0.3,-0.6)

(0.2,-0.3)

0.4)

x(0.3,

0.5)

Fig. 6.11 Restrained dominating set of G u(0.5, −0.3)

G

y(0.6, −0.1)

0.5)

(0.3,-0.2) ) ,-0.1 (0.3

w(0.3,

(0.5,-0.1)

w(0.3, −0.5)

(0. 3,0.2 )

x(0.5, −0.2)

-0.3 )

) .1

,-0

.4 (0

.2) -0

y(0.6, −0.1)

2)

.5, (0

(0.5,-0.1)

,-0. (0.4

(0.4,-0.2)

(0.4,-0.3)

v(0.4, −0.6)

(0.3 ,

z (0.7, −0.2)

v(0.4, −0.6)

5) ,-0. (0.3

1) ,-0. (0.6

z (0.7, −0.2) (0.5 ,-0.2 )

u(0.5, −0.3)

(0.5 ,-0. 2)

x(0.5, −0.2)

Gc

Fig. 6.12 Global RD set of G

Example 6.14 Consider a bipolar fuzzy graph G and its complement G c on X = {u, v, w, x, y, z} as shown in Fig. 6.12. By direct calculations, it is easy to see that S1 = {v, y}, S2 = {u, x}, and S3 = {w, z} are global RD sets of G. The global RD number of G is γgr (G) = 2. Definition 6.39 An edge x y of a bipolar fuzzy graph G = (A, B) is called an effecp p tive edge if B(x y) = (μ A (x) ∧ μ A (y), μnA (x) ∨ μnA (y)). Otherwise, it is called a noneffective edge. Theorem 6.16 Let G = (A, B) be a complete bipolar fuzzy graph on X , then γr (G) = |X |. p

Proof Suppose that G = (A, B) is a complete bipolar fuzzy graph. Then μ B (uv) = p p min(μ A (u), μ A (v)) and μnB (uv) = max(μnA (u), μnA (v)), i.e., each edge in G is an effective edge. Let S be the minimal RD set of G. Then every vertex in X − S dominates a vertex in S and also a vertex in X − S. Thus, every vertex dominates all

other vertices. Hence, γr (G) = |X |. Note 6.1 For a complete bipartite bipolar fuzzy graph, the RD number is γr (G) = |X 1 | + |X 2 |.

268

6 Domination in Bipolar Fuzzy Graphs

Theorem 6.17 Let G = (A, B) be a bipolar fuzzy graph on X with |X | = n, then γgr (G) = n if and only if G = K n or K nc . Proof Let G = K n be a complete bipolar fuzzy graph on X , then for each edge p p p uv ∈ E, μ B (uv) = min{μ A (u), μ A (v)} and μnB (uv) = max{μnA (u), μnA (v)}. Thus, c the complement K n , having no edges, is a null bipolar fuzzy graph. Each vertex in the null bipolar fuzzy graph K nc is an isolated vertex. Therefore, RD set of K n and K nc is the vertex set X . Hence, global RD set of G = K n is equal to set X , i.e., γgr (G) = O(G ∗ ). If G = K nc , then the RD set of K n is the singleton set {v}, for all v ∈ X , but it is not the RD set of K nc . Since every vertex in K nc is an isolated vertex, the only RD set of both K n and K nc is the vertex set X. Conversely, assume that γgr (G) = n = O(G ∗ ) but G = K n or G = K nc , then there exists a vertex v which is not dominated by u. Thus, X − {v} is a global RD

set containing n − 1 vertices. Hence, γgr (G) = O(G ∗ ) which is a contradiction. Theorem 6.18 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs with X 1 ∩ X 2 = ∅, then the strong product G = G 1  G 2 remains connected even after removal of all noneffective edges in it. Proof Suppose that G = G 1  G 2 = (A, B) is a strong product of two bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ). Let e = (u, u 1 )(v, v1 ) be a p p p noneffective edge in G, i.e., μ B ((u, u 1 )(v, v1 )) < min{μ A (u, u 1 ), μ A (v, v1 )} and μnB ((u, u 1 )(v, v1 )) > max{μnA (u, u 1 ), μnA (v, v1 )}. Let G  = G − e and assume that G  is disconnected. The edge e disconnects the bipolar fuzzy graph into more than one components. Thus, there does not exist any path between (u, u 1 ) and p (v, v1 ) except the edge e = (u, u 1 )(v, v1 ) in G  . It implies that μ B ((u, u 1 )(v, v1 )) = p p min{μ A (u, u 1 ), μ A (v, v1 )} and μnB ((u, u 1 )(v, v1 )) = max{μnA (u, u 1 ), μnA (v, v1 )}

which is a contradiction. Hence, G  = G − e is connected. Theorem 6.19 The strong product G 1  G 2 of two complete bipolar fuzzy graphs is a complete bipolar fuzzy graph. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs such that E is the set of edges in G 1  G 2 . If (u, u 1 )(u, v1 ) ∈ E, then since G 2 is complete, p

p

p

μ B1 B2 ((u, u 1 )(u, v1 )) = min{μ X 1 (u), μ B2 (u 1 v1 )} p

p

p

= min{μ X 1 (u), min{μ A2 (u 1 ), μ A2 (v1 )} p

p

= min{(μ B1 B2 )(u, u 1 ), (μ B1 B2 )(u, v1 )}. If (u, u 1 )(v, u 1 ) ∈ E, then since G 1 is complete, p

p

p

μ B1 B2 ((u, u 1 )(v, u 1 )) = min{μ B1 (u, v), μ X 2 (u 1 )} p

p

p

= min{min{μ X 1 (u), μ A1 (v)}, μ A2 (u 1 )} p

p

= min{(μ B1 B2 )(u, u 1 ), (μ B1 B2 )(v, u 1 )}.

6.2 Types of Domination in Bipolar Fuzzy Graphs

w1 (0.5, −0.3)

(0.

) 0.1 2,(0.

) 0.1 2,-

(0. 2,0

.1)

.3) ,-0 ((v, v1 )(0.4, −0.3))

4 (0.

(0. 4,

(u, w1 ), (0.5, −0.3)

-0. 3)

. (0

) 0.1 2,-

(v, w1 )(0.4, −0.3)

G2

G1

(0. 2,0.1 )

(w , v1 )(0.3, −0.3)

.2)

(u, v1 )(0.4, −0.3)

v(0.4, −0.5)

(w, u1 )(0.3, −0.5)

(0. 2,0

(0.4,-0.3)

G1

(v, u1 ), (0.3, −0.5)

(u, u1 )(0.2, −0.2)

(0.2,-0.2)

(0.2,-0.1)

w(0.3, −0.5)

u1 (0.3, −0.5)

v1 (0.4, −0.3)

(0.4,-0.3)

u(0.6, −0.3)

269

(w, w1 )(0.3, −0.3)

G2

Fig. 6.13 Direct product of G 1 and G 2

If (u, u 1 )(v, v1 ) ∈ E, then since G 1 and G 2 are complete, p

p

p

μ B1 B2 ((u, u 1 )(v, v1 )) = min{μ B1 (u, v), μ B2 (u 1 v1 )} p

p

p

p

= min{min{μ X 1 (u), μ X 1 (v)}, min{μ X 2 (u 1 ), μ X 2 (v1 )} p

p

= min{(μ B1 B2 )(u, u 1 ), (μ B1 B2 )(v, v1 )}.

Similarly, it is so for other cases. Hence, G 1  G 2 is a complete bipolar fuzzy graph. Theorem 6.20 If a vertex u dominates a vertex v in G 1 = (A1 , B1 ) and the vertex u 1 dominates a vertex v1 in G 2 = (A2 , B2 ), then the vertex (u, v) does not dominate the vertex (u 1 , v1 ) in G 1 G 2 = (A, B). Proof Assume that u dominates v in G 1 . Then there exists an effective edge uv p p p in G 1 , i.e., μ B1 (uv) = min{μ A1 (u), μ A1 (v)} and μnB1 (uv) = max{μnA1 (u), μnA1 (v)}. p Similarly, assume that a vertex u 1 dominates a vertex v1 in G 2 such that μ B2 (u 1 v1 ) = p p min{μ A2 (u 1 ), μ A2 (v1 )} and μnB2 (u 1 v1 ) = max{μnA2 (u 1 ), μnA2 (v1 )}. Now by Definition 1.18, there does not exist any edge between the vertices (u, u 1 ) and (v, v1 ) in G 1 G 2 , p i.e., μ B ((u, u 1 )(v, v1 )) = 0 and μnB ((u, u 1 )(v, v1 )) = 0. Thus, (u, u 1 ) does not dom inate (v, v1 ) in G 1 G 2 . Note 6.2 If a vertex u dominates a vertex v in G 1 and a vertex u 1 dominates a vertex v1 in G 2 , then the vertex (u, u 1 ) dominates the vertex (v, v1 ) in G 1 × G 2 . It can be seen in Fig. 6.13 Example 6.15 Consider a bipolar fuzzy graph as shown in Fig. 6.13. From Fig. 6.13, it is clear that the vertex u dominates v in G 1 and v1 dominates w1 in G 2 . Also, the vertex (u, v1 ) dominates the vertex (v, w1 ) in G 1 × G 2 .

270

6 Domination in Bipolar Fuzzy Graphs

Theorem 6.21 Let G 1 and G 2 be bipolar fuzzy graphs on a non-empty sets X 1 and X 2 , respectively, then γe (G 1 × G 2 ) ≤ min(|S1 × X 1 |, |S2 × X 2 |), where S1 and S2 are ED sets of G 1 and G 2 , respectively. Proof Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs with direct product G 1 × G 2 = (A, B). Suppose S1 and S2 are ED sets of G 1 and G 2 , respectively, with minimum cardinality. Then for every vertex v ∈ X 1 − S1 , there exists a p p p vertex u ∈ S1 such that |deg p (v) − deg p (u)| ≤ 1, μ B1 (uv) = min{μ A1 (u), μ A1 (v)} n n n n n and |deg (v) − deg (u)| ≥ 1, μ B1 (uv) = max{μ A1 (u), μ A1 (v)}. Similarly, for every vertex x ∈ X 2 − S2 , there exists a vertex w ∈ S2 such that |deg p (w) − deg p (x)| ≤ 1, p p p μ B2 (wx) = min{μ A2 (w), μ A2 (x)} and |deg n (w) − deg n (x)| ≥ 1, μnB2 (wx) = n n max{μ A2 (w), μ A2 (x)}. That is, u dominates v in G 1 and w dominates x in G 2 . Thus, by Note 6.2, the vertex (u, w) dominates the vertex (v, x) in G 1 × G 2 . So p p p |deg p ((u, w)) − deg p ((v, x))| ≤ 1, μ B ((u, w)(v, x)) = min{μ A (u, w), μ A (v, x)} n n n n and |deg ((u, w)) − deg ((v, x))| ≥ 1, μ B ((u, w)(v, x)) = max{μ A (u, w), μnA (v, x)}. Hence, γe (G 1 × G 2 ) ≤ min(|S1 × X 1 |, |S2 × X 2 |).

Theorem 6.22 Let S1 and S2 be k-dominating sets of connected bipolar fuzzy graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ), respectively, then 1. G 1 G 2 is connected. 2. If S1 is a connected k-dominating set of G 1 , then S1 × X 2 is a connected kdominating set of G 1 G 2 . 3. If S2 is a connected k-dominating set of G 2 , then X 1 × S2 is a connected kdominating set of G 1 G 2 . Proof To prove that G 1 G 2 is connected, consider any two arbitrary distinct vertices (u, w), (v, x) of X 1 × X 2 . Then by Definition 1.18, there exists a bipolar fuzzy path between these two vertices in the following cases. 1. If u = v, then, since G 2 is a connected bipolar fuzzy graph, there exists a p bipolar fuzzy path P : w, w1 , w2 , . . . , x such that μ B2 (r s) > 0 and μnB2 (r s) < 0 p for any two vertices r, s of bipolar fuzzy path P. Thus, μ B ((u, w)(u, x)) = p p min{μ A1 (u), μ B2 (wx)} and μnB ((u, w)(u, x)) = max{μnA1 (u), μnB2 (wx)}. Therefore, P  : (u, w), (u, w1 ), (u, w2 ), . . . , (u, x) is the bipolar fuzzy path between (u, w), (v, x) in G 1 G 2 . 2. If w = x, then, since G 1 is a connected bipolar fuzzy graph, there exists a p bipolar fuzzy path Q : u, u 1 , u 2 , . . . , v such that μ B1 (uv) > 0 and μnB1 (uv) < 0 p for any two vertices r, s of bipolar fuzzy path Q. Thus, μ B ((u, w)(v, w)) = p p min{μ B1 (u, v), μ A2 (w)} and μnB ((u, w)(v, w)) = max{μnB1 (u, v), μnA2 (w)}. Therefore, Q  : (u, w), (u 1 , w), (u 2 , w), . . . , (v, w) is the bipolar fuzzy path between (u, w), (v, x) in G 1 G 2 . 3. If u = v, w = x, then by case 1, there exists a bipolar fuzzy path between the vertices (u, w) and (u, x) in G 1 G 2 . Also by case 2, there exists a bipolar fuzzy

6.2 Types of Domination in Bipolar Fuzzy Graphs

271

path between the vertices (u, w) and (v, w) in G 1 G 2 . Thus, the union of these two disjoint bipolar fuzzy paths is a bipolar fuzzy path between the vertices (u, w) and (v, x) in G 1 G 2 . Now, if S1 and S2 are k-dominating sets of G 1 and G 2 , respectively, then γk (G 1 G 2 ) = min{|S1 × X 2 |, |S1 × X 2 |}. Hence, X 1 × S2 and S1 × X 2 are k-dominating sets of

G 1 G 2 and the connectivity can be proved similarly. Theorem 6.23 Let G 1 and G 2 be bipolar fuzzy graphs on non-empty sets X 1 and X 2 , respectively. Let S1 and S2 be k-dominating sets of G 1 and G 2 , then (1) S1 × X 2 is an independent k-dominating set of G 1 G 2 if and only if S1 is kindependent and, p

p

1. μ B1 (uv) < μ A2 (w) and μnB1 (uv) > μnA2 (w), for u, v ∈ S1 , w ∈ X 2 , p p 2. μ B2 (wx) < μ A1 (u) and μnB2 (wx) > μnA1 (u), for u ∈ S1 , w, x ∈ X 2 , p p p 3. μ B2 (wx) < min{μ A2 (w), μ A2 (x)} and μnB2 (wx) > max{μnA2 (w), μnA2 (x)}, for w, x ∈ X 2 . (2) X 1 × S2 is an independent k-dominating set of G 1 G 2 if and only if S2 is kindependent and, p

p

1. μ B1 (uv) < μ A2 (w) and μnB1 (uv) > μnA2 (w), for u, v ∈ X 1 , w ∈ S2 , p p 2. μ B2 (wx) < μ A1 (u) and μnB2 (wx) > μnA1 (u), for u ∈ X 1 , w, x ∈ S2 , p p p 3. μ B1 (uv) < min{μ A1 (u), μ A1 (v)} and μnB1 (uv) > max{μnA1 (u), μnA1 (v)}, for u, v ∈ X 1 . Proof To prove that every two distinct vertices (u, w), (v, x) in S1 × X 2 are not adjacent. If u = v, then p

p

p

μ B ((u, w)(u, x)) = min{μ A1 (u), μ B2 (wx)} p

p

p

< min{μ A1 (u), min{μ A2 (w), μ A2 (x)}} p

p

< min{μ X (u, w), μ X (u, x)}, μnB ((u, w)(u, x)) = max{μnX 1 (u), μnY2 (wx)} > max{μnX 1 (u), max{μnX 2 (w), μnX 2 (x)}} > max{μnX (u, w), μnX (u, x)}. If w = x, the result is obtained by independence of u, v in S1 . If u = v, w = x, p then μ B ((u, w)(v, x)) = 0 and μnB ((u, w)(v, x)) = 0. Hence, (u, w), (v, x) are not adjacent in G 1 G 2 . Conversely, suppose that (1) and (2) are false. That is, there exist vertices y, z ∈ X 2 p p p such that μ B2 (yz) = min{μ A2 (y), μ A2 (z)} and μnB2 (yz) = max{μnA2 (y), μnA2 (z)}. Let u ∈ S1 , then

272

6 Domination in Bipolar Fuzzy Graphs p

p

p

μ B ((u, y)(u, z)) = min{μ A1 (u), μ B2 (yz)} p

p

p

= min{μ A1 (u), min{μ A2 (y), μ A2 (z)}} p

p

= min{μ X (u, y), μ X (u, z)}, n μ B ((u, y)(u, z)) = max{μnA1 (u), μnB2 (yz)} = max{μnA1 (u), max{μnA2 (y), μnA2 (z)}} = max{μnX (u, y), μnX (u, z)}. p

Thus, S1 × X 2 is not independent. Hence, condition (1) is true, i.e., μ B2 (yz) < p p min{μ A2 (y), μ A2 (z)} and μnB2 (yz) > max{μnA2 (y), μnA2 (z)}. Similarly (2) can be proved.



6.3 Irredundance in Bipolar Fuzzy Graphs In this section, the notions of private neighbor, lower and upper irredundant sets, and irredundance number are discussed in relation with independence and domination numbers of bipolar fuzzy graphs. Definition 6.40 Let G = (A, B) be a bipolar fuzzy graph and S be a set of vertices. A vertex v is said to be a private neighbor of u ∈ S with respect to S if N [v] ∩ S = {u}. Furthermore, a private neighborhood of u ∈ S with respect to S is denoted by a set as P N [u, S] = {v | N [v] ∩ S = {u}}. Also, P N [u, S] = N [u] − N [S − {u}]. Definition 6.41 A vertex u in a subset S ⊆ X is said to be a redundant vertex if P N [u, S] = ∅. Equivalently, u is redundant in S if N [u] ⊆ N [S − {u}]. Otherwise u is said to be an irredundant vertex. Definition 6.42 A subset S ⊆ X is said to be an irredundant set if P N [u, S] = ∅, for every vertex u in S. Definition 6.43 An irredundant set S ⊆ X in a bipolar fuzzy graph G = (A, B) is called a maximal irredundant set if for every vertex v ∈ X − S, the set S ∪ {v} is not an irredundant set which means that there is at least one vertex w ∈ S ∪ {u} which does not have any private neighbor. Definition 6.44 The maximum cardinality among all maximal irredundant sets is called the upper irredundance number and is denoted by I R B (G). Definition 6.45 The minimum cardinality among all maximal irredundant sets is called the lower irredundance number and is denoted by ir B (G). Example 6.16 Consider a bipolar fuzzy graph G = (A, B) on X = {v1 , v2 , v3 , v4 , v5 , v6 } as shown in Fig. 6.14. Clearly, S1 = {v5 , v6 }, S2 = {v2 , v3 , v5 }, S3 = {v1 , v4 , v6 }, S4 = {v1 , v2 }, and S5 = {v3 , v4 } are maximal irredundant sets, P N [v5 ,

6.3 Irredundance in Bipolar Fuzzy Graphs v1 (0.3, −0.6)

v2 (0.4, −0.5) (0 .2,

−0 .3)

−0 .3)

v6 (0.3, −0.4) (0.3, −0.3)

) 0.3 ,− 2 . (0

(0.3,

0.5)

0.3)

v3 (0.5,

v5 (0.5, −0.3)

v4 (0.4,

(0.3, −0.3)

(0.3, −0.3)

(0.3−, 0.3)

(0 .2,

273

) 0.3 − .2, (0 0.5)

Fig. 6.14 Irredundant sets of G

S1 ] = {v2 , v3 }, P N [v6 , S1 ] = {v1 , v4 }, P N [v2 , S2 ] = {v1 }, P N [v3 , S2 ] = {v4 }, P N [v5 , S2 ] = {v6 }, and so on. The set {v1 , v2 } has minimum cardinality among all maximal irredundant sets and ir B (G) = 2, {v2 , v3 , v5 } has maximum cardinality among all maximal irredundant sets and I R B (G) = 3. Theorem 6.24 Let G = (A, B) be a bipolar fuzzy graph on X , then ir B (G) ≤ O(G ∗ ) − δ(G ∗ ). Proof Let S be an irredundant set in a bipolar fuzzy graph G = (A, B). Let v be a vertex in S which is a strong neighbor to a set of vertices in S with cardinality |k|. Since the neighborhood degree of v in G ∗ is at least δ(G ∗ ), v must be a strong neighbor to a set of vertices in X − S with cardinality δ(G ∗ ) − |k|. Here δ(G ∗ ) − |k| ≤ |X − S|. If |k| = 0, then δ(G ∗ ) ≤ |X − S|. That is, δ(G ∗ ) ≤ O(G ∗ ) − ir B (G) or ir B (G) ≤ O(G ∗ ) − δ(G). If |k| > 0, then each neighbor of v in S must be distinct and has a private neighbor in X − S. Hence, |X − S| ≥ (δ(G ∗ ) − |k|) + |k| = δ(G ∗ ). Thus,

ir B (G) ≤ O(G ∗ ) − δ(G ∗ ). Corollary 6.2 Every maximal independent set in a bipolar fuzzy graph G is a maximal irredundant set, that is, ir B (G) ≤ i B (G) ≤ d B (G) ≤ I R B (G). Theorem 6.25 Let G = (A, B) be a bipolar fuzzy graph on X , then ir B (G) + I R B (G) ≤ 2(O(G ∗ ) − δ(G∗)). The equality holds if and only if O(G ∗ ) − δ(G ∗ ) divides O(G ∗ ). Proof By Theorem 6.24 and Corollary 6.2, ir B (G) ≤ I R B ≤ O(G ∗ ) − δ(G ∗ ) ∗

(6.1) ∗

⇒ ir B (G) + I R B (G) ≤ 2(O(G ) − δ(G )).

274

6 Domination in Bipolar Fuzzy Graphs

If O(G ∗ ) − δ(G ∗ ) divides O(G ∗ ), then there exists a complete multipartite bipolar fuzzy graph, each of whose defining independent sets is of size O(G ∗ ) − δ(G ∗ ). Hence, i B (G) = I R B (G) = O(G ∗ ) − δ(G ∗ ). It follows that ir B (G) + I R B (G) = 2(O(G ∗ ) − δ(G ∗ )). Conversely, if ir B (G) + I R B (G) = 2(O(G ∗ ) − δ(G ∗ )), from Eq. (6.1), i B (G) = I R B (G) = O(G ∗ ) − δ(G ∗ ). It follows that every vertex of G lies in a maximal independent set of cardinality O(G ∗ ) − δ(G ∗ ). Since δ(G ∗ ) is the minimum neighborhood degree, every vertex in such a maximal independent set is a strong neighbor to every vertex outside the set. It follows that the vertex set X can be partitioned into maximal independent sets of size O(G ∗ ) − δ(G ∗ ), and hence O(G ∗ ) − δ(G ∗ )

divides O(G ∗ ). It follows that G is a complete multipartite bipolar fuzzy graph.

6.4 Applications of Domination in Decision-Making Problems In this section, the applications of domination in bipolar fuzzy graphs are discussed in facility location problem, finding the set of representatives, and in transmission tower location problem .

6.4.1 Facility Location Problem Suppose that a provincial government has decided to build some colleges which must serve all the towns in a province. In order to decide the location of a college in a town, it is very important to analyze the educational situation in a town. The educational situation depends on many factors including the availability of skilled teachers, availability of transportation, urbanization, rate of literacy, flow rate, and drop-out rate of students. Now the government wants to locate a college in such a way that every town has a college or is a neighbor of a town which has a college, i.e., the government wants to build a minimum number of colleges to save money. We can present this problem as a bipolar fuzzy graph G = (T, R), where T is a bipolar fuzzy set of towns presenting the literacy rate and drop-out rate as positive and negative membership degrees and R is a bipolar fuzzy set of roads between the towns. The positive and negative membership degrees can be calculated as p

p

p

μ B (uv) ≤ min{μ A (u), μ A (v)}, μnB (uv) ≥ max{μnA (u), μnA (v)}. Consider the example of a bipolar fuzzy graph of various towns as shown in Fig. 6.15. By direct computations, it is easy to check that the minimum dominating set of a bipolar fuzzy graph is S = {T onopah, Beatt y, Oasis}. Thus, the government

6.4 Applications of Domination in Decision-Making Problems

275

(0.6,-0.1)

Manhattan (0.6,-0.4)

-0.2) (0.3,

Dyer (0.5,-0.3)

T onopah (0.8,-0.1)

3, (0.

(0. 2,0.1 ) (0.4,-0.3)

(0.3 ,-0. 2) (0.5,-0.1) (0.3,-0.1)

Beatty (0.8,-0.2)

Lida (0.6,-0.4)

-0. 2)

(0.5,-0.2)

(0.4,-0.2) DeepSprings (0.4,-0.2)

Goldf ield

(0.2,-0.2)

(0.6,-0.2)

(0.5,-0.2)

2) -0.

Oasis (0.7,-0.2)

.1) ,-0 5 . (0

4, (0.

.2) (0.5,-0

SilverP eak (0.5,-0.5)

W armSprings

(0. 7,

2) -0.

1) ,-0. (0.4

(0 .4 ,-0 .1 )

GoldP oint (0.7,-0.3)

Fig. 6.15 Bipolar fuzzy graph of towns

needs to build colleges in three towns to save money and provide college facility to all towns of the province.

6.4.2 Representatives in Youth Development Council A bipolar fuzzy graph can be used to find the set of representatives in a youth development council of a university. Consider a group of students who wants to become a member of the youth development council. Each student has some good and some bad leadership qualities. Bad leadership is characterized by lack of enthusiasm, poor communication skills, incompetency, inflexibility, and poor decision- making skills. A member of the youth development council must be a good listener, honest and fair, good communicator, and approachable. All these qualities are fuzzy in nature. We want to form a council with as a few members as possible. Form a council in such a way that every member not in the council has something in common with the member in the council, that is, finding a minimum dominating set. The method for computing the youth development council is explained in Algorithm 6.4.1.

276

6 Domination in Bipolar Fuzzy Graphs (0.6,-0.1)

U sama (0.6,-0.2)

Aiza (0.7,-0.1)

1) -0.

(0.4 ,-0. 1)

3, (0.

-0. 1)

(0 .4,

T aha (0.8,-0.2)

-0 .1)

(0. 4, (0.4,-0.1)

1) -0.

0,2) (0.5,-

) .1 ,-0 .5 (0

Mif ra (0.5,-0.3)

6, (0.

N adir (0.5,-0.5)

Adan (0.5,-0.3)

(0.3,0.2)

Laiba (0.4,-0.5)

(0.4,-0.2)

(0.3,-0.2)

Rayan (0.6,-0.3)

Moez (0.8,-0.1)

(0. 4,0.2 )

(0.6,-0.1)

2) ,-0. (0.5

Dia (0.6,-0.4)

Fig. 6.16 Bipolar fuzzy graph of representatives

Algorithm 6.4.1 Computing the representative of youth development council. 1. Input the bipolar fuzzy A set of students u 1 , u 2 , . . . , u n . 2. Compute the bipolar fuzzy graph G = (A, B) from given data. p p p 3. Find the vertices u i such that μ B (u i , u j ) = min{μ A (u i ), μ A (u j )} and μnB (u i , u j ) = max{μnA (u i ), μnA (u j )}, i = j. 4. Form the set Si ⊆ X of vertices u i . 5. If ∪ j {u j } = X − Si , then Si is a dominating set otherwise, Si is not a dominating set. 6. Repeat the steps from 3 to 5 and find all the dominating sets Si of X. 7. The decision is Si if Si is a minimal dominating set. Consider the example of a bipolar fuzzy graph as shown in Fig. 6.16. The nodes of the bipolar fuzzy graph represent the students and their leadership qualities in terms of the positive membership degree and its counter property, bad leadership by a negative membership degree, e.g., Usama has 60% leadership qualities and lacks 20% leadership qualities. The edges of a bipolar fuzzy graph represent the common qualities between the students. The positive and negative degree of membership can be interpreted as the percentage of common leadership qualities and the qualities which are not common between the students, e.g., Usama and Aiza have 60% qualities in common but 10% qualities are different. From Fig. 6.16, it is clear that the dominating set of a bipolar fuzzy graph G is S = {T aha, Moez, Ai za, Mi f ra} and it is also the minimal dominating set of G. The domination number of a bipolar fuzzy graph G is γ(G) = 4. Thus, Taha, Aiza, Moez, and Mifra are the members of a youth development council.

6.4 Applications of Domination in Decision-Making Problems (0.5,-0.2)

Morris Town

(0.6,-0.3)

Newark

(0.7,-0.3) (0.4,-0.1)

Washington

(0.6,-0.2)

(0.6,-0.2)

) 0.1 5,(0.

Flemington

(0.7,-0.2)

(0.5,-0.2)

-0. 1)

Trenton

(0.6,-0.2)

(0.6,-0.3)

(0.4,-0.2) Asbury Park

(0.5,-0.4)

3) -0.

Millville

(0.8,-0.1) (0.5,-0.1)

Cape May

(0.5,-0.5)

) 0.2 5,(0.

Camden

3, (0.

) ,-0.1 (0.3

0.2) (0.4,-

(0.8,-0.2)

,-0. (0.6

2)

(0.6,-0.4)

(0.4,-0.1)

Wrights Town

0.2) (0.5,-

(0.5,-0.1)

(0. 4,

Wood Bridge

2) -0.

(0.6,-0.2)

-0 .1)

4, (0.

Phillipsburg

(0.6,-0.2)

.1) -0

) .2 ,-0 .5 (0

.4, (0

(0.6 ,-0. 2)

(0.5,-0.4)

(0 .4,

.1) (0.5,-0

) 0.2 4,(0.

277

(0.7,0.2 ) Atlantic City

(0.7,-0.3)

(0.5,-0.2)

Fig. 6.17 Bipolar fuzzy graph of cities

6.4.3 Transmission Tower Location Problem Consider the problem of locating transmission towers in selected cities so that the transmission can be broadcasted to all cities. The selection of the location of a transmission tower depends on many factors including the availability of electrical power, telecommunication network connectivity, etc. Also, for better transmission signals, it is necessary to avoid electromagnetic interferences. The strength of the transmission signal decreases as the distance between two towers increases. Since each transmission tower has a limited transmission range, there is a need to locate several transmission towers so that the transmission can reach to each city. But transmission towers are costly. Therefore, we want to locate as few transmission towers as possible. This problem can be modeled by using a bipolar fuzzy graph, and an example is given in Fig. 6.17. The vertices of the bipolar fuzzy graph represent the cities. The positive and the negative membership degrees of the vertices can be interpreted as the availability of electrical power and electromagnetic interferences, respectively, e.g., Washington has 50% availability of electrical power and 40% electromagnetic interferences. The edge between two cities indicates that a transmission from a tower at one of these cities reaches the other. The positive and the negative membership degree of the edges represent the transmission strength and transmission range, respectively. The problem is to find the minimum number of towers so that each city either has a tower or is a

278

6 Domination in Bipolar Fuzzy Graphs

neighbor of a city that has a tower. This is equivalent to finding a minimum dominating set of locations that covers all the cities. By routine calculations, it is easy to see that the minimum dominating set is S = {Flemington, N ewar k, Millville, Camden}. Thus, it is sufficient to place the transmission towers at these four cities for better transmission coverage and minimum cost. The method for finding the desired set of cities is presented in Algorithm 6.4.2. Algorithm 6.4.2 Transmission tower location problem 1 Begin 2 Input the set of locations(vertices) X = {a1 , a2 , . . . , an } p 3 Input the positive {u i : i = 1, 2, 3, . . . , n} and negative {u in : i = 1, 2, 3, . . . , n} membership values of vertices p 4 Input the adjacency matrices A p = [u i j ]n×n and An = [u inj ]n×n of positive and negative membership values 5 Take S = ∅ and T = ∅ 6 do i from 1 to n 7 do j from 1 to n p p p 8 if(i = j and u i j = min{u i , u j }, u inj = max{u in , u nj })then 9 ai ∈ S and a j ∈ T 10 end if 11 if(S ∪ T = X )then 12 Print that S is a dominating set 13 goto step 19 14 else if (i = n)then 15 Print that there is no dominating set 16 end if 17 end do 18 end do 19 Stop

6.5 Conclusions The term domination has a wide range of applications in graph theory for the analysis of bipolar fuzzy information. In this chapter, we have discussed different concepts of dominating and independent sets in bipolar fuzzy graphs and used these concepts to solve real-world problems for finding the minimum number of locations in any bipolar fuzzy network and a team of representatives for any organization. We have presented certain properties concerning the relationship of domination, total domination, and independence numbers in complete bipolar fuzzy graphs, complete bipartite bipolar fuzzy graphs, and products of bipolar fuzzy graphs. We have studied the notions of EN degree, equitable isolated vertices, private neighbors, irredundant

6.5 Conclusions

279

sets, and irredundance number in bipolar fuzzy graphs and elaborated their relation with independence and domination numbers.

Exercises 6 Let G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) be two bipolar fuzzy graphs on X 1 and X 2 , respectively, then prove or disprove the following statements. 



1. If D1 and D2 are dominating sets of G 1 and G 2 , then d(G 1 ∪ G 2 ) = d(G 1 ) +   d(G 2 ) − |D1 ∩ D2 |. 2. If for y1 ∈ X 1 , A1 (y1 ) = (0, 0), and y2 dominates z 2 in G 2 , then (y1 , y2 ) dominates (y1 , z 2 ) in G 1 G 2 . 3. If for z 2 ∈ Y2 , A2 (z 2 ) = (0, 0), and y1 dominates z 1 in G 1 , then (y1 , z 2 ) dominates (z 1 , z 2 ) in G 1 G 2 .   4. If D1 and D2 are minimal dominating sets of G 1 and G 2 , respectively, then    D1 × X 2 and X 1 × D2 are dominating sets of G 1 G 2 and d(G 1 G 2 ) ≤ |D1 ×  X 2 | ∧ |X 1 × D2 |.     5. Let D1 and D2 be the dominating sets of G 1 and G 2 , respectively, then D1 × D2   is a dominating set of the direct product G 1 × G 2 and d(G 1 × G 2 ) = |D1 × D2 |. 6. If y1 dominates z 1 in G 1 and y2 dominates z 2 in G 2 , then (y1 , z 1 ) dominates (y2 , z 2 ) in G 1 × G 2 . 7. If X 1 ∩ X 2 = ∅, then I (G 1 ∪ G 2 ) = I (G 1 ) + I (G 2 ). 8. If N1 and N2 are maximal independent sets of G 1 and G 2 , respectively, and X 1 ∩ X 2 = ∅, then I (G 1 G 2 ) = |N1 × N2 | + |N |, where N = {(yi , z i ) : yi ∈ X 1 \ N1 , z i ∈ X 2 \ N2 , yi yi+1 ∈ E 1 , z i z i+1 ∈ E 2 , i = 1, 2, 3, . . .}.     9. If D1 and D2 are minimal dominating sets of G 1 and G 2 , then X 1 × X 2 \ D1 × D2 is a maximal independent set of G 1 × G 2 and I (G 1 × G 2 ) = n 1 n 2 − d(G 1 × G 2 ), where n 1 and n 2 are the number of vertices in G 1 and G 2 , respectively.

References 1. Akram, M.: Bipolar fuzzy graphs. Inf. Sci. 181(24), 5548–5564 (2011) 2. Akram, M.: Bipolar fuzzy graphs with applications. Knowl.-Based Syst. 39, 1–8 (2013) 3. Akram, M., Dudek, W.A.: Regular bipolar fuzzy graphs. Neural Comput. Appl. 21(supp 1), 197–205 (2012) 4. Akram, M., Alshehri, N., Davvaz, B., Ashraf, A.: Bipolar fuzzy digraphs in decision support systems. J. Multivalued Log. Soft Comput. 27(5–6), 531–551 (2016) 5. Akram, M., Waseem, N.: Novel applications of bipolar fuzzy graphs to decision making problems. J. Appl. Math. Comput. 56(1–2), 73–91 (2018) 6. Akram, M., Waseem, N.: Novel decision making method based on domination in m-polar fuzzy graphs. Commun. Korean Math. Soc. 32(4), 1077–1097 (2017) 7. Akram, M., Waseem, N., Davvaz, B.: Certain types of domination in m-polar fuzzy graphs. J. Multivalued Log. Soft Comput. 29(5), 619–646 (2017)

280

6 Domination in Bipolar Fuzzy Graphs

8. Bauer, D.B., Harary, F., Nieminen, J., Suffel, C.L.: Domination alteration sets in graphs. Discret. Math. 47, 153–161 (1983) 9. Ball, W.W.R.: Mathematical Recreations and Problems of Past and Present Times. Macmillan and Company, New York (1892) 10. Cockayne, E.J., Favaron, O., Payan, C., Thomason, A.C.: Contribution to the theory of domination and irredundance in graphs. Discret. Math. 33(3), 249–258 (1981) 11. De Jaenisch, C.F.: Applications de l’Analyse Mathematique an Jeu des Echecs (Applications of mathematical analysis in chess games). Petrograd (1862) 12. Guy, R.K.: Unsolved Problems in Number Theory, vol. 1. Springer, Berlin (1981) 13. Haynes, T.W., Hedetniemi, S., Slater, P.: Fundamentals of domination in graphs. CRC Press, Boca Raton (2013) 14. Karunambigai, M.G., Akram, M., Palanivel, K., Sivasankar, S.: Domination in bipolar fuzzy graphs. In: IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–6 (2013) 15. Nagoorgani, A., Chandrasekaran, V.T.: Domination in fuzzy graphs. Adv. Fuzzy Sets Syst. I(1), 17–26 (2006) 16. Nagoorgani, A., Vadivel, P.: Fuzzy independent dominating set. Adv. Fuzzy Sets Syst. 2(1), 99–108 (2007) 17. Ore, O.: Theory of Graphs, vol. 38. American Mathematical Society Publications, Providence (1962) 18. Parvathi, R., Thamizhendhi, G.: Domination in intuitionistic fuzzy graphs. Notes Intuit. Fuzzy Sets 16(2), 39–49 (2010) 19. Somasundaram, A., Somasundaram, S.: Domination in fuzzy graphs. Pattern Recognit. Lett. 19(9), 787–791 (1998) 20. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998)

Chapter 7

Bipolar Fuzzy Circuits

In this chapter, we discuss the notions of bipolar fuzzy rank function, bipolar fuzzy vector spaces, bipolar fuzzy basis, bipolar fuzzy matroids, bipolar fuzzy circuits, and describe certain characterizations concerning the linear independence of linear bipolar fuzzy matroids, uniform bipolar fuzzy matroids, partition bipolar fuzzy matroids, and cycle bipolar fuzzy matroids. We establish the relation of crisp matroids and bipolar fuzzy matroids using (α, β)−cuts. We study the concepts of closure of bipolar fuzzy matroids, M−induced matroid sequence, fundamental sequence, circuit rectangles, and put special emphasis on bipolar fuzzy circuits. We also introduce the concepts of matroids and circuits in a soft environment and bipolar fuzzy soft environment. We present certain applications of bipolar fuzzy matroids and bipolar fuzzy soft matroids in decision support systems and network analysis. The main results of this chapter are from [25].

7.1 Introduction Matroid theory laid down its foundations in 1935 after the work of Whitney [28]. This theory constitutes a useful approach for linking major ideas of linear algebra, graph theory, combinatorics, and many other areas of Mathematics. Matroid theory has been a focus for active research during the last few decades. The resurgence of work in combinatorics during the last few decades has resulted in an increased interest in matroids, and matroid theory remains a focus for much active research. In 1988, Goetschel [10] presented the approach of fuzzification to matroid theory and discussed the uncertain behavior of matroids. The same authors introduced the concept of bases of fuzzy matroids, fuzzy matroid structures and greedy algorithm in fuzzy matroids, fuzzy circuits, fuzzy rank functions and fuzzy matroid sums in [10–16]. Hsueh [17] proposed one possible fuzzification of matroids which entended the independence axioms of matroids from set systems to fuzzy set systems and © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_7

281

282

7 Bipolar Fuzzy Circuits

preserves the most basic properties of matroids. Novak [20] presented two assertions concerning a regular pair of crisp matroids in order to clarify an important step of the proof of Lemma 2.5 of the paper by Goetschel and Voxman [11]. Novak [21, 22] also presented the concepts of fuzzy pre-independence set system, fuzzy independence set system, fuzzy pre-matroid, and hereditary fuzzy pre-matroid. Shi [26, 27] generalized the concept of closed fuzzy pre-matroids to L-fuzzy set theory when L is a complete lattice, called L-fuzzifying matroid and (L , M)-fuzzy matroid. It comes as no surprise that bipolarity in data and information plays a vital role in various fields of science and technology. As an extension of the fuzzy set theory, Zhang [30] introduced the theory of bipolar fuzzy sets (also known as YinYang bipolar fuzzy sets, where Yang depicts the positive side while Yin represents the negative side in a system), which illustrate the bipolar behavior of objects. The coexistence, equilibrium, and harmony of the two sides are considered a key for the mental and physical health of a person as well as for the stability and prosperity of a social system. In recent years, bipolar fuzzy sets seem to have been studied and applied a bit enthusiastically and a bit increasingly. The concept of bipolar fuzzy sets was applied to graphs in [2, 3, 5, 7]. The idea of bipolar fuzzy matroids and bipolar fuzzy circuits is discussed in [25] with a number of real-world applications in decision support systems.

7.2 Bipolar Fuzzy Circuits In this section, the notion of bipolar fuzzy vector spaces, bipolar fuzzy matroids, and bipolar fuzzy circuits is given with certain properties. The term crisp matroid has various equivalent definitions but we are using here the simplest definition of a matroid. Definition 7.1 If X is a non-empty set and I is any subset of P(X ), the power set of X , satisfying the following conditions: 1. If B1 ∈ I and B2 ⊂ B1 , then B2 ∈ I, 2. If B1 , B2 ∈ I and |B1 | < |B2 |, then there exists B3 ∈ I such that B1 ⊂ B3 ⊆ B1 ∪ B2 , then the pair M = (X, I ) is called a matroid and I is known as the family of independent subsets of M. Definition 7.2 If M = (X, I ) is a matroid, then the mapping R : P(X ) → {0, 1, 2, . . . , |X |} defined by R(B) = max{|E||E ⊆ B, E ∈ I } is a rank function for M. If B ∈ P(X ), R is known as the rank of B.

7.2 Bipolar Fuzzy Circuits

283

Definition 7.3 A bipolar fuzzy set A on a non-empty set X is an object of the form p p A = {(x, μ A (x), μnA (x)) | x ∈ X }, where μ A : X → [0, 1] and μnA : X → [−1, 0] are mappings. p The positive membership degree μ A (x) denotes the truth or satisfaction degree of an element x to a certain property corresponding to the bipolar fuzzy set, A and μnA (x) represents the satisfaction degree of an element x to some counter property of p the bipolar fuzzy set A. If μnA (x) = 0 and μ A (x) = 0, it is the situation that x is not p satisfying the property of A but satisfying the counter property to A. If μ A (x) = 0 n and μ A (x) = 0, it is the case when x has only positive satisfaction for A. It is possible p for x to be such that μ A (x) = 0 and μnA (x) = 0 when x satisfies the property of A as well as its counter property in some part of X . Definition 7.4 Let A be a bipolar fuzzy set on a non-empty set X . A bipolar fuzzy relation B on A is a mapping B : A → A such that p

p

p

μ B (x y) ≤ μ A (x) ∧ μ A (y) and μnB (x y) ≥ μnA (x) ∨ μnA (y),

for all x, y ∈ X.

B is also a bipolar fuzzy relation in X defined by the mapping B : X × X → [0, 1] × [−1, 0]. p

Definition 7.5 The support of a bipolar fuzzy set A = (μ A , μnA ) is denoted by supp(A) and defined as supp(A) = supp p (A) ∪ supp n (A), where supp p (A) = p {x | μ A (x) > 0}, supp n (A) = {x | μnA (x) < 0}. We call supp p (A) as positive support and supp n (A) as negative support. p

Definition 7.6 Let A = (μ A , μnA ) be a bipolar fuzzy set on X and α ∈ [0, 1]. The α-cut of a bipolar fuzzy set A is denoted by Aα and defined as Aα = Aαp ∪ Anα , where Aαp = {x | μαp (x) ≥ α}, Anα = {x | μnα (x) ≤ −α}. We call Aαp as positive α-cut and p Anα as negative α-cut. The height of a bipolar fuzzy set A = (μ A , μnA ) is defined p p as h(A) = max{μ A (x)|x ∈ X }. The depth of a bipolar fuzzy set A = (μ A , μnA ) is defined as d(A) = min{μnA (x)|x ∈ X }. Definition 7.7 A bipolar polar fuzzy graph on a non-empty set X is a pair G = (A, B), where A : X → [0, 1] × [−1, 0] is a bipolar fuzzy set on the set of vertices X and B : X × X → [0, 1] × [−1, 0] is a bipolar polar fuzzy relation in X such that p

p

p

μ B (x y) ≤ μ A (x) ∧ μ A (y) and μnB (x y) ≥ μnA (x) ∨ μnA (y),

for all x, y ∈ X.

Note that B(x y) = (0, 0) for all x y ∈ X × X − E, where E ⊆ X × X is the set of edges. A is called a bipolar fuzzy vertex set of G, and B is a bipolar fuzzy edge set of G. A bipolar fuzzy relation B on A is symmetric if B(x y) = B(yx) for all x, y ∈ X . Example 7.1 Let A be a bipolar fuzzy set on the set of vertices X = {x1 , x2 , x3 } and B be a bipolar fuzzy relation in X as shown in Tables 7.1 and 7.2, respectively. The bipolar fuzzy graph G = (A, B) is shown in Fig. 7.1.

284

7 Bipolar Fuzzy Circuits

Table 7.1 Bipolar fuzzy set A on {x1 , x2 , x3 } A x1 p μ A (x) μnA (x)

0.5 −0.4

x2

x3

0.6 −0.5

0.7 −0.7

Table 7.2 Bipolar fuzzy relation B in {x1 , x2 , x3 } B x1 x2 x1 x2 x3

(0,0) (0.2,−0.3) (0.2,−0.3)

Fig. 7.1 Bipolar fuzzy graph

x3

(0.2,−0.3) (0,0) (0.6,−0.4)

(0.2,−0.3) (0.6,−0.4) (0,0)

0.4) .5, − x 1(0

(0.2, −0.3)

x3 (0.7,

x2 (0.6, −0.5)

(0.2 , −0 .3)

.4) , −0 (0.6

0.7)

n Definition 7.8 The order or cardinality of a bipolarfuzzyset A = (μ p , μ  ), on a p n p n non-empty set X , is defined as |A| = (|A| , |A| ) = x∈X μ A (x), μ A (x) .

Definition 7.9 Let (t p , t n ), (r p , r n ) ∈ [0, 1] × [−1, 0] be two ordered pairs, then we denote (t p , t n ) < (r p , r n ) if t p < r p and t n > r n . If A and B are two bipolar fuzzy sets, then 1. |A| < |B| if |A| p < |B| p and |A|n > |B|n , 2. |A| ≤ |B| if |A| p ≤ |B| p and |A|n ≥ |B|n . Definition 7.10 Let G = (A, B) be a bipolar fuzzy graph on X and (α, β) ∈ [0, 1] × p [−1, 0]. Define (α, β)-cut of a bipolar fuzzy set A as A(α,β) = {x ∈ X | μ A (x) ≥ n α, μ A (x) ≤ β}. The crisp graph (α,β) , B(α,β) ) is called an (α, β)-level  G (α,β) = (A p graph of G, where B(α,β) = x y ∈ E | μ B (x y) ≥ α, μnB (x y) ≤ β is (α, β)-cut of B. Definition 7.11 Let β be any bipolar fuzzy set on X and t = (t p , t n ), t p ∈ [0, 1], t n ∈ [−1, 0], then

7.2 Bipolar Fuzzy Circuits

285 p

β t = {y ∈ X | μβ (y) ≥ t p , μnβ (y) ≤ t n }, p

p

R p (β) = {μβ (y) | μβ (y) > 0}, R n (β) = {μnβ (y) | μnβ (y) < 0}. Definition 7.12 A bipolar fuzzy vector space over a field F is defined as a pair p (X, B), where B = (μ B , μnB ) : X → [0, 1] × [−1, 0] is a mapping and X is a vector space over F such that for all c, d ∈ F and y, z ∈ X , p

p

p

μ B (cy + dz) ≥ min{μ B (y), μ B (z)}, μnB (cy + dz) ≤ max{μnB (y), μnB (z)}. Example 7.2 Let X be a vector space of 2 × 1 column vectors  t over R. Define a mapping B : X → [0, 1] × [−1, 0] such that for each z = x y , ⎧  t ⎨ (1, −1), z = 0 0    B(z) = ( 31 , − 23 ), z = x 0 t or z = 0 y t . ⎩ (1, −1), x = 0 and y = 0  t First show that (X, B) is a bipolar fuzzy vector space. For z = 0 0 , the case is trivial. So the following cases are to be discussed.  t  t Case 1: Consider two column vectors z = x y and u = u v , then for any scalars c and d,  cx + du B(cz + du) = B . cy + dv If either exactly one of c or d is zero or both are non-zero, then cx + du = 0 and p p p cy + dv = 0 and so μ B (cz + du) = 1 ≥ min{μ B (z), μ B (u)} and μnB (cz + du) = −1 ≤ max{μnB (z), μnB (u)}.  t  t  t Case 2: If z = x 0 and u = 0 v , then B(cz + du) = B( cx dv ). If both c and d are non-zero, then p

p

p

μ B (cz + du) = 1 > min{μ B (z), μ B (u)} and μnB (cz + du) = −1 < max{μnB (z), μnB (u)}. If exactly one of c or d is zero, then 1 p p = min{μ B (z), μ B (u)} and 3 2 μnB (cz + du) = − = max{μnB (z), μnB (u)}. 3 p

μ B (cz + du) =

Hence, (X, B) is a bipolar fuzzy vector space.

286

7 Bipolar Fuzzy Circuits

Definition 7.13 Let (X, B) be a bipolar fuzzy vector space over F. A set of vectors {yk }nk=1 is known as bipolar fuzzy linearly independent in (X, B) if 1. {yk }nk=1 is linearly independent, n n n    p μnB (ck yk )), for all {ck }nk=1 ⊂ F and distinct yi ∈ 2. B( ck yk ) = ( μ B (ck yk ), k=1

X , 1 ≤ i ≤ n.

k=1

k=1

Definition 7.14 A set of vectors B = {yk }nk=1 is said to be a bipolar fuzzy basis in (X, B) if condition 2 of Definition 7.13 holds and B is a basis in X . Proposition 7.1 If (X, B) is a bipolar fuzzy vector space, then any set of vectors with distinct positive and negative degrees of membership is linearly independent and bipolar fuzzy linearly independent. Proof This proposition can be proved by induction on the number of vectors, n, with distinct degree of membership. For n = 1, {y1 }, the statement is trivial. Assume that the statement is true for k vectors. Let {y j }k+1 j=1 be the set of vectors with distinct positive and negative degrees of membership. Suppose {y j }k+1 j=1 are not linearly independent and hence not bipolar fuzzy linearly independent. Then there exist scalars k  c1 , c2 , . . . , ck such that yk+1 = c j y j and j=1

⎛ B(yk+1 ) = ⎝

k 

p μ B (c j y j ),

j=1 p

k  j=1

⎞ μnB (c j y j )⎠ = (

k  j=1

p

μ B (y j ),

k 

μnB (y j )).

j=1

p

It clearly shows that μ B (yk+1 ) ∈ {μ B (y j )}kj=1 and μnB (yk+1 ) ∈ {μnB (y j )}kj=1 , a contradiction to the fact that yk+1 has distinct positive and negative degrees of membership. Thus, {y j }k+1 j=1 are linearly independent. The bipolar fuzzy linear independence of the vectors can be proved similarly. Proposition 7.2 Let (X, B) be a bipolar fuzzy vector space, then p

1. B(0) = (max y∈X μ B (y), min y∈X μnB (y)), 2. B(ay) = B(y) for all a ∈ F \ {0} and y ∈ X , p p 3. If B(y) = B(z) for some y, z ∈ X , then B(y + z) = (μ B (y) ∧ μ B (z), μnB (y) ∨ n μ B (z)). Proof The proof of conditions 7.2.1 and 7.2.3 is enough. p p 1. For any y ∈ X , B(0) = B(0y) = (μ B (0y), μnB (0y)). By Definition 7.12, μ B (0) = p p p p μ B (0y) ≥ μ B (y), for every y ∈ X . Hence, μ B (0) = max y∈X μ B (y). Similarly, it can be proved that μnB (0) = min y∈X μnB (y). 3. Suppose for y, z ∈ X , B(y) = B(z). Without loss of generality, assume that p p p p p p μ B (y) > μ B (z). By Definition 7.12, μ B (y + z) ≥ μ B (y) ∧ μ B (z) = μ B (z). Also, p p p p p p Hence, μB μ B (z) = μ B ((y + z) − y) ≥ μ B (y + z) ∧ μ B (y) = μ B (y + z). p (y + z) = μ B (z). Other cases can be proved similarly.

7.2 Bipolar Fuzzy Circuits

287

Remark 7.1 If B is a bipolar fuzzy basis of (X, B), then the membership value of every element  of X can be calculated from the membership values of basis elements, i.e., if u = nk=1 ck u k , then  B(u) = B  =

n 

 ck u k

 =

k=1 n 

k=1

p

μ B (u k ),

n  k=1

n 

p μ B (ck u k ),

n 

 μnB (ck u k )

k=1



μnB (u k ) .

k=1

Definition 7.15 Let X be a non-empty finite set and I ⊆ BF(X ) be a family of bipolar fuzzy subsets; BF(X ) is a bipolar fuzzy power set of X , satisfying the following conditions. 1. If β1 ∈ I, β2 ∈ BF(X ), and β2 ⊂ β1 , then β2 ∈ I. 2. If β1 , β2 ∈ I and |supp(β1 )| < |supp(β2 )|, then there exists β3 ∈ I such that a. β1 ⊂ β3 ⊆ β1 ∪ β2 , p b. m p (β3 ) ≥ min{m p (β1 ), m p (β2 )}, m p (βi ) = min{μβi (y)|y ∈ supp(βi )}, i = 1, 2, 3, c. m n (β3 ) ≤ max{m n (β1 ), m n (β2 )}, m n (βi ) = max{μnβi (y)|y ∈ supp(βi )}, i = 1, 2, 3. Then the pair M(X ) = (X, I) is called a bipolar fuzzy matroid on X , and I is a family of independent bipolar fuzzy subsets of M(X ). Note 7.1 {α : α ∈ BF(X ), α ∈ / I} is the family of dependent bipolar fuzzy subsets in M(X ). A minimal dependent bipolar fuzzy set is called a bipolar fuzzy circuit. Equivalently, a bipolar fuzzy set α ∈ BF(X ) is a bipolar fuzzy circuit in M if α ∈ /I and β ∈ I for each β ⊂ α. The family of all bipolar fuzzy circuits is represented by Cr (M). A bipolar fuzzy n−circuit is a bipolar fuzzy circuit which has n number of elements. Since the elements of I are those members of BF(X ) which are mutually disjoint with the members of Cr (M), a bipolar fuzzy matroid can be uniquely determined from Cr (M). Definition 7.16 Let X be a non-empty universe. For any bipolar fuzzy matroid, the bipolar fuzzy rank function λr : BF(X ) → [0, ∞) × (−∞, 0] is defined as    λr (β) = λrp (β), λrn (β) = max |γ| p , min |γ|n , γ⊆β

γ⊆β

where β ∈ I. Clearly, the bipolar fuzzy rank function of a bipolar fuzzy matroid possesses the following properties. p

p

1. If β1 , β2 ∈ BF(X ) and β1 ⊆ β2 , then λr (β1 ) ≤ λr (β2 ) and λrn (β1 ) ≥ λrn (β2 ), p 2. If β ∈ BF(X ), then λr (β) ≤ |β| p and λrn (β) ≥ |β|n , 3. If β ∈ I, then λr (β) = |β|.

288

7 Bipolar Fuzzy Circuits

The concept of certain types of bipolar fuzzy matroids is described in the following examples. Example 7.3 A simple example of a bipolar fuzzy matroid, known as a uniform bipolar fuzzy matroid is defined as I = {β ∈ BF(X ) | |supp(β)| ≤ k}. If |X | = n and k is any positive integer, then the uniform bipolar fuzzy matroid is denoted by Uk,n = (X, I). The bipolar fuzzy circuit of Uk,n is a bipolar fuzzy subset β such that |supp(β)| = k + 1. Consider the example of a uniform bipolar fuzzy matroid M = (X, I), where X = {e1 , e2 , e3 } and I = {β ∈ BF(X ) : |supp(β)| ≤ 2} such that for any β ∈ BF(X ), β(y) = τ(y), for all y ∈ X , where ⎧ ⎨ (0.2, −0.3), y = e1 τ(y) = (0.4, −0.5), y = e2 . ⎩ (0.1, −0.3), y = e3 I ={∅, {(e1 , 0.2, −0.3)}, {(e2 , 0.4, −0.5)}, {(e3 , 0.1, −0.3)}, {(e1 , 0.2, −0.3), (e2 , 0.4, −0.5)},

{(e2 , 0.4, −0.5), (e3 , 0.1, −0.3)}, {(e1 , 0.2, −0.3), (e3 , 0.1, −0.3)}}.

(7.1)

The bipolar fuzzy circuit of M is {(e1 , 0.2, −0.3), (e2 , 0.4, −0.5), (e3 , 0.1, −0.3)}. For β = {(e2 , 0.4, −0.5), (e1 , 0.2, −0.3)}, λr (β) = (0.6, −0.8). Example 7.4 A bipolar fuzzy matroid derived from a bipolar fuzzy matrix is called a linear bipolar fuzzy matroid. If X is the set of all column labels of a bipolar fuzzy matrix and βx represents a bipolar fuzzy submatrix having those columns which are labeled by X , then the bipolar fuzzy linear matroid is defined as I = {βx ∈ BF(X ) | columns of βx are bipolar fuzzy linearly independent}, BF(X ) is the power set of all bipolar fuzzy submatrices labeled by elements in X. For any βx ∈ BF(X ), |βx | =

 r  c c p sup μβx (aki ), inf μnβx (aki ) , βx = [ai j ]r ×c . k=1

i=1

i=1

Let X = {1, 2, 3, 4} be a set of column labels of bipolar fuzzy 2 × 1 vectors over R such that for any βx ∈ BF(X ), βx (y) = A(y), where ⎡

⎤ 1 2 3 4 A = ⎣(0.2, −0.3) (0.4, −0.5) (0.6, −0.7) (0.8, −0.9)⎦ . (0.3, −0.4) (0.5, −0.6) (0.7, −0.8) (0.9, −1.0) Take I = {∅, {1}, {2}, {4}, {1, 2}, {2, 4}}, then M(A) = (A, I) is a bipolar fuzzy matroid on X . The family of dependent bipolar fuzzy subsets of M(A) is {{3}, {1, 3}, {1, 4}, {2, 3}, {3, 4}} ∪ {η : η ⊆ A, |supp(η)| ≥ 3}. For β = {2, 4}, μr (β) = (1.7, −1.9).

7.2 Bipolar Fuzzy Circuits

289

y1 (0 .2, − 0.3)

0.7) .6, − 0 ( y4

y5 (0.3, −0.4)

y3 (0.2, −0.3)

y2 (0.2, −0.3)

Fig. 7.2 3−polar fuzzy multigraph

Example 7.5 A bipolar fuzzy matroid in which the universe X is partitioned into sets δ1 , δ2 , . . . , δr such that I = {β ∈ BF(X ) | |supp(β) ∩ δi | ≤ li , for all 1 ≤ i ≤ r } for given positive integers l1 , l2 , . . . , lr is called a partition bipolar fuzzy matroid. The circuit of a partition bipolar fuzzy matroid is a bipolar fuzzy subset α such that |supp(α) ∩ δi | = li + 1. Example 7.6 A significant class of bipolar fuzzy matroids derived from bipolar fuzzy graphs is known as a cycle bipolar fuzzy matroid. The detail is illustrated in Proposition 7.3. The cycle bipolar fuzzy matroid derived using Proposition 7.3 is denoted by M = (X, I), where X = {y1 , y2 , . . . , yn } is the set of edges and I is defined as I = {β ∈ BF(X ) | β is a bipolar fuzzy set on U ⊆ X }. Clearly, β ∈ I is an independent set of M(G) if and only if supp(β) is not the set of edges of any crisp cycle. Alternatively, the elements of M are bipolar fuzzy graphs β such that supp(β) is a crisp forest. As an example, consider a cycle bipolar fuzzy matroid (X, I), where X = {y1 , y2 , y3 , y4 , y5 } and for any β ∈ I, β(y) = D(y), (C, D) is a bipolar fuzzy multigraph with edge set X as shown in Fig. 7.2. By Proposition 7.3, Cr (M) = {{(y5 , 0.3, −0, 4)}, {(y2 , 0.2, −0.3), (y3 , 0.2, −0.3)}, {(y1 , 0.2, −0.3), (y2 , 0.2, −0.3), (y4 , 0.6, −0.7)}, {(y1 , 0.2, −0.3), (y3 , 0.2, −0.3)}}. I = {∅, {(y1 , 0.2, −0.3)}, {(y2 , , 0.2, −0.3)}, {(y3 , 0.2, −0.3)}, {(y4 , 0.6, −0.7)}, {(y1 , 0.2, −0.3), (y2 , 0.2, −0.3)}, {(y1 , 0.2, −0.3), (y4 , 0.6, −0.7)}, {(y2 , 0.2, −0.3), (y4 , 0.6, −0.7)}, {(y1 , 0.2, −0.3), (y3 , 0.2, −0.3)}, {(y3 , 0.2, −0.3), (y4 , 0.6, −0.7)}}. For β = {(y2 , 0.2, −0.3), (y4 , 0.6, −0.7)}, λr (β) = (0.8, −1.0). Proposition 7.3 For any bipolar fuzzy matroid (X, I), if Cr (M) is the class of bipolar fuzzy edge sets α such that supp(α) is the set of edges of a cycle, then Cr (M) is the class of bipolar fuzzy circuits on M. Example 7.7 For any bipolar fuzzy graph G = (C, D) and t = (t p , t n ), 0 ≤ t p ≤ 1, −1 ≤ t n ≤ 0 define p E t = {yz ∈ supp(D)|μ D (yz) > t p , μnD (yz) < t n }, Ft = {F | F is a forest in the crisp graph (X, E t )}, I t = {E(F)|F ∈ Ft }, where E(F) is the edge set of F. Then (E t , I t ) is a matroid for each t.

290

7 Bipolar Fuzzy Circuits

Define J = {β ∈ BF(X )| β t ∈ I t , for each t = (t p , t n ), 0 ≤ t p ≤ 1, −1 ≤ t n ≤ 0}, then (X, J ) is a cycle bipolar fuzzy matroid. Theorem 7.1 Let M = (X, I) be a bipolar fuzzy matroid and, for each t = (t p , t n ), 0 ≤ t p ≤ 1, −1 ≤ t n ≤ 0, define I t = {β t | β ∈ I}. Then (X, I t ) is a matroid on X . Proof We prove conditions 1 and 2 of Definition 7.1. Assume that β1t ∈ I t and α ⊆ β1t . Define a bipolar fuzzy set β2 ∈ BF(X ) as  β2 (y) =

t, y∈α . (0, 0), otherwise

Clearly, β2 ⊆ β1 , β2 ∈ I, and β2t = α, therefore α ∈ I t . To prove condition 2, let α1 , α2 ∈ I t and |α1 | < |α2 |. Then there exist β1 and β2 such that β1t = α1 and β2t = α2 . Define βˆ1 and βˆ2 as   t, y ∈ α t, y ∈ α2 1 βˆ1 (y) = βˆ2 (y) = . (0, 0), otherwise (0, 0), otherwise It is clear that supp(βˆ1 ) < supp(βˆ2 ). As M is a bipolar fuzzy matroid, there exists β3 such that βˆ1 ⊆ β3 ⊆ βˆ1 ∪ βˆ2 . Since  βˆ1 ∪ βˆ2 (y) =

t, y ∈ α1 ∪ α2 , (0, 0), otherwise

there exists a set α3 such that  β3 (y) =

t, y ∈ α3 . (0, 0), otherwise

Also, α1 ⊆ α3 ⊆ α1 ∪ α2 , α3 ∈ I t . Hence, M t is a matroid on X . Remark 7.2 Let M = (X, I) be a bipolar fuzzy matroid and, for each t = (t p , t n ), 0 ≤ t p ≤ 1, −1 ≤ t n ≤ 0, M t = (X, I t ) be a matroid on a finite set X as given in p Theorem 7.1. As X is finite, there is a finite sequence t 0 , t 1 , t 2 , . . . , t m , 0 < t1 < p p t2 < · · · < tm , 0 > t1n > t2n > · · · > tmn > −1 such that M t i = (X, I t i ) is a crisp matroid, for each 1 ≤ i ≤ m, and p

1. t 0 = (0, 0), tm ≤ 1 and tmn ≥ −1, p 2. Iw = ∅ if w = (w p , w n ), 0 ≤ w p ≤ tm and tmn ≤ w n ≤ 0, p p n , then Iw = Is , 3. If for s = (s p , s n ), ti < w p , s p < ti+1 and tin > w n , s n > ti+1 0 ≤ i ≤ m − 1, p p p n n > s n > ti+2 , then Iw ⊃ Is , 0 ≤ 4. If ti < w < ti+1 < s p < ti+2 and tin > w > ti+1 i ≤ m − 2.

7.2 Bipolar Fuzzy Circuits

291

The sequence t 0 , t 1 , t 2 , . . . , t m is known as a fundamental sequence of M. Let t¯i = 21 (t i−1 + t i ) for 1 ≤ i ≤ m. The decreasing sequence of crisp matroids M t 1 ⊃ M t 2 ⊃ · · · ⊃ M t m is known as M−induced matroid sequence. Definition 7.17 Let t 0 , t 1 , . . . , t m be the fundamental sequence of a bipolar fuzzy matroid. For any t = (t p , t n ), 0 < t p ≤ 1 and −1 ≤ t n < 0, define I t = I t i , where p p n p t i = 21 (t i−1 + t i ) and ti−1 < t p ≤ ti , ti−1 > t n ≥ tin . If t p > tm and t n < tmn take I t = I t . Define I = {β ∈ BF(X ) : β t ∈ I t , for each t = (t p , t n ), 0 < t p ≤ 1 and − 1 ≤ t n < 0}. Then M = (X, I) is known as closure of M = (X, I). Example 7.8 We now explain the concept of closure by an example of a bipolar fuzzy uniform matroid M = (X, I), where X = {y1 , y2 , y3 } and I = {β ∈ BF(X ) : |supp(η)| ≤ 1} such that for any β ∈ BF(X ), β(y) = τ (y), for all y ∈ X , where ⎧ ⎪ ⎨(0.2, −0.3), y = y1 τ(y) = (0.3, −0.4), y = y2 . ⎪ ⎩ (0.4, −0.5), y = y3   I = ∅, {(y1 , 0.2, −0.3)}, {(y2 , 0.3, −0.4)}, {(y3 , 0.4, −0.5)} . The fundamental sequence of M is {t 0 = (0, 0), t 1 = (0.2, −0.3), t 2 = (0.3, −0.4), t 3 = (0.4, −0.5)}. From routine calculations, t 1 = (0.1, −0.15), t 2 = (0.25, −0.35), t 3 = (0.35, −0.45). Now I t 1 = {{y1 }, {y2 }, {y3 }}, I t 2 = {{y2 }, {y3 }}, I t 3 = {{y3 }}. Hence, the closure of I can be defined as I = {∅, {(y1 , 0.2, −0.3)}, {(y2 , 0.3, −0.4)}, {(y1 , 0.2, −0.3), (y2 , 0.3, −0.4)}, {(y1 , 0.2, −0.3), (y3 , 0.4, −0.5)}, {(y2 , 0.3, −0.4), (y3 , 0.4, −0.5)}, {(y3 , 0.4, −0.5)}}. Theorem 7.2 The closure M = (X, I) of a bipolar fuzzy matroid M = (X, I) is also a bipolar fuzzy matroid. The proof of Theorem 7.2 is a clear consequence of Theorem 7.1. Definition 7.18 A bipolar fuzzy matroid with fundamental sequence t 0 , t 1 , . . . , t m p p is known as a closed bipolar fuzzy matroid if for each t = (t p , t n ), ti−1 < t p ≤ ti n n n and ti−1 > t ≥ ti , I t = I t i . Remark 7.3 The closure M of a bipolar fuzzy matroid M is closed and the smallest closed bipolar fuzzy matroid containing M is M. Also note that the fundamental sequences of M and M are same. Theorem 7.3 Let M = (X, I) be a bipolar fuzzy matroid and suppose β ∈ BF(X ), then β ∈ I if and only if for each t = (t p , t n ), β t ∈ I t , where t p ∈ R p (β), t n ∈ R n (β). Theorem 7.3 follows from the proof of Theorem 7.1.

292

7 Bipolar Fuzzy Circuits

Theorem 7.4 If M = (X, I) is a closed fuzzy matroid, then each dependent bipolar fuzzy set in M properly contains a dependent bipolar fuzzy subset. p

p

Proof Assume that α is a dependent bipolar fuzzy set in M and R p (α) = {t1 , t2 , . . . , p p p p tm }, R n (α) = {t1n , t2n , . . . , tmn }, where t1 < t2 < · · · < tm and t1n > t2n > · · · > tmn . By / I t i . Since M is a closed Theorem 7.3, there exists t i , 1 ≤ i ≤ m, such that α t i ∈ bipolar fuzzy matroid, there is an ε = (ε p , εn ), ε p > 0, εn < 0 such that for each p p t = (t p , t n ), ti − ε p ≤ t p ≤ ti , tin − εn ≥ t n ≥ tin , α t = α t i and I t = I t i . Let γ be a proper bipolar fuzzy subset of α defined by  γ(y) =

p

t i − ε, μαp (y) ≥ ti and μnα (y) ≤ tin . (0, 0), otherwise

/ I t i −ε and hence by Theorem 7.3, γ ∈ / I. Note that γ t i −ε ∈ As a consequence of Theorem 7.10, we adopt the following definition of a bipolar fuzzy circuit. Definition 7.19 Let M = (X, I) be a bipolar fuzzy matroid. Then α ∈ BF(X ) is a bipolar fuzzy circuit in M if α ∈ / I and α\\z ∈ I for each z ∈ supp(α), where α\\z is defined as  α(y), y = z (α\\z)(y) = . (0, 0), y = z Now characterizing the properties of bipolar fuzzy circuits in the form of crisp circuits. Theorem 7.5 (Characterization Theorem) Assume that M = (X, I) is a bipop p p lar fuzzy matroid and α ∈ BF(X ). Let R p (α) = {t1 , t2 , . . . , tm } and R n (α) = p p p n n n n n n {t1 , t2 , . . . , tm }, where t1 < t2 < · · · < tm and t1 > t2 > · · · > tm . Then α is a bipolar fuzzy circuit in M if and only if (1) α t 1 is a crisp circuit in (X, I t 1 ), (2) α t i ∈ I t i , for 2 ≤ i ≤ m. Proof Let α be a bipolar fuzzy circuit, then α ∈ / I. By Theorem 7.3, there exists at least one i, 1 ≤ i ≤ m, (say, i = i  ) such that α t i  ∈ / I t i  . We now show that i  = 1. On p p  contrary, assume that 2 ≤ i ≤ m. Since t1 < ti  and t1n > tin , α t i  ⊂ α t 1 . Then there / I t i  . It follows from Theorem 7.3 exists, z ∈ α t 1 \ α t i  . Clearly, (α\\z) t i  = α t i  ∈ that α\\z ∈ / I which is a contradiction to the supposition that α is a bipolar fuzzy circuit. Thus, for 2 ≤ i ≤ m, α t i  ∈ I t i  and it establishes (2). / I t 1 . If α t 1 is not a circuit, then there exist We now prove (1). It is clear that α t 1 ∈ / I t 1 . If x ∈ α t 1 \ B, then B ⊆ (α\\x) t 1 ⇒ (α\\x) t 1 ∈ / It 1 . B ⊂ α t 1 such that B ∈

7.2 Bipolar Fuzzy Circuits

293

So, by Theorem 7.3, α\\x ∈ / I, a contradiction to the fact that α is a bipolar fuzzy circuit and it establishes (1). Conversely, let α ∈ BF(X ) satisfy (1) and (2). By Theorem 7.3, α ∈ / I. Suppose p that y ∈ supp(α), then μαp (y) ≥ t1 and μnα (y) ≤ t1n . Hence, (α\\y) t 1 = α t 1 \ y ∈ I t 1 and for 2 ≤ i ≤ m α t i ⊇ (α\\y) t i ∈ I t i . Clearly, by Theorem 7.3, α\\y ∈ I and so α is a bipolar fuzzy circuit. We can say that a bipolar fuzzy circuit α is elementary if it is an elementary bipolar fuzzy set, i.e., α is single valued on supp(α). Corollary 7.1 Let M = (X, I) be a bipolar fuzzy matroid, then α is a bipolar fuzzy circuit in M if and only if α = λ ∪ γ, where λ is an elementary bipolar fuzzy circuit, γ ∈ I and supp(γ) ⊂ supp(λ). p

p

p

Proof Assume that α is a bipolar fuzzy circuit in M, R p (α) = {t1 , t2 , . . . , tm }, and p p p R n (α) = {t1n , t2n , . . . , tmn }, where t1 < t2 < · · · < tm and t1n > t2n > · · · > tmn . Define λ and γ as 

 t 1, y ∈ supp(α) α(y), if y ∈ α t 2 λ(y) = . and γ(y) = (0, 0), other wise (0, 0), other wise Consequently, λ is an elementary bipolar fuzzy circuit, γ ∈ I, supp(γ) ⊂ supp(λ) and α = λ ∪ γ. Conversely, let α = λ ∪ γ, λ is elementary, γ ∈ I and supp(γ) ⊂ supp(λ). If p p p p p p R p (α) = {t1 , t2 , . . . , tm } and R n (α) = {t1n , t2n , . . . , tmn }, where t1 < t2 < · · · < tm p p p n n n n p n p and t1 > t2 > · · · > tm , then R (λ) = {t1 }, R (λ) = {t1 }, R (γ) = {t2 , . . . , tm }, and R n (γ) = {t2n , . . . , tmn }. It follows from Theorem 7.5, that 1) α t 1 is a circuit in I t 1 and for 2 ≤ i ≤ m, α t i ∈ I t i . Thus, by Theorem 7.5, α is a bipolar fuzzy circuit. Note 7.2 Corollary 7.1 implies that every bipolar subset of a bipolar fuzzy circuit is either a bipolar fuzzy circuit or an independent bipolar fuzzy set. Corollary 7.2 Suppose that M = (X, I) is a bipolar fuzzy matroid and γ ∈ BF(X ). If γ ∈ / I, then there exists a bipolar fuzzy circuit α such that α ⊆ γ. p

p

p

Proof Let R p (γ) = {t1 , t2 , . . . , tm } and Rn (γ) = {t1n , t2n , . . . , tmn }. By Theorem 7.3, / I t i . Let B ⊆ γ t i be a circuit in (X, I t i ) and define there exists ti such that γ t i ∈ α ∈ BF(X ) as  ti , if y ∈ B α(y) = . (0, 0), otherwise p

Then α ⊆ γ and R p (α) = {ti }, R n (α) = {tin }, supp(α) = B and hence α is a bipolar fuzzy circuit. Theorem 7.6 If α1 and α2 are bipolar fuzzy circuits in a bipolar fuzzy matroid M and α1 ⊆ α2 , then supp(α1 ) = supp(α2 ).

294

7 Bipolar Fuzzy Circuits

Proof Let α1 and α2 be two bipolar fuzzy circuits in M = (X, I). On contrary, assume that supp(α1 ) ⊂ supp(α2 ), then there exists y ∈ X such that y ∈ supp(α2 ) \ supp(α1 ). Moreover, α1 ⊆ α2 \\y ∈ I and hence, α1 ∈ I, a contradiction to the supposition that α1 is a bipolar fuzzy circuit.

7.2.1 Circuit Rectangles In this section, the concept of circuit rectangles and circuit function is discussed in detail. Definition 7.20 The truncation of a bipolar fuzzy set β at level t = (t p , t n ) is a bipolar fuzzy set β t∗ defined as  β t∗ (y)

=

p

β(y), if μβ (y) ≤ t p , μnβ (y) ≥ t n . p (0, 0), μβ (y) > t p , μnβ (y) < t n

If M = (X, I) is a bipolar fuzzy matroid, then define η = (η p , η n ) : BF(X ) → [0, 1] × [−1, 0] by η p (β) = max{t p | β t∗ is not a bipolar fuzzy circuit in M}, η n (β) = min{t n | β t∗ is not a bipolar fuzzy circuit in M}. The function η is known as a circuit function for β. Example 7.9 Consider the example of a bipolar fuzzy uniform matroid as given in Example 7.3. Take β = {(e1 , 0.2, −0.3), (e2 , 0.4, −0.5), (e3 , 0.1, −0.3)}. 1. For t = (0.2, −0.3), β t∗ = {(e1 , 0.2, −0.3), (e3 , 0.1, −0.3)}. 2. For t = (0.1, −0.3), β t∗ = {(e3 , 0.1, −0.3)}. 3. For t = (0.4, −0.5), β t∗ = β. Clearly, η(β) = (0.2, −0, 3). If we take β = {(e1 , 0.2, −0.3), (e2 , 0.4, −0.5)}, then η(β) = (1, −1). Theorem 7.7 Let M = (X, I) be a bipolar fuzzy matroid and η is a circuit function for β. Then the following conditions hold. 1. If β ∈ BF(X ) is not a bipolar fuzzy circuit, then η(β) = (1, −1). 2. If β is a bipolar fuzzy circuit, then η p (β) ≥ m p (β) and η n (β) ≤ m n (β). Proof 1. Since β is not a bipolar fuzzy circuit, for t = (1, −1), β t∗ = β. Consequently, η(β) = (1, −1). p p p p 2. Let R p (β) = {t1 , t2 , . . . , tm }, Rn (β) = {t1n , t2n , . . . , tmn }, and t i = (ti , tin ), 1 ≤ ∗ i ≤ m. For t ∈ {t 1 , t 2 , . . . , t m−1 }, β t is not a bipolar fuzzy circuit and for t ∈ {t | p p t p ≥ tm , tn ≤ tmn }, β t∗ is a bipolar fuzzy circuit. For t p < t1 and t n > t1n , β t∗ = ∅. p p n n Hence, η (β) ≥ m (β) and η (β) ≤ m (β).

7.2 Bipolar Fuzzy Circuits

295

Definition 7.21 Let M = (X, I) be a closed bipolar fuzzy matroid and α be a bipolar fuzzy circuit in M. Let I (α) = [m n (α), 0) × (0, m p (α)] which is called circuit rectangle for α. Theorem 7.8 Let α be a bipolar fuzzy circuit in a closed bipolar fuzzy matroid M = (X, I). Let I (α) = [m n (α), 0) × (0, m p (α)] be a circuit rectangle for α, then 1. if t ∈ I (α), then α t is a circuit in M t = (X, I t ), 2. if t ∈ / I (α), then α t ∈ I t . Proof By Theorem 7.1, α = δ ∪ γ, where δ is an elementary bipolar fuzzy circuit, γ ∈ I, and supp(γ) ⊂ supp(δ). Clearly, R p (δ) = {m p (α)} and R n (α) = {m n (α)}. If t p > m p (α) and t n < m n (α), then α t = γ t ∈ I t . If 0 < t p ≤ m p (α) and 0 > t n ≥ m n (α), then α t = supp(δ). Hence, α t is a circuit in M t if t ∈ I (α).

7.3 Bipolar Fuzzy Soft Circuits It is worth noting that uncertainty arising from various domains has a different nature and cannot be captured within a single mathematical framework. To solve complicated problems in economics, engineering, and environment, classical methods cannot be used successfully because of their difficulties, possibly, the inadequacy of a parametrization tool. Soft set theory was proposed by Molodtsov [19] in 1999 to deal with uncertain, fuzzy, not clearly defined objects in a parametric manner. Molodtsov’s soft sets provide a new way of coping with uncertainty from the viewpoint of parametrization in various domains, including the smoothness of functions, game theory, operations research, Riemann integration, probability theory, and measurement theory [19]. A soft set is a parameterized family of sets— intuitively, this is “soft” because the boundary of the set depends on the parameters. Formally, a soft set over a universal set X and set of parameters P is a pair ( f, A) = {(e, f (e)) | e ∈ A, f (e) ∈ P(X )} where A ⊆ P and f : A → P(X ) is a function from A to the power set of X . For each e ∈ A, the set f (e) is called the value set of e in ( f, A). Each parameter is a word or a sentence. In most cases, parameters are considered to be attributes, characteristics, or properties of objects in X . The pair (X, P) is also known as a soft universe and is defined as the set of all soft sets over X with attributes from P. Definition 7.22 A soft graph on a non-empty set X is a 3−tuple (X, E, A) such that for each e ∈ A, (X (e), E(e)) is a graph, where X (e) ⊆ X and E(e) ⊆ X (e) × X (e). Definition 7.23 A soft matroid on a non-empty set X is a 3−tuple M = (X, I, A), where A is the set of parameters and for each e ∈ A, I (e) ⊆ P(X ) satisfies the following conditions. 1. If B1 (e) ∈ I (e) and B2 (e) ⊂ B1 (e), then B2 (e) ∈ I (e). 2. If B1 (e), B2 (e) ∈ I (e) and |B1 (e)| < |B2 (e)|, then there exists B3 (e) ∈ I such that B1 (e) ⊂ B3 (e) ⊆ B1 (e) ∪ B2 (e).

296

7 Bipolar Fuzzy Circuits

Note that (I, A) = {I (e) | e ∈ A} is known as the family of independent soft subsets of M. Definition 7.24 Let X be a non-empty set and P be a set of parameters. Let BF(X ) denote the set of all bipolar fuzzy sets on X . A pair ( f, A) is called a bipolar fuzzy soft set on X , where A ⊆ P is a set of parameters and f : A → BF(X ) is a setvalued mapping, called a bipolar fuzzy approximation function. For any e ∈ A, p f (e) = {(x, μ f (e) (x), μnf (e) (x)) | x ∈ X }. (X, P) is the collection of all bipolar fuzzy soft sets on X with attributes from P and is said to be bipolar fuzzy soft class. Definition 7.25 Let ( f, A) and (g, B) be two bipolar fuzzy soft sets over (X, P), then a bipolar fuzzy soft relation from ( f, A) to (g, B) is a bipolar fuzzy soft subset of ( f, A) × ( f, B). In other words, a bipolar fuzzy soft relation from ( f, A) to (g, B) is of the form (R, C), where C ⊆ A × B and R(a, b) ⊆ ( f, A) × ( f, B), for all (a, b) ∈ C. Definition 7.26 A bipolar fuzzy soft graph on a non-empty set X is an ordered triplet G = (φ, ψ, A) such that 1. A is a non-empty set of parameters. 2. (φ, A) is a bipolar fuzzy soft set on X . 3. (ψ, A) is a bipolar fuzzy soft relation in X , that is, ψ : A → BF(X × X ), where BF(X × X ) is the set of all bipolar fuzzy power sets on X × X . 4. (φ(e), ψ(e)) is a bipolar fuzzy graph for all e ∈ A, that is, p

p

p

μψ(e) (x y) ≤ μφ(e) (x) ∧ μφ(e) (y) and μnψ(e) (a)(x y) ≥ μnφ(e) (x) ∨ μnφ(e) (y), for all x, y ∈ X. p

The bipolar fuzzy graph (φ(e), ψ(e)) is denoted by H (e). Note that μψ(e) (x y) = μnψ(e) (x y) = 0, for all x y ∈ X × X − E, for each e ∈ A. (φ, A) is called a bipolar fuzzy soft vertex set and (ψ, A) is called a bipolar fuzzy soft edge set. A bipolar fuzzy soft graph is a parameterized family of bipolar fuzzy graphs. Example 7.10 Let A = {e1 , e2 } be a set of parameters and (φ, A) be a bipolar fuzzy soft set on X = {x1 , x2 , x3 , x4 } with bipolar fuzzy approximation function φ : A → BF(X ) defined by φ(e1 ) = {(x1 , 0.3, −0.4), (x2 , 0.5, −0.2), (x3 , 0.7, −0.1), (x4 , 0.9, 0.0)}, φ(e2 ) = {(x1 , 0.7, −0.1), (x2 , 0.3, −0.4), (x3 , 0.4, −0.4), (x4 , 0.6, −0.2)}. Let (ψ, A) be a bipolar fuzzy soft relation in X with bipolar fuzzy approximation function ψ : A → BF(X × X ) defined by ψ(e1 ) = {(x1 x2 , 0.2, −0.1), (x2 x3 , 0.4, −0.1), (x1 x3 , 0.3, −0.1), (x2 x4 , 0.2, 0.0), (x3 x4 , 0.4, 0.0)}, ψ(e2 ) = {(x1 x2 , 0.2, −0.1), (x2 x3 , 0.2, −0.3), (x1 x3 , 0.4, −0.1), (x1 x4 , 0.5, −0.1), (x2 x4 , 0.1, −0.2)}. Clearly, H (e1 ) = (φ(e1 ), ψ(e1 )) and H (e2 ) = (φ(e2 ), ψ(e2 )) are bipolar fuzzy graphs corresponding to the parameters e1 and e2 , respectively, as shown in Fig. 7.3.

7.3 Bipolar Fuzzy Soft Circuits

297

x2 (0.5, −0.2)

x1 (0.7, −0.1)

(0.4, −0.1)

1 0.

(0.2, 0.0)

(0.3, −0.1)

)

,− .4 (0

(0.4, 0.0) x4 (0.9, 0.0)

x3 (0.7, −0.1)

x2 (0.3, −0.4)

(0.2, −0.1)

(0.2, −0.1)

(0

)

,− .2 (0

3 0.

.5 ,−

x3 (0.4, −0.4)

0. 1

)

(0.1, −0.2)

x1 (0.3, −0.4)

x4 (0.6, −0.2)

H(e2 )

H(e1 )

Fig. 7.3 Bipolar fuzzy soft graph G = {H (e1 ), H (e2 )}. Table 7.3 Bipolar fuzzy soft vertex set φ x1 x2 e1 e2

(0.3, −0.4) (0.7, −0.1)

(0.5, −0.2) (0.3, −0.4)

Table 7.4 Bipolar fuzzy soft edge set ψ x1 x2 x2 x3 e1 e2

(0.2, −0.1) (0.2, −0.1)

(0.4, −0.1) (0.2, −0.3)

x3

x4

(0.7, −0.1) (0.4, −0.4)

(0.9, 0.0) (0.6, −0.2)

x1 x3

x1 x4

x2 x4

x3 x4

(0.3, −0.1) (0.4, −0.1)

(0.0, 0.0) (0.5, −0.1)

(0.2, 0.0) (0.1, −0.2)

(0.4, 0.0) (0.0, 0.0)

Hence, G = {H (e1 ), H (e2 )} is a bipolar fuzzy soft graph on X . The tabular representation of a bipolar fuzzy soft graph G = {H (e1 ), H (e2 )} is given in Tables 7.3 and 7.4. Definition 7.27 Let G = (φ, ψ, A) be a bipolar fuzzy soft graph on X and t = (t p , t n ) ∈ [0, 1] × [−1, 0]. For any e ∈ A, define t-cut of a bipolar fuzzy set φ(e) p as φ(e) t = {x ∈ X | μφ(e) (x) ≥ t p , μnφ(e) (x) ≤ t n }. Then (φ t , A) = {φ(e) t | e ∈ A} is called a t-cut of a bipolar fuzzy soft set (φ, A). The soft graph G t = (φ t , ψ t , A) is called a t-level ! soft graph of G, where (ψ t , A) = {ψ(e) t "| e ∈ A} is a t-cut of (ψ, A) p and ψ t (e) = x y ∈ E | μψ(e) (x y) ≥ t p , μnψ(e) (x y) ≤ t n . Definition 7.28 A bipolar fuzzy soft matroid on a non-empty finite set X is a 3-tuple M(X ) = (X, I, A), where A is the set of parameters and (I, A) is a soft family of bipolar fuzzy subsets satisfying the following conditions. For each e ∈ A, 1. If β1 (e) ∈ I(e), β2 (e) ∈ BF(X ) and β2 (e) ⊂ β1 (e), then β2 (e) ∈ I(e). 2. If β1 (e), β2 (e) ∈ I(e) and |supp (β1 (e))| < |supp (β2 (e))|, then there exists β3 (e) ∈ I(e) such that a. β1 (e) ⊂ β3 (e) ⊆ β1 (e) ∪ β2 (e), p b. m p (β3 (e)) ≥ m p (β1 (e)) ∧ m p (β2 (e)), m p (βi (e)) = min{μβi (e) (y)|y ∈ supp (βi (e))}, i = 1, 2, 3,

298

7 Bipolar Fuzzy Circuits

c. m n (β3 (e)) ≤ m n (β1 (e)) ∨ m n (β2 (e)), supp (βi (e))}, i = 1, 2, 3.

m n (βi (e)) = max{μnβi (e) (y)|y ∈

That is, for each e ∈ A, the pair Me (X ) = (X, I(e)) is a bipolar fuzzy matroid on X . (I, A) is a family of independent bipolar fuzzy soft subsets of M(X ). Note 7.3 For each e ∈ A, {α(e) | α(e) ∈ BF(X ), α(e) ∈ / I(e)} is the family of dependent bipolar fuzzy subsets in M(e). Let Cr (M(e)) be a family of minimal dependent bipolar fuzzy subsets in Me (X ), then Cr (M(e)) = {Cr (M(e)) | e ∈ A} is called a bipolar fuzzy soft circuit. Equivalently, a bipolar fuzzy soft set (α, A) is a bipolar fuzzy soft circuit in M if (α, A) ∈ / (I, A) and, for each e ∈ A, if β(e) ⊂ α(e) then β(e) ∈ I(e). Since the elements of (I, A) are those bipolar fuzzy soft sets which are mutually disjoint with the members of Cr (M(e)), a bipolar fuzzy soft matroid can be uniquely determined from Cr (M). Example 7.11 A simple example of a bipolar fuzzy soft matroid on a non-empty with set X with n elements, known as a uniform bipolar fuzzy soft matroid, is denoted by Uk,n = (X, I, A), where for each e ∈ A, I(e) = {β(e) ∈ BF(X ) | |supp(β(e))| ≤ k}. The bipolar fuzzy soft circuit of Uk,n is a bipolar fuzzy soft subset (α, A) such that for each e ∈ A, |supp(α(e))| = k + 1. Consider the example of a uniform bipolar fuzzy matroid U2,3 = (X, I, A) = {(X, I(e1 )), (X, I(e2 ))}, where X = {y1 , y2 , y3 }, A = {e1 , e2 }, I(ei ) = {β(ei ) ∈ BF(X ) | |supp(β(ei ))| ≤ 2} and each β(ei ) ∈ BF(X ), i = 1, 2, is defined as β(e1 )(y) = {(y1 , 0.2, −0.3), (y2 , 0.4, −0.5), (y3 , 0.1, −0.3)} , β(e2 )(y) = {(y1 , 0.3, −0.2), (y2 , 0.5, −0.4), (y3 , 0.3, −0.1)} . I(e1 ) ={∅, {(y1 , 0.2, −0.3)}, {(y2 , 0.4, −0.5)}, {(y3 , 0.1, −0.3)}, {(y1 , 0.2, −0.3), (y2 , 0.4, −0.5)}, {(y2 , 0.4, −0.5), (y3 , 0.1, −0.3)}, {(y1 , 0.2, −0.3), (y3 , 0.1, −0.3)}} I(e2 ) ={∅, {(y1 , 0.3, −0.2)}, {(y2 , 0.5, −0.4)}, {(y3 , 0.3, −0.1)}, {(y1 , 0.3, −0.2), (y2 , 0.5, −0.4)}, {(y2 , 0.5, −0.4), (y3 , 0.3, −0.1)}, {(y1 , 0.3, −0.2), (y3 , 0.3, −0.1)}}.

The bipolar fuzzy soft circuit of U2,3 is {α(e1 ), α(e2 )}, where α(e1 ) = {(y1 , 0.2, −0.3), (y2 , 0.4, −0.5), (y3 , 0.1, −0.3)} and α(e2 ) = {(y1 , 0.3, −0.2), (y2 , 0.5, −0.4), (y3 , 0.3, −0.1)}. Example 7.12 A significant class of bipolar fuzzy soft matroids derived from bipolar fuzzy soft graphs is known as a cycle bipolar fuzzy soft matroid M = (X, I, A). For each e ∈ A, I(e) = {β(e) | supp(β(e)) is not the edge set of a cycle}. (α, A) is a bipolar fuzzy soft circuit if, for each e ∈ A, supp(α(e)) is the edge set of a cycle. Theorem 7.9 Let M = (X, I, A) be a bipolar fuzzy soft matroid and, for each t = (t p , t n ), 0 ≤ t p ≤ 1, −1 ≤ t n ≤ 0, define (I t , A) = {(β t , A) | (β, A) ∈ (I, A)}. Then (X, I t , A) is a soft matroid on X .

7.3 Bipolar Fuzzy Soft Circuits

299

Proof It is given that (X, I, A) is a bipolar fuzzy soft matroid on X . For each t = (t p , t n ) ∈ [0, 1] × [−1, 0], the t-cut of (I, A) is defined as (I t , A) = {(β t , A) | (β, A) ∈ (I, A)}, where (β t , A) = {β t (e) | e ∈ A}. To prove that (X, I t , A) is a soft matroid, it is to be shown that, for each e ∈ A, (X, I(e) t ) is a matroid. We prove conditions 1 and 2 of Definition 7.23. Assume that for each e ∈ A, β1 (e) t ∈ I(e) t and α(e) ⊆ β(e)1t . Define a bipolar fuzzy set β2 (e) ∈ BF(X ) as 

t, y ∈ α(e) . (0, 0), otherwise

β2 (e)(y) =

Clearly, β2 (e) ⊆ β1 (e), β2 (e) ∈ I(e) and β2 (e) t = α(e) therefore, α(e) ∈ I(e) t . To prove condition 2, let α1 (e), α2 (e) ∈ I(e) t and |α1 (e)| < |α2 (e)|. Then there exist β1 and β2 such that β1 (e) t = α1 (e) and β2 (e) t = α2 (e). Define βˆ1 (e) and βˆ2 (e) as   t, y ∈ α t, y ∈ α2 (e) (e) 1 βˆ1 (e)(y) = and βˆ2 (e)(y) = . (0, 0), otherwise (0, 0), otherwise It is clear that supp(βˆ1 (e)) ⊂ supp(βˆ2 (e)). As (X, I(e)) is a bipolar fuzzy matroid, there exists β3 (e) such that βˆ1 (e) ⊆ β3 (e) ⊆ βˆ1 (e) ∪ βˆ2 (e). Since  (βˆ1 (e) ∪ βˆ2 (e))(y) =

t, y ∈ α1 (e) ∪ α2 (e) . (0, 0), otherwise

Therefore, there exists a set α3 (e) such that  β3 (e)(y) =

t, y ∈ α3 (e) . (0, 0), otherwise

Also, α1 (e) ⊆ α3 (e) ⊆ α1 (e) ∪ α2 (e), α3 (e) ∈ I(e) t . Hence, (X, I(e) t ) is a matroid on X for each e ∈ A and so, (X, I t , A) is a soft matroid. Remark 7.4 Let M = (X, I, A) be a bipolar fuzzy soft matroid and, for each t = (t p , t n ), 0 ≤ t p ≤ 1, −1 ≤ t n ≤ 0, M t = (X, I t , A) is a soft matroid on a finite set X as given in Theorem 7.9. As X is finite, for every e ∈ A there is a finite sequence p p p S(e) = {t 0 , t 1 , t 2 , . . . , t m }, 0 < t1 < t2 < · · · < tm , 0 > t1n > t2n > · · · > tmn > −1 such that M(e) t i = (X, I(e) t i ) is a crisp matroid, for each 1 ≤ i ≤ m, and p

1. t 0 = (0, 0), tm ≤ 1 and tmn ≥ −1, p 2. I(e)w = ∅ if w = (w p , w n ), 0 ≤ w p ≤ tm and tmn ≤ w n ≤ 0, p p n p n p p , then I(e)w = 3. If for s = (s , s ), ti < w , s < ti+1 and tin > w n , s n > ti+1 I(e)s , 0 ≤ i ≤ m − 1, p p p n n > s n > ti+2 , then I(e)w ⊃ 4. If ti < w < ti+1 < s p < ti+2 and tin > w > ti+1 I(e)s , 0 ≤ i ≤ m − 2.

300

7 Bipolar Fuzzy Circuits

The sequence S(e) = {t 0 , t 1 , t 2 , . . . , t m } is known as a fundamental sequence of M(e) and the family of all such sequences S = {S(e) | e ∈ A} is called a soft fundamental sequence of M. Note that if t 0 , t 1 , t 2 , . . . , t m is the fundamental sequence for every M(e), e ∈ A, that is, the fundamental sequence is the same for every M(e), then t 0 , t 1 , t 2 , . . . , t m is called the fundamental sequence for the bipolar fuzzy soft matroid M = (X, I, A). Definition 7.29 A bipolar fuzzy soft matroid with fundamental sequence t 0 , t 1 , . . . , p t m is known as a closed bipolar fuzzy soft matroid if for each t = (t p , t n ), ti−1 < p n n p n t ≤ ti and ti−1 > t ≥ ti , I t = I t i . Theorem 7.10 If M = (X, I, A) is a closed fuzzy soft matroid, then each dependent bipolar fuzzy soft set in M properly contains a dependent bipolar fuzzy soft subset. Proof Let α be a dependent bipolar fuzzy soft set in M. Then for each e ∈ A, p p p α(e) is a dependent bipolar fuzzy set in M(e) and R p (α(e)) = {t1 , t2 , . . . , tm }, p p p n n n n n n n R (α(e)) = {t1 , t2 , . . . , tm }, where t1 < t2 < · · · < tm and t1 > t2 > · · · > tm . By / I(e) t i . Since M(e) is a Theorem 7.3, there exists t i , 1 ≤ i ≤ m, such that α(e) t i ∈ closed bipolar fuzzy matroid, there is an ε = (ε p , εn ), ε p > 0, εn < 0 such that for p p each t = (t p , t n ), ti − ε p ≤ t p ≤ ti , tin − εn ≥ t n ≥ tin , α(e) t = α(e) t i and I(e) t = I(e) t i . Let γ(e) be a proper bipolar fuzzy subset of α(e) defined by  (γ(e)) (y) =

p

p

t i − ε, μα(e) (y) ≥ ti and μnα(e) (y) ≤ tin . (0, 0), otherwise

/ I(e) t i −ε and by Theorem 7.3, γ(e) ∈ / I(e), for each e ∈ A. Note that γ(e) t i −ε ∈ Hence, γ ∈ / I. Corollary 7.3 Let M = (X, I, A) be a bipolar fuzzy soft matroid. If γ ∈ / I, then there exists a bipolar fuzzy soft circuit α such that α(e) ⊆ γ(e), for each e ∈ A. p

p

p

Proof For e ∈ A, let R p (γ(e)) = {t1 , t2 , . . . , tm } and Rn (γ(e)) = {t1n , t2n , . . . , tmn }. / I(e) t i . Let B(e) ⊆ γ(e) t i be a By Theorem 7.3, there exists t i such that γ(e) t i ∈ circuit in (X, I(e) t i ) and define α(e) ∈ BF(X ) as  (α(e)) (y) =

ti , if y ∈ B . (0, 0), otherwise p

Then α(e) ⊆ γ(e) and R p (α(e)) = {ti }, R n (α(e)) = {tin }, supp(α(e)) = B and hence α(e) is a bipolar fuzzy circuit, for each e ∈ A. Thus, α is a bipolar fuzzy soft circuit.

7.3 Bipolar Fuzzy Soft Circuits

301

Theorem 7.11 If α1 and α2 are bipolar fuzzy soft circuits in a bipolar fuzzy soft matroid M = (X, I, A) and α1 (e) ⊆ α2 (e), for each e ∈ A, then supp (α1 (e)) = supp (α2 (e)). Proof Let α1 and α2 be two bipolar fuzzy soft circuits in M = (X, I, A), then α1 (e) and α2 (e) are two bipolar fuzzy circuits in M(e) = (X, I(e)). On contrary, assume that supp (α1 (e)) ⊂ supp (α2 (e)), then there exists y ∈ X such that y ∈ supp (α2 (e)) \ supp (α1 (e)). Moreover, α1 (e) ⊆ α2 (e)\\y ∈ I(e) and hence, α1 (e) ∈ I(e), a contradiction to the supposition that α1 (e) is a bipolar fuzzy circuit, for each e ∈ A.

7.4 Decision Support Systems Bipolar fuzzy matroids have intriguing applications with regards to combinatorics, graph theory, algebra, and many domains in addition to Mathematics. Bipolar fuzzy matroids are important to study the uncertain and conflicting behavior of objects if the data have incomplete and bipolar information. In this section, the applications of bipolar fuzzy matroids are discussed for a decision support system for network analysis and the ordering of machines.

7.4.1 Ordering of Machines Using Bipolar Fuzzy Matroids The theory of bipolar fuzzy matroids have been found useful in decision support systems to find the ordering of m assignments to be fulfilled in k days if each assignment requires entire day consideration. All assignments are available at 0 time and each assignment has a profit p p associated with it, and a penalty p n to be paid if p it is not completed at deadline d. The profit p j can be achieved if each assignment j is finished at the cutoff time d j . The problem is basically to find a bipolar fuzzy ordering of assignments to maximize the total profit. It doesn’t look like a bipolar fuzzy matroid problem because the bipolar fuzzy matroid problem requires to find an optimal bipolar fuzzy subset, but this problem asks to find an ideal schedule. It is, however, a bipolar fuzzy matroid problem because the penalty and profit of any ordering can be obtained by a bipolar fuzzy subset of assignments that are on or before time. For a bipolar fuzzy subset S of deadlines {d1 , d2 , . . . , dn } corresponding to tasks T = {t1 , t2 , . . . , tn }, if there is a ordering such that every task in S is on or before time, and all tasks out of S are late. The procedure for the selection of tasks is given in Algorithm 7.4.1 whose time complexity is O(n 2 ).

302

7 Bipolar Fuzzy Circuits

Table 7.5 Decision Support System Task Deadline Profit Penalty t1 t2 t3 t4

d1 d2 d3 d4

0.8 0.5 0.6 0.9

0.7 0.6 0.8 0.5

Task

Deadline

Profit

Penalty

t5 t6 t7 t8

d5 d6 d7 d8

0.6 0.7 0.8 0.5

0.8 0.3 0.4 0.1

Algorithm 7.4.1 (Selection of tasks to maximize profit) 1. Input the deadline di corresponding to each task ti and set X = {(ti , di ) | 1 ≤ i ≤ n}. p 2. Input the profit pi and penalty pin corresponding to each task ti . n 3. do i from 1 → 2 4. supp(βi ) ∈ P(X ) 5. do j from 1 → |supp(βi )| 6. y j ∈ supp(βi ) p

μ (y j )+μn (y j )+1

βi cost(βi ) = cost(βi ) + βi 2 end do end do Calculate cost= max{cost(βi ) | 1 ≤ i ≤ 2n } Choose a subset βi such that cost = cost(βi ). Construct a bipolar fuzzy matroid M = (X, I) of tasks and deadlines such that I = {β | supp(β) ⊆ supp(βi )}. 13. To perform only |supp(β)| number of tasks on time, any bipolar fuzzy subset of |supp(β)| elements can be chosen from the bipolar fuzzy matroid to gain maximum profit.

7. 8. 9. 10. 11. 12.

As an example consider a set of eight tasks t1 , t2 , . . . , t8 with deadlines d1 , d2 , . . . , d8 . p If a task ti is completed on or before the deadline, it gains a profit pi , and if it is late the penalty to be paid is pin . The percentage profit and penalty associated with each task is given in Table 7.5. The family of a particular number of bipolar fuzzy tasks that maximize the profit is a bipolar fuzzy matroid. Any bipolar fuzzy subset S of p  p + pin + 1 tasks will maximize the profit if (ti ,di )∈S i is maximum. The bipolar 2 fuzzy matroid is given as {∅, {((t4 , d4 ), 0.9, −0.5)}, {((t6 , d6 ), 0.7, −0.3)}, {((t7 , d7 ), 0.8, −0.4)}, {((t8 , d8 ), 0.5, −0.1)}, {((t4 , d4 ), 0.9, −0.5), ((t6 , d6 ), 0.7, −0.3)}, {((t4 , d4 ), 0.9, −0.5), ((t7 , d7 ), 0.8, −0.4)}, {((t4 , d4 ), 0.9, −0.5), ((t8 , d8 ), 0.5, −0.1)}, {((t6 , d6 ), 0.7, −0.3), ((t7 , d7 ), 0.8, −0.4)}, {((t6 , d6 ), 0.7, −0.3), ((t8 , d8 ), 0.5, −0.1)}, {((t7 , d7 ), 0.8, −0.4), ((t8 , d8 ), 0.5, −0.1)}, {((t4 , d4 ), 0.9, −0.5), ((t6 , d6 ), 0.7, −0.3), ((t7 , d7 ), 0.8, −0.4)}, {((t4 , d4 ), 0.9, −0.5), ((t6 , d6 ), 0.7, −0.3), ((t8 , d8 ), 0.5, −0.1)}, ((t8 , d8 ), 0.5, −0.1)}, {((t6 , d6 ), 0.7, {((t4 , d4 ), 0.9, −0.5), ((t7 , d7 ), 0.8, −0.4), −0.3), ((t7 , d7 ), 0.8, −0.4), ((t8 , d8 ), 0.5, −0.1)}, {((t4 , d4 ), 0.9, −0.5), ((t6 , d6 ), 0.7, −0.3), ((t7 , d7 ), 0.8, −0.4), ((t8 , d8 ), 0.5, −0.1)}}

7.4 Decision Support Systems

303

If d1 < d2 < · · · < d8 , then the subset supp(S) = {d4 , d6 , d7 , d8 } will maximize the profit. If we want to perform only 3 tasks on time, then any bipolar fuzzy subset of three elements can be chosen from the bipolar fuzzy matroid and similarly.

7.4.2 Network Analysis Bipolar fuzzy matroids can be used in network analysis problems to determine the minimum number of connections for wireless communication. The procedure for the selection of minimum number of locations from a wireless connection is explained in the following steps. 1. Input the n number of locations L 1 , L 2 , . . . , L n of the wireless communication network. 2. Input the adjacency matrix ξ = [L i j ]n 2 of locations. 3. From this adjacency matrix, arrange the positive membership values in increasing order and negative membership values in decreasing order. 4. Select an edge having minimum positive membership value and maximum negative membership value. 5. Repeat Step 4 so that the selected edge does not create any circuit with previous selected edges. 6. Stop the procedure if the connection between every pair of locations is set up. The use of bipolar fuzzy matroids in network analysis is explained in the following example. Bipolar fuzzy graph in Fig. 7.4 represents a wireless communication between five locations L 1 , L 2 , L 3 , L 4 , L 5 . The positive degree of each edge shows the signal damage through this path, and negative degree indicates the strong signal strength in sending a message from one location to the other. Each pair of vertices is connected by an edge. But, in general, we do not need connections among all the vertices because the vertices linked indirectly will also have a message service between them, i.e., if there is a connection from L 2 to L 3 and L 3 to L 4 , then we can send a message from L 2 to L 4 even if there is no edge between L 2 and L 4 . The problem is to find a set of edges such that we are able to send a message between every two vertices under the condition that the signal disturbance is minimum. The procedure is as follows: Arrange the positive membership values of edges in increasing order and negative membership values in decreasing order as {(0.5, −0.28), (0.6, −0.33), (0.6, −0.37), (0.7, −0.41), (0.7, −0.44), (0.7, −0.46), (0.7, −0.48), (0.8, −0.5), (0.8, −0.51), (0.8, −0.53)}. At each step, select an edge having minimum positive membership value and maximum negative membership so that it does not create any circuit with previous selected edges. The bipolar fuzzy set of the selected edges is {(L 3 L 4 , 0.5, −0.28), (L 3 L 5 , 0.6, −0.33), (L 1 L 5 , 0.6, −0.37), (L 2 L 4 , 0.7, −0.41), (L 1 L 4 , 0.7, −0.46)}.

304

7 Bipolar Fuzzy Circuits

(0. 8, L5

.5) , −0 (0.8

7) 0.3 − , 6 (0.

−0 .51 )

L1

(0. 7,

−0 .46 ) L2

4) −0.4 (0.7, (0.

1) 0.4 − , 7

(0. 6,

(0.8 , −0 .53)

(0.7, −0.48)

−0 .33 )

(0.5, −0.28)

L4

L3

Fig. 7.4 Wireless communication

7.4.3 Ordering of Tasks Using Bipolar Fuzzy Soft Matroids The theory of bipolar fuzzy soft matroids is also useful in decision support systems to find the ordering of n products’ manufacturing in k days if all the products require equal time. Let e1 and e2 be two parameters under consideration for the ordering of tasks, where e1 represents profit (and loss) and e2 represents the demand (and supply). p 1. Each product has a profit p j (e1 ) if it is completed within the required time, n and a loss of p j (e1 ) if it is not completed within the deadline d j . p 2. Each product has a demand p j (e2 ) associated with it, and supply of p nj (e2 ) within the deadline d j . As an example, consider a set of four products P = {P1 , P2 , P3 , P4 } with deadlines d1 , d2 , d3 , d4 . The percentage profit, loss, demand, and supply associated with each product is given in Table 7.6. The family of a particular number of bipolar fuzzy soft sets that maximize the profit is a bipolar fuzzy soft matroid. If we consider that only two products can be manu-

Table 7.6 Bipolar fuzzy soft information Task Deadline Profit Loss P1 P2 P3 P4

d1 d2 d3 d4

0.8 0.5 0.6 0.9

0.7 0.6 0.8 0.5

Product

Deadline

Demand

Supply

P1 P2 P3 P4

d1 d2 d3 d4

0.6 0.7 0.8 0.5

0.8 0.3 0.4 0.1

7.4 Decision Support Systems

305

factured within d days, then a uniform bipolar fuzzy soft matroid can be constructed. The family of independent bipolar fuzzy subsets corresponding to parameters e1 and e2 are calculated as  I(e1 ) = β | β ⊂ {(P  3 , 0.5, −0.6), (P2 , 0.5, −0.6), (P3 , 0.6, −0.8), (P4 , 0.9, −0.5)}, |supp(β)| ≤ 2 ,  I(e2 ) = β | β ⊂ {(P  3 , 0.6, −0.8), (P2 , 0.7, −0.3), (P3 , 0.8, −0.4), (P4 , 0.5, −0.1)}, |supp(β)| ≤ 2 . Any bipolar fuzzy subset S(ei ) of products will maximize the profit if p  p j (ei )+ pnj (ei )+1 , i ∈ {1, 2}, is maximum. The problem can also be solved P j ∈S(ei ) 2 similarly if the numbers of days for completing the tasks are 3 or 4.

7.5 Conclusions In this chapter, we have discussed the notions of bipolar fuzzy vector spaces and bipolar fuzzy matroids and, linked major ideas of linear algebra and bipolar fuzzy graph theory in the form of linear bipolar fuzzy matroids, uniform bipolar fuzzy matroids, partition bipolar fuzzy matroids, and cycle bipolar fuzzy matroids. We put special emphasis on fundamental properties of bipolar fuzzy rank function, bipolar fuzzy circuits, and linear independence of bipolar fuzzy matroids. We have established the relation of crisp matroids and bipolar fuzzy matroids using (α, β)−cuts and studied the concept of the closure of bipolar fuzzy matroids, M−induced matroid sequence, fundamental sequence, and circuit rectangles. We have also introduced the concepts of matroids and circuits in a soft environment and a bipolar fuzzy soft environment. We have studied the importance of the presented concepts in decision support systems and network analysis.

Exercises 7 1. A bipolar fuzzy basis for a bipolar fuzzy matroid M = (X, I) is a maximal member in I. Let M be a fuzzy matroid, then M is closed if and only if for each β ∈ I, there is a bipolar fuzzy basis γ ∈ I such that β ⊆ γ. 2. If λr and λr are bipolar fuzzy rank functions of M = (X, I) and M = (X, I), respectively, then λr = λr . 3. Let M = (X, I) be a bipolar fuzzy matroid with fundamental sequence p p p t 0 , t 1 , . . . , t n . M is said to be regular if whenever for t i = (ti , tin ), ti < t j , n n ti > t j and β is a basis of (X, I t i ), there is a basis γ of (X, I t j ) such that γ ⊆ β. Prove that M is regular if and only if all fuzzy bases for M have the same cardinality.

306

7 Bipolar Fuzzy Circuits

4. Let M = (X, I) be a bipolar fuzzy matroid with fundamental sequence t 0 , t 1 , . . . , t n . If β is a bipolar fuzzy basis of M, then prove that β t i is a basis for (X, I t i ), for each 1 ≤ i ≤ n. 5. Let M = (X, I) be a closed regular bipolar fuzzy matroid with fundamental sequence t 0 , t 1 , . . . , t n . If for i < j, βi is a basis of (X, I t i ) and β j is a basis of (X, I t j ), then |βi | > |β j |.

References 1. Abdullah, S., Aslam, M., Ullah, K.: Bipolar fuzzy soft sets and its applications in decision making problem. J. Intell. Fuzzy Syst. 27(2), 729–742 (2014) 2. Akram, M.: Bipolar fuzzy graphs. Inf. Sci. 181(24), 5548–5564 (2011) 3. Akram, M.: Bipolar fuzzy graphs with applications. Knowl.-Based Syst. 39, 1–8 (2013) 4. Akram, M.: Bipolar fuzzy soft Lie algebras. Quasigroups Relat. Syst. 21(1), 1–10 (2013) 5. Akram, M., Dudek, W.A.: Regular bipolar fuzzy graphs. Neural Comput. Appl. 21(supp 1), 197–205 (2012) 6. Akram, M., Feng, F., Borumand Saeid, A., Leoreanu-Fotea, V.: A new multiple criteria decisionmaking method based on bipolar fuzzy soft graphs. Iran. J. Fuzzy Syst. 15(4), 73–92 (2018) 7. Akram, M., Alshehri, N., Davvaz, B., Ashraf, A.: Bipolar fuzzy digraphs in decision support systems. J. Mult.-Valued Log. Soft Comput. 27(5–6), 531–551 (2016) 8. Feng, F., Liu, X., Leoreanu-Fotea, V., Jun, Y.B.: Soft sets and soft rough sets. Inf. Sci. 181(6), 1125–1137 (2011) 9. Kharal, A., Ahmad, B.: Mappings on soft classes. New Math. Nat. Comput. 7(03), 471–481 (2011) 10. Goetschel Jr., R., Voxman, W.: Fuzzy matroids. Fuzzy Sets Syst. 27(3), 291–302 (1988) 11. Goetschel, Jr., R., Voxman, W.: Bases of fuzzy matroids. Fuzzy Sets Syst. 31(2), 253–261 (1989) 12. Goetschel, Jr., R., Voxman, W.: Fuzzy circuits. Fuzzy Sets Syst. 32(1), 35–43 (1989) 13. Goetschel, Jr., R., Voxman, W.: Fuzzy matroids and a greedy algorithm. Fuzzy Sets Syst. 37(2), 201–214 (1990) 14. Goetschel, Jr., R., Voxman, W.: Fuzzy matroid structures. Fuzzy Sets Syst. 41(3), 343–357 (1991) 15. Goetschel, Jr., R., Voxman, W.: Fuzzy rank functions. Fuzzy Sets Syst. 42(2), 245–258 (1991) 16. Goetschel, Jr., R., Voxman, W.: Fuzzy matroid sums and a greedy algorithm. Fuzzy Sets Syst. 52(2), 189–200 (1992) 17. Hsueh, Y.-C.: On fuzzification of matroids. Fuzzy Sets Syst. 53(3), 319–327 (1993) 18. Maji, P.K., Biswas, R., Roy, A.R.: Fuzzy soft sets. J. Fuzzy Math. 9(3), 589–602 (2001) 19. Molodtsov, D.: Soft set theory-first results. Comput. Math. Appl. 37(4–5), 19–31 (1999) 20. Novak, L.A.: A comment on “Bases of fuzzy matroids” Fuzzy Sets and Systems 31 (1989) 253–261. Fuzzy Sets Syst. 87(2), 251–252 (1997) 21. Novak, L.A.: On fuzzy independence set systems. Fuzzy Sets Syst. 91(3), 365–374 (1997) 22. Novak, L.A.: On Goetschel and Voxman fuzzy matroids. Fuzzy Sets Syst. 117(3), 407–412 (2001) 23. Roy, A.R., Maji, P.K.: A fuzzy soft set theoretic approach to decision making problems. J. Comput. Appl. Math. 203(2), 412–418 (2007) 24. Sarwar, M., Akram, M.: New applications of m-polar fuzzy matroids. Symmetry 9, 319 (2017). https://doi.org/10.3390/sym9120319

References

307

25. Sarwar, M., Akram, M.: Bipolar fuzzy circuits with applications. J. Intell. Fuzzy Syst. 34(1), 547–58 (2018) 26. Shi, F.-G.: A new approach to the fuzzification of matroids. Fuzzy Sets Syst. 160(5), 696–705 (2009) 27. Shi, F.-G.: (L , M)-fuzzy matroids. Fuzzy Sets Syst. 160(16), 2387–2400 (2009) 28. Whitney, H.: On the abstract properties of linear dependence. Am. J. Math. 57(3), 509–533 (1935) 29. Wilson, R.J.: An introduction to matroid theory. Am. Math. Mon. 80(5), 500–525 (1973) 30. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 31. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998) 32. Zhang, W.-R., Zhang, J.H., Shi, Y., Chen, S.S.: Bipolar linear algebra and YinYang-N-element cellular networks for equilibrium-based biosystem simulation and regulation. J. Biol. Syst. 17(4), 547–576 (2009)

Chapter 8

Energy of Bipolar Fuzzy Graphs

In this chapter, we study the concept of energy of bipolar fuzzy graphs and present certain formulae, lower and upper bounds of Laplacian energy, signless Laplacian energy, dominating energy, out-dominating energy, double dominating energy, and double out-dominating energy of bipolar fuzzy graphs and bipolar fuzzy digraphs. We elaborate the presented concept of energy and its extensions with numerical and graphical examples. Using bipolar fuzzy preference relations, we present multicriteria decision-making models based on the energy of bipolar fuzzy graphs in business partnerships and smooth communication problems. The main results of this chapter are from [3, 4, 14].

8.1 Introduction Several matrices can be associated with a graph such as the adjacency matrix (denoted by A) or the Laplacian matrix L = D − A, where D is the diagonal matrix of the degrees. The properties of the spectrum of a graph are related to the properties of a graph. The area of graph theory that deals with these concepts is called the spectral graph theory. There are numerous applications of graph spectra in Chemistry, Physics, Computer Science, Biology, and Mathematics: 1. The first mathematical paper on graph spectra was motivated by the membrane vibration problem, i.e., by approximative solving of partial differential equations. The main applications of graph spectra in Chemistry is the application in a theory of unsaturated conjugated hydrocarbons known as the Hckel ¨ molecular orbital theory. 2. In Physics: Treating the membrane vibration problem by approximative solving of the corresponding partial differential equation leads to the consideration of eigenvalues of a graph. The dimer problem is related to the investigation of the thermodynamic properties of a system of diatomic molecules (dimers) adsorbed on the surface of a crystal. The most favorable points for the adsorption of atoms on such © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_8

309

310

8 Energy of Bipolar Fuzzy Graphs

a surface form a two-dimensional lattice, and a dimer can occupy two neighboring points. A graph can be associated with a given adsorption surface. The vertices of the graph represent the points which are the most favorable for adsorption. Two vertices are adjacent if and only if the corresponding points can be occupied by a dimer. An arrangement of dimers on the surface determines a 1-factor in the corresponding graph, and vice versa. Thus, the dimer problem is reduced to the task of determining the number of 1-factors in a graph. The enumeration of 1-factors involves the consideration of walks in corresponding graphs and graph eigenvalues. 3. One of the oldest applications (from the 1970s) of eigenvalues in Computer Science is related to graphs called expanders. Avoiding a formal definition, we shall say that a graph has good expanding properties if each subset of the vertex set of small cardinality has a set of neighbors of large cardinality. Expanders and some related graphs (called enlargers, magnifiers, concentrators, and superconcentrators, just to mention some specific terms) appear in the treatment of several problems in Computer Science (for example, communication networks, error-correcting codes, optimizing memory space, computing functions, and sorting algorithms). 4. The largest eigenvalue (λ) of a matrix plays an important role in modeling virus propagation in computer networks. The smaller the largest eigenvalue, the larger the robustness of a network against the spread of viruses. In fact, it was shown that the epidemic threshold in spreading viruses is proportional to λ1 . Motivated by this fact, that graphs with minimal λ among other graphs are determined with a given number of vertices and edges, and having a given diameter. 5. Combinatorial matrix theory studies matrices with several digraphs. Many results and techniques from the theory of graph spectra can be applied for the foundation and development of matrix theory. The largest eigenvalue of a minimal spanning tree of the complete weighted graphs, with distances between cities serving as weights, can be used as a complexity index for the traveling salesman problem. 6. Networks appearing in biology have been analyzed by spectra of the normalized graph. Research and development networks (R&D networks) are studied by the largest eigenvalue of the adjacency matrix. A concept related to the spectrum of a graph is that of the energy of the graph. In relevance to π−electron energy of molecules, the concept of energy of a graph G was studied by Gutman [13], in 1978, as the sum of absolute values of the eigenvalues of the adjacency matrix of G. Some structural properties can be deduced from the spectrum but in general, we cannot determine the nature of a graph from its adjacency or Laplacian spectrum alone. The energy of a graph is very important in this sense. A lot of work, on the energy of graphs and its various types, has been done by Gutman [7–9]. Anjali and Mathew [5] introduced the concept of energy of fuzzy graphs as the sum of absolute values of the eigenvalues of the adjacency matrix of the fuzzy graph. Praba et al. [15] extended this concept to intuitionistic fuzzy graphs and discussed lower and upper bounds of the energy of intuitionistic fuzzy graphs. Sharbaf and Fayazi [16] studied the notions of the Laplacian spectrum and Laplacian energy in fuzzy graphs. Naz et al. [14] introduced the concept of energy under bipolar

8.1 Introduction

311

fuzzy environment and its applications to decision making problems. Akram et al. [3] discussed Laplacian and singless Laplacian energy of bipolar fuzzy graphs and bipolar fuzzy digraphs. For further terminologies and studies on energy of fuzzy graphs and their extensions, readers are referred to [1, 2, 6, 10–12, 17–19].

8.2 Energy of Bipolar Fuzzy Graphs If G is a graph (digraph) with n vertices and m edges, then its adjacency matrix A(G) is the n × n matrix whose i jth entry is the number of edges joining vertices i and j. The eigenvalues λ1 , λ2 , . . . , λn of the adjacency matrix of G are the eigenvalues of G. The set {λ1 , λ2 , . . . , λn } is the spectrum of G and is denoted by Spec(G). The eigenvalues of a graph satisfy the following relations. n 

λi = 0,

i=1

n 

λi2 = 2m.

i=1

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute n  values of the eigenvalues of G, i.e., E(G) = |λi |. A graph with all isolated vertices i=1

K nc has zero energy while the complete graph K n with n vertices has energy 2(n − 1). − → − → The energy of a digraph G , denoted by E( G ), is defined as the sum of the absolute n  − → − → values of the real part of eigenvalues of G , i.e., E( G ) = |Re(z i )|. i=1

Definition 8.1 The degree of a vertex x in a bipolar  fuzzy graph G is defined as a p p n p n pair deg(x) = (deg (x), deg (x)), where deg (x) = y∈X μ B (x y) and deg (x) =  n y∈X μ B (x y). Definition 8.2 Let G = (A, B) be a bipolar fuzzy graph on X . If all vertices of G have the same degree, then G is called a regular bipolar fuzzy graph , that is, deg(x) = deg(y), for all x, y ∈ X . Definition 8.3 The total degree of a vertex x in a bipolar fuzzy graph G is denoted by T deg(x) = (T deg p (x), T degn (x)) and defined as T deg p (x) =



p

p

p

μ B (x y) + μ A (x) = deg p (x) + μ A (x),

x y∈E

T degn (x) =



μnB (x y) + μnA (x) = degn (x) + μnA (x).

x y∈E

If each vertex of G has the same total degree, then G is called a totally regular bipolar fuzzy graph.

312

8 Energy of Bipolar Fuzzy Graphs u4 (0.2, −0.3)

u3 (0.5, −0.6)

(0 .2 ,− 0. 1)

.3) , −0 (0.4

u2 (0.4, −0.2)

(0 .2,

(0.1, −0.2)

(0.2 , −0 .1)

.1) −0 , .3 (0

−0 .3)

u5 (0.6, −0.4)

3) 0. − , .4 (0

u1 (0.7,

0.5)

Fig. 8.1 Bipolar fuzzy graph G

Definition 8.4 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . The adjacency matrix A(G) of G is defined as a square matrix A(G) = [ai j ]n×n , where p ai j = (μ B (u i u j ), μnB (u i u j )), u i , u j ∈ X . The adjacency matrix can also be written as A(G) = (A p (G), An (G)), where A p (G) and An (G) are the adjacency matrices corresponding to positive and negative membership values of the edges, respectively. Example 8.1 Consider a bipolar fuzzy graph G = (A, B) on X = {u 1 , u 2 , u 3 , u 4 , u 5 } as shown in Fig. 8.1. The adjacency matrix A(G) of Fig. 8.1 is shown in Eq. (8.1). ⎡

⎤ (0, 0) (0.2, −0.1) (0.4, −0.3) (0.2, −0.1) (0.4, −0.3) ⎢ (0.2, −0.1) (0, 0) (0.3, −0.1) (0, 0) (0, 0) ⎥ ⎢ ⎥ ⎥ . (8.1) (0.4, −0.3) (0.3, −0.1) (0, 0) (0.1, −0.2) (0, 0) ⎥ A(G) = ⎢ ⎢ ⎣ (0.2, −0.1) (0, 0) (0.1, −0.2) (0, 0) (0.2, −0.3) ⎦ (0.4, −0.3) (0, 0) (0, 0) (0.2, −0.3) (0, 0) Definition 8.5 Let G = (A, B) be a bipolar fuzzy graph on X with A p (G) and An (G) as the adjacency matrices of positive and negative membership values of the edges, respectively. If λ1 ≥ λ2 ≥ · · · ≥ λn are the eigenvalues of A p (G) in nonincreasing order then the spectrum of A p (G) is Spec p (G) = {λ1 , λ2 , . . . , λn }, called μ p -spectrum of G. If δ1 ≤ δ2 ≤ · · · ≤ δn are the eigenvalues of An (G) in nondecreasing order, then the spectrum of An (G) is Specn (G) = {δ1 , δ2 , . . . , δn }, called μn -spectrum of G. The spectrum of G is denoted by spec(G) and defined as a set Spec(G) = {(λ1 , δ1 ), (λ2 , δ2 ), . . . , (λn , δn )}. Definition 8.6 The energy of a bipolar fuzzy graph is defined as an ordered pair E(G) = (E p (G), E n (G)), where E p (G) and E n (G) are the sum of absolute values of eigenvalues ofA p (G) and An (G), respectively. It can also be written as E(G) =

n n   |λi |, |δi | . i=1

i=1

8.2 Energy of Bipolar Fuzzy Graphs

313

Theorem 8.1 Let G = (A, B) be a bipolar fuzzy graph with adjacency matrix p A(G) = [(ai j , ainj )]n×n , then n 

1.

i=1 n 

2.

λi = 0, (λi )2 =

i=1 n 

3.

(δi )2 =

i=1

n 

δi i=1 n n   i=1 j=1 i= j n  n  i=1 j=1 i= j

= 0, p

p

(ai j )(a ji ),

(ainj )(a nji ).

Proof 1. As the sum of eigenvalues of a matrix is equal to trace of the matrix, n 

λi ,

i=1

n 

δi

=



p

(aii , aiin ) = (0, 0).

u i ∈X

i=1

2. Similarly, the sum of squares of eigenvalues of the matrices A p (G) and An (G) are equal to the trace of (A p (G))2 and (An (G))2 , respectively, so n 

 p p p p p p p (λi )2 = (a11 )2 + a12 × a21 + a13 × a31 + · · · + a1n × an1 ]

i=1

 p p p p p p p + a21 × a12 + (a22 )2 + a23 × a32 + · · · + a2n × an2 ] .. .  p p p p p p p 2 ) ] + an1 × a1n + an2 × a2n + an3 × a3n + · · · + (ann =

n n  n   p p p (aii )2 + (ai j )(a ji ). i=1

=

i=1 j=1 i= j

n n   p p (ai j )(a ji )

p

∵ aii = 0, for all u i ∈ X.

i=1 j=1 i= j

Condition 3 can be proved on the same lines as of that condition 2.



Example 8.2 The spectrum and energy of a bipolar fuzzy graph G, given in Fig. 8.1, are as follows: Spec(G) = {(−0.5661, −0.6219), (−0.2767, −0.1029), (−0.1504, 0.0814), (0.2075, 0.1361), (0.7857, 0.5074)},

E(G) = (1.9864, 1.4498).

Further,

5 

λi = −0.5661 − 0.2767 − 0.1504 +

i=1

0.2075 + 0.7857 = 0, 5 5   δi = −0.6219 − 0.1029 + 0.0814 + 0.1361 + 0.5074 = 0. λi2 = 1.0800 =

i=1

i=1

314

8 Energy of Bipolar Fuzzy Graphs

2(0.54) = 2 5  i=1

n  5  i=1 j=1 i= j

p

p

(ai j )(a ji ),

δi2 = 0.6800 = 2(0.34) = 2

n  5  i=1 j=1 i= j

(ainj )(a nji ).

The upper and lower bounds of energy of a bipolar fuzzy graph G in terms of the number of vertices and the sum of squares of positive membership and negative membership values of edges are discussed in the following theorems. Theorem 8.2 Let G = (A, B) be a bipolar fuzzy graph with n vertices. If A(G) = p p [(ai j , ainj )]n×n , Q = det(A p (G)) = det([ai j ]n×n ), R = det(An (G)) = det([ainj ]n×n ), then    2 p p p p 1. (ai j )(a ji ) + n(n − 1)Q n ≤ E p (G) ≤ n (ai j )(a ji ). i= j

2.



i= j

i= j

(ainj )(a nji )

  p + n(n − 1)R ≤ E (G) ≤ n (ainj )(a ji ). 2 n

n

i= j

Proof The Cauchy Schwarz inequality is

n 

2 u i vi

i=1

≤

n  i=1



u i2

n  i=1

vi2

 .

1. In Cauchy Schwarz inequality, substituting u i = 1, vi = |λi | and using Theorem 8.1, it follows that

 n

2 |λi |

i=1

⇒ E p (G) =

≤



   n n  p p 1 (λi )2 = n (ai j )(a ji ) i=1

n 

i=1

  p p |λi | ≤ n (ai j )(a ji ) i= j

i=1

n 

i= j

2 |λi |

i=1

=

n 

|λi |2 +



|λi ||λ j |.

(8.2)

i= j

i=1

= j, 1 ≤ i, j ≤ n. Clearly, the arithConsider the set of n(n − 1) values |λi ||λ j |, i 1 |λi ||λ j | and the geometric mean is metic mean of these values is A.M = n(n−1) G.M =



i= j

i= j

1

n(n−1)

|λi ||λ j |

. As arithmetic mean is always greater than or equal to

geometric mean,

⎛ ⎞ 1 n n

2

1 n(n−1) n n(n−1)     2 1 2(n−1) ⎝ ⎠ |λi ||λ j | ≥ |λi ||λ j | = |λi | = |λi | = Qn. n(n − 1) i= j

i= j

i=1

i=1

(8.3) Equation (8.2) takes the form:

8.2 Energy of Bipolar Fuzzy Graphs

n 

2 |λi |

i=1

⇒ E p (G) =

≥

n 

315



2

|λi |2 + n(n − 1)Q n

2

p

i= j

i=1

n 

p

(ai j )(a ji ) + n(n − 1)Q n

 2 p p |λi | ≥ (ai j )(a ji ) + n(n − 1)Q n . i= j

i=1



The other case can be proven with similar arguments.

Theorem 8.3 Let G = (A, B) be a bipolar fuzzy graph with n vertices and A p (G) = p [ai j ]n×n , An (G) = [ainj ]n×n ) be the adjacency matrices of positive and negative mem  p (ai j )2 and n ≤ 2 (ainj )2 , then bership values G. If n ≤ 2 1≤i< j≤n

1≤i< j≤n

 ⎧ 

2 ⎫  ⎬ ⎨    2  p p p , (ai j )2 + (n − 1) 2 (ai j )2 − (ai j )2 1. E p (G) ≤ n2 ⎭ ⎩ 1≤i< j≤n n 1≤i< j≤n 1≤i< j≤n  ⎧ 

2 ⎫  ⎨ ⎬    2 2  2. E n (G) ≤ (ainj )2 + (n − 1) 2 (ainj )2 − (ainj )2 . ⎩ 1≤i< j≤n ⎭ n 1≤i< j≤n n 1≤i< j≤n

Proof If A = [ai j ]n×n is a symmetric matrix with zero trace then λmax ≥

2 n

 1≤i< j≤n

ai j , where λmax is the maximum eigenvalue If A(G) is the adjacency matrix of  of A. p ai j . By Theorem 8.1, a bipolar fuzzy graph G then λ1 ≥ n2 1≤i< j≤n

n 

λi2 = 2

i=1 n 



p

(ai j )2

1≤i< j≤n

λi2 = 2

i=2



p

(ai j )2 − λ21 .

(8.4)

1≤i< j≤n

Applying the Cauchy–Schwarz inequality to the vectors (1, 1, . . . , 1) and (|λ1 |, |λ2 |, . . . , |λn |) with n − 1 entries, we obtain E p (G) − λ1 =

n 

  n   |λi | ≤ (n − 1) |λi |2 .

i=2

i=2

From Eqs. (8.4) and (8.5),  ⎛    E p (G) − λ1 ≤ (n − 1) ⎝2

 1≤i< j≤n

⎞ (ai j )2 − λ21 ⎠ p

(8.5)

316

8 Energy of Bipolar Fuzzy Graphs  ⎛    p E (G) ≤ λ1 + (n − 1) ⎝2

⎞



p (ai j )2

− λ21 ⎠.

(8.6)

1≤i< j≤n



The function F(x) = x + (n − 1) 2

2 n

1≤i< j≤n

1





2

1≤i< j≤n

p (ai j )2

!



, 2

1≤i< j≤n

1  2 p (ai j )2

p (ai j )2

−

x2

decreases on the interval 

. Also, n ≤ 2

1≤i< j≤n

p (ai j )2 , 1



2

≤

p

1≤i< j≤n

(ai j )2

n

therefore,   2  n



p

(ai j )2 ≤

1≤i< j≤n

2 n



p

(ai j )2 ≤

1≤i< j≤n

2 n



 p

ai j ≤ λ1 ≤



2

1≤i< j≤n

p

(ai j )2 .

1≤i< j≤n

Equation (8.6) takes the form as

E p (G) ≤

2 n

 ⎧ ⎛   ⎨    2 p 2 p 2 (ai j ) + (n − 1) 2 (ai j ) − ⎝ ⎩ n

1≤i< j≤n

1≤i< j≤n

 1≤i< j≤n

⎞2 ⎫ ⎬ p (ai j )2 ⎠ . ⎭ 

The other part can be proved similarly.

− → − → Definition 8.7 Let G = (A, B ) be a bipolar fuzzy digraph with n vertices and − → − → p A p ( G ) = [ai j ]n×n and An ( G ) = [ainj ]n×n as the nonsymmetric adjacency matrices of positive and negative membership values, respectively. Let z 1 , z 2 , . . . , z n and − → − → w1 , w2 , . . . , wn be the eigenvalues of A p ( G ) and An ( G ), respectively. The energy − → − → − → − → of G is defined as an ordered pair E( G ) = (E p ( G ), E n ( G )) such that   − → − → E p( G ) = |Re(z i )| and E n ( G ) = |Re(wi )|. n

n

i=1

i=1

− → − → Example 8.3 Let G = (A, B ) be a bipolar fuzzy digraph on X = {u 1 , u 2 , u 3 , u 4 , u 5 } as given in Fig. 8.2. − → − → The corresponding adjacency matrix A( G ) of G is given in Eq. (8.7). ⎡

(0, 0) (0.6, −0.1) (0, 0) (0, 0) ⎢ (0, 0) (0, 0) (0.1, −0.2) (0, 0) ⎢ − → (0.2, −0.4) (0, 0) (0, 0) (0.2, −0.1) A( G ) = ⎢ ⎢ ⎣ (0.4, −0.1) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0.3, −0.2)

⎤ (0, 0) (0, 0) ⎥ ⎥ (0, 0) .⎥ ⎥ (0, 0) ⎦ (0, 0)

(8.7)

8.2 Energy of Bipolar Fuzzy Graphs

317 u3 (0.3, −0.5)

(0 .2,

u2 (0.9, −0.3)

(0.6, −0.1)

−0 .1)

(0.2, − 0.4)

.2) −0 , .1 (0

u4 (0.5, −0.2)

1) 0. − , .4 (0

u1 (0.7, −0.4)

(0.3, −0.2)

u5 (0.4,

0.7)

Fig. 8.2 Bipolar fuzzy digraph

− → The spectrum and energy of a bipolar fuzzy digraph G , given in Fig. 8.2, are − → Spec( G ) = {(0, 0), (0.3031, −0.2077), (−0.0432 + 0.2669i, 0.0914 + 0.1739i), − → (−0.0432 − 0.2669i, 0.0914 − 0.1739i), (−0.2166, 0.0250)} and E( G ) = (0.6061, 0.4154).

8.3 Laplacian Energy of Bipolar Fuzzy Graphs In this section, certain bounds and formulae of Laplacian energy and signless Laplacian energy of bipolar fuzzy graphs are presented. Definition 8.8 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . The p degree matrix of G is defined as a square matrix D(G) = [di j ]n×n = [di j , dinj ]n×n such that " deg(vi ), if i = j, vi ∈ X, di j = (0, 0) otherwise. Definition 8.9 The Laplacian matrix of a bipolar fuzzy graph G = (A, B) is p a matrix of the form L(G) = [li j , linj ]n×n = D(G) − A(G), where D(G) is the degree matrix of G and A(G) is the adjacency matrix G. The Laplacian spectrum L Spec(G) = (L Spec p (G), L Specn (G)) of G can be defined using the eigenvalues of L p (G) and L n (G) as explained in Definition 8.6. Example 8.4 Consider a bipolar fuzzy graph G = (A, B) on X = {v1 , v2 , v3 , v4 , v5 , v6 , v7 } as shown in Fig. 8.3. The adjacecy matrix, degree matrix, and Laplacian matrix of the bipolar fuzzy graph G are computed as:

318

8 Energy of Bipolar Fuzzy Graphs v2 (0.3, −0.9)

.4) , −0 (0.1

(0 .6,

0.3)

v1 (0.5, 0.6)

v7 (0.1,

−0 .2)

v6 (0.8, −0.4)

.2) −0 , .1 (0

(0.1, −0.2 )

(0. 1,

) 0.3 ,− 1 . (0

−0 .3)

(0.1, −0.1)

0.3)

v4 (0.7,

5) 0. ,− 2 . (0

v3 (0.1, −0.7)

(0.1, −0.6)

5, (0.

v5 (0.6,

.2) −0

0.5)

Fig. 8.3 Bipolar fuzzy graph G ⎛

⎞ (0.1, −0.4) (0, 0) (0, 0) (0, 0) (0.1, −0.2) (0.1, −0.6) (0, 0) (0, 0) (0, 0) (0, 0) ⎟ ⎟ (0, 0) (0.1, −0.2) (0, 0) (0.1, −0.3) (0, 0) ⎟ ⎟ (0.1, −0.2) (0, 0) (0.5, −0.2) (0, 0) (0, 0) ⎟ ⎟ (0, 0) (0.5, −0.2) (0, 0) (0.6, −0.3) (0.1, −o.1) ⎟ ⎟ (0.1, −0.3) (0, 0) (0.6, −0.3) (0, 0) (0.1, −0.2) ⎠ (0, 0) (0, 0) (0.1, −0.1) (0.1, −0.2) (0, 0)

⎛

⎞ (0.4, −1.1) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0.3, −1.1) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) ⎟ ⎟ (0, 0) (0, 0) (0.4, −1.5) (0, 0) (0, 0) (0, 0) (0, 0) ⎟ ⎟ (0, 0) (0, 0) (0, 0) (0.6, −0.4) (0, 0) (0, 0) (0, 0) ⎟ ⎟ (0, 0) (0, 0) (0, 0) (0, 0) (1.2, −0.6) (0, 0) (0, 0) ⎟ ⎟ (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0.8, −0.8) (0, 0) ⎠ (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0.3, −0.5)

(0, 0) (0.2, −0.5) ⎜ (0.2, −0.5) (0, 0) ⎜ ⎜ (0.1, −0.4) (0.1, −0.6) ⎜ A(G) = ⎜ (0, 0) ⎜ (0, 0) ⎜ (0, 0) (0, 0) ⎜ ⎝ (0, 0) (0, 0) (0.1, −0.2) (0, 0)

⎜ ⎜ ⎜ ⎜ D(G) = ⎜ ⎜ ⎜ ⎜ ⎝

⎛

⎞ (0.4, −1.1) (−0.2, 0.5) (−0.1, 0.4) (0, 0) (0, 0) (0, 0) (−0.1, 0.2) ⎜ (−0.2, 0.5) (0.3, −1.1) (−0.1, 0.6) ⎟ (0, 0) (0, 0) (0, 0) (0, 0) ⎜ ⎟ ⎜ (−0.1, 0.4) (−0.1, 0.6) (0.4, −1.5) (−0.1, 0.2) ⎟ (0, 0) (−0.1, 0.3) (0, 0) ⎜ ⎟ ⎟. L(G) = ⎜ (0, 0) (0, 0) (−0.1, 0.2) (0.6, −0.4) (−0.5, 0.2) (0, 0) (0, 0) ⎜ ⎟ ⎜ (0, 0) (0, 0) (0, 0) (−0.5, 0.2) (1.2, −0.6) (−0.6, 0.3) (−0.1, 0.1) ⎟ ⎜ ⎟ ⎝ (0, 0) ⎠ (0, 0) (−0.1, 0.3) (0, 0) (−0.6, 0.3) (0.8, −0.8) 0, 0) (−0.1, 0.2) (0, 0) (0, 0) (0, 0) (−0.1, 0.1) (0, 0) (0.3, −0.5)

By usual calculations, L Spec p (G) = {0.0228, 0.1360, 0.2882, 0.4875, 0.5859, 0.6907, 1.7888}, L Specn (G)) = {−2.0102, −1.5903, −1.0182, −0.5622, −0.4730, −0.3027, −0.0434}, and L Spec(G) = {(0.0228, −2.0102), (0.1360, −1.5903), (0.2882, −1.0182), (0.4875, −0.5622), (0.5859, −0.4730), (0.6907, −0.3027), (1.7888, −0.0434)}. Theorem 8.4 Let& G = (A, B) be a bipolar fuzzy graph on X and L(G) = % p [li j ]n×n , [linj ]n×n be the Laplacian matrix of G. If α1 ≥ α2 ≥ · · · ≥ αn and β1 ≤ β2 ≤ · · · ≤ βn are the eigenvalues of L p (G) and L n (G), respectively, then 1.

n  i=1

αi =

n  i=1

deg p (vi ) ,

n  i=1

βi =

n  i=1

deg n (vi ),

8.3 Laplacian Energy of Bipolar Fuzzy Graphs

2.

n 

(αi )2 =

i=1

n  i=1

p

(dii )2 +

n  n  i=1 j=1 i= j

p

p

319

(li j )(l ji ),

n 

(βi )2 =

i=1

n  i=1

(diin )2 +

n  n  i=1 j=1 i= j

(linj )(l nji ).

Proof 1. The trace of L p (G) is equal to sum of its eigenvalues therefore, n 

αi = tr (L p (G)) =

i=1

n n n    p p p (dii − aii ) = dii = deg p (vi ). i=1

i=1

i=1

Similarly, it is so for the other case. 2. Generally, the Laplacian matrix L p (G) can be written as follows: ⎡

p

d11 p ⎢ l21 ⎢ L p (G) = ⎢ . ⎣ .. p

p

l12 p d22 .. . p

... ... .. .

p

l1n p l2n .. .

⎤ ⎥ ⎥ ⎥. ⎦

p

ln1 ln2 . . . dnn

The sum of squares of eigenvalues of the matrix L p (G) is equal to the trace of (L p (G))2 so, n 

αi2 = tr ((L p (G))2 )

i=1 n 

 p p p p p p p (αi )2 = (d11 )2 + l12 × l21 + l13 × l31 + · · · + l1n × an1 ]

i=1

 p p p p p p p + l21 × l12 + (d22 )2 + l23 × l32 + · · · + l2n × ln2 ] .. .  p p p p p p p 2 + ln1 × l1n + ln2 × l2n + ln3 × l3n + · · · + (dnn ) ] =

n n  n   p p p (dii )2 + (li j )(l ji ) i=1

=

i=1 j=1 i= j

n   p p p (dii )2 + 2 (li j )(l ji ). i=1

1≤i< j≤n

The other part corresponding to negative membership values can be easily proved.  Definition 8.10 Let % & G = (A, B) be a bipolar fuzzy graph on X and L(G) = p n [li j ]n×n , [li j ]n×n be the Laplacian matrix of G. Let α1 ≥ α2 ≥ · · · ≥ αn and β1 ≤ β2 ≤ · · · ≤ βn be the eigenvalues of L p (G) and L n (G), respectively. If

320

8 Energy of Bipolar Fuzzy Graphs

2 n

πi = αi −



p

li j and ϑi = βi −

1≤i< j≤n

2 n



linj ,

1≤i< j≤n

then the Laplacian energy of G is defined as an ordered pair L E(G) = (L E p (G), n n   |πi | and L E n (G) = |ϑi |. L E n (G)) such that L E p (G) = i=1

i=1

Example 8.5 The Laplacian energy of a bipolar fuzzy graph G = (A, B) shown in Fig. 8.3 is L E(G) = (2.6737, 4.0374). Corollary 8.1 Let % & G = (A, B) be a bipolar fuzzy graph on X and L(G) = p n [li j ]n×n , [li j ]n×n be the Laplacian matrix of G. Let α1 ≥ α2 ≥ · · · ≥ αn and p n β1 ≤ β2 ≤· · · ≤ βn be the eigenvalues of Ln (G) and L (G), respectively. If πi = p 2 2 li j and ϑi = βi − n li j , then αi − n 1≤i< j≤n

n 

1. 2. 3.

i=1 n  i=1 n  i=1

1≤i< j≤n

πi = 0 , (πi )2 = (ϑi ) = 2

n  i=1 n  i=1

n 

ϑi = 0,

i=1 p

(dii −

2 n

(diin

2 n

−





p

1≤i< j≤n



1≤i< j≤n

l i j )2 + 2 p l i j )2

+2

p

1≤i< j≤n n  1≤i< j≤n

p

(li j )(l ji ), (linj )(l nji ).

Example 8.6 Consider a bipolar fuzzy graph G = (A, B) on X = {v1 , v2 , v3 , v4 , v5 , v6 , v7 } as given in Fig. 8.3. Here, L E p (G) = 2.6737, L E n (G) = 4.0374, L E(G) = 7 Moreover, 7(2.6737, 4.0374). π = 0, l i=1 i=1 ϑi = 0, 7 7    p p p p 2 π = 2.0800 = (dii − 27 l i j )2 + 2 (li j )(l ji ), i=1 i n i=1

ϑl2 = 3.1199 =

i=1 7 

i=1

1≤i< j≤7

(diin −

2 7

 1≤i< j≤7

1≤i< j≤7

linj )2 + 2

 1≤i< j≤7

(linj )(l nji ).

Theorem 8.5 Let G = (A, B) be a bipolar fuzzy graph with n vertices. If L(G) = p [(li j , linj )]n×n is the Laplacian matrix of G, then 

2     n p 2 p 2  p p li j , 1. L E (G) ≤ 2n 1≤i< j≤n (li j ) + n i=1 deg (vi ) − n     n  n 2 n 2. L E (G) ≤ 2n 1≤i< j≤n (li j ) + n i=1 deg n (vi ) −

Proof The Cauchy Schwarz inequality is

n  i=1

2 u i vi

≤

1≤i< j≤n

2 n

n  i=1

2

 1≤i< j≤n



u i2

.

linj

n  i=1

vi2

 .

1. In Cauchy Schwarz inequality, substituting u i = 1, vi = |πi | and using Corollary 8.1, we obtain

8.3 Laplacian Energy of Bipolar Fuzzy Graphs n 

2 |πi |

i=1

Hence L E p (G) =

≤

n  i=1

321

⎛

n

n   2 p 2 1 (πi ) = n ⎝ (dii − n i=1



i=1

n 

    n p |πi | ≤ n (aii )2 + 2n



i=1

i=1

1≤i< j≤n

p l i j )2

+



⎞ p p (li j )(l ji )⎠ .

i = j

1≤i< j≤n

p

(li j )2 .

 Theorem 8.6 Let G = (A, B) be a bipolar fuzzy graph on X , then 

2   n  p 2 p 1  2  p p 1. L E (G) ≥ 2 li j , deg (vi ) − n 1≤i< j≤n (li j ) + 2 2. L E n (G) ≥ 2

i=1

   

n 2 1≤i< j≤n (li j )

Proof Take K =

n  i=1

p

(dii −

2 n

+



n 

1 2

1≤i< j≤n

deg n (vi ) −

i=1

p

1≤i< j≤n

l i j )2 + 2



2 n

 1≤i< j≤n

p

1≤i< j≤n

2 .

linj

p

(li j )(l ji ) then from Definition

8.10 and Corollary 8.1, n 

2 |πi |

=

i=1

n 



|πi |2 + 2

i=1

|πi π j | ≥ 2K .

1≤i< j≤n



Hence the result. Theorem 8.7 Let G = (A, B) be a bipolar fuzzy graph, then 1.

 ⎛    n  p p L E (G) ≤ |π1 | + (n − 1) ⎝2 1≤i< j≤n (li j )2 + i=1 deg p (vi ) −

2 n

2.

 ⎛    n  n L E (G) ≤ |ϑ1 | + (n − 1) ⎝2 1≤i< j≤n (linj )2 + i=1 deg n (vi ) −

2 n

 1≤i< j≤n

 1≤i< j≤n

⎞

2

− π12 ⎠.

p li j

⎞

2 linj

− ϑ21 ⎠.

Proof From Corollary 8.1, n 

πi2

i=2

=

n  i=1

p

(dii −

2 n

 1≤i< j≤n

p

l i j )2 + 2



p

p

(li j )(l ji ).

Applying the Cauchy–Schwarz inequality to vectors . . . , 1) and (|π1 |, |π2 |, . . . , |πn |) with n − 1 entries, it clearly follows that E (G) − |π1 | = p

n  i=2

  n   |πi | ≤ (n − 1) |πi |2 , i=2

(8.8)

1≤i< j≤n

(1, 1,

322

8 Energy of Bipolar Fuzzy Graphs

 ⎛  ⎛  n   ⎜ ⎝d p − 2 E p (G) − |π1 | ≤  (n − 1) ⎝ ii n i=1

 1≤i< j≤n

⎞

⎞2



li j ⎠ + 2 p

⎟ p p (li j )(l ji ) − π12 ⎠.

1≤i< j≤n

(8.9) As L p (G) is a symmetric matrix, required result is obtained.



Theorem 8.8 Let G = (A, B) be a k-regular bipolar fuzzy graph, then  & %  p p 1. L E (G) ≤ |π1 | + (n − 1) 2 1≤i< j≤n (li j )2 − π12 ,  & %  n 2. L E (G) ≤ |ϑ1 | + (n − 1) 2 1≤i< j≤n (linj )2 − ϑ21 . Proof Let G be a k−regular bipolar fuzzy graph, then deg(u i ) = k = (k p , k n ), for p 2 p li j , then each u i ∈ X . Without any loss of generality, assume that k = n 1≤i< j≤n  p p p deg(u i ) = dii = n2 li j . Substituting the value of dii , 1 ≤ i ≤ n, in Eq. (8.9), 1≤i< j≤n

required result is obtained.



The concept of the Laplacian energy of bipolar fuzzy digraphs is studied briefly as a generalization of the Laplacian energy of bipolar fuzzy graphs. − → − → Definition 8.11 Let G = (A, B ) be a bipolar fuzzy digraph on X . The out-degree − → − → − → − → − →p matrix of G is defined as D out ( G ) = ((D out ) p (G), (D out )n (G)) = ([ d i j ]n×n , − → − → − → [ d inj ]n×n ), where (D out ) p (G) and (D out )n (G) are nonsymmetric diagonal matrices such that " " p n deg− deg− − →n − →p → (vi ) if i = j, vi ∈ X → (vi ) if i = j, vi ∈ X G G and d i j = . d ij = 0 otherwise 0 otherwise − → − → Definition 8.12 The Laplacian matrix of a bipolar fuzzy digraph G = (A, B ) is − → − → − → − →p − → − → − → defined as L( G ) = (L p ( G ), L n ( G )) = ([ l i j ]n×n , [ l inj ]n×n ) = D out ( G ) − A( G ), − → − → where D out ( G ) and A( G ) are the out-degree matrix and adjacency matrix of the − → bipolar fuzzy digraph G . Definition 8.13 The spectrum of a Laplacian matrix of a bipolar fuzzy digraph − → − → − → − → G is defined as an ordered pair L Spec( G ) = (L Spec p ( G ), L Specn ( G )), where − → − → − → L Spec p ( G ) and L Specn ( G ) can be computed using the eigenvalues of L p ( G ) and − → L n ( G ), respectively, as explained in Definition 8.6. − → − →p − → − → Theorem 8.9 Let G = (A, B ) be a bipolar fuzzy digraph and L p ( G ) = [ l i j ]n×n − → − → − → and L n ( G ) = [ l inj ]n×n be the Laplacian matrices of G corresponding to positive

8.3 Laplacian Energy of Bipolar Fuzzy Graphs v1 (0.6, −0.8)

) 0.2 3, − . 0 (

−0 .3)

0.6)

(0.6, −0.2)

v2 (0.7, −0.8)

(0.4, −0.7) (0. 2,

v4 (0.9,

323

(0. 4,

(0.5,

0.7)

0.7)

(0.5,

v5 (0.5, −0.4) ) 0.1 3, − . 0 (

−0 .4)

v3 (0.8,

0.9)

− → Fig. 8.4 Bipolar fuzzy digraph G

and negative membership values. If ' α1 ≥ ' α2 ≥ · · · ≥ ' αn and ' μ1 ≤ ' μ2 ≤ · · · ≤ ' μn → → p − n − are the eigenvalues of L ( G ) and L ( G ), respectively, then n 

Re(' αi ) =

i=1

n   −  − →p →n l i j and l ij. Re(' μi ) = 1≤i, j≤n

i=1

1≤i, j≤n

− → − → Definition 8.14 The Laplacian energy of a bipolar fuzzy digraph G = (A, B ) is ( ) n n − → − → − → defined as an ordered pair L E( G ) = (L E p ( G ), L E n ( G )) = i=1 |θi |, i=1 |γi | such that θi = Re(' αi ) −

1  − 1  − →p →n l i j and γi = Re(' μi ) − l . n 1≤i, j≤n n 1≤i, j≤n i j

n − → − → Corollary n 8.2 Let G = (A, B ) be a bipolar fuzzy digraph on X then i=1 θi = 0 and i=1 γi = 0. The proof of Corollary 8.2 is a direct consequence of Definition 8.14. − → − → Example 8.7 Consider a bipolar fuzzy digraph G = (A, B ) on X = {v1 , v2 , v3 , v4 , v5 } as shown in Fig. 8.4. The adjacency matrix, out-degree matrix, and the Lapla− → cian matrix of G are given as follows: ⎛

⎞ (0, 0) (0.4, −0.7) (0, 0) (0, 0) (0.2, −0.3) ⎜ (0, 0) (0, 0) (0, 0) (0, 0) (0.3, −0.2) ⎟ ⎜ ⎟ − → ⎜ (0, 0) (0, 0) ⎟ A( G ) = ⎜ (0, 0) (0.5, −0.6) (0, 0) ⎟ ⎝ (0.6, −0.2) (0, 0) (0.5, −0.7) (0, 0) (0, 0) ⎠ (0, 0) (0, 0) (0.4, −0.4) (0.3, −0.1) (0, 0)

324

8 Energy of Bipolar Fuzzy Graphs

⎛

⎞ (0.6, −1.0) (0, 0) (0, 0) (0, 0) (0, 0) ⎜ (0, 0) (0.3, −0.2) (0, 0) (0, 0) (0, 0) ⎟ ⎜ ⎟ − → out ⎜ (0, 0) (0.5, −0.6) (0, 0) (0, 0) ⎟ D ( G ) = ⎜ (0, 0) ⎟ ⎝ (0, 0) (0, 0) (0, 0) (1.1, −0.9) (0, 0) ⎠ (0, 0) (0, 0) (0, 0) (0, 0) (0.7, −0.5) ⎛

⎞ (0.6, −1.0) (−0.4, 0.7) (0, 0) (0, 0) (−0.2, 0.3) ⎜ (0, 0) (0.3, −0.2) (0, 0) (0, 0) (−0.3, 0.2) ⎟ ⎜ ⎟ − → ⎜ (−0.5, 0.6) (0.5, −0.6) (0, 0) (0, 0) ⎟ L( G ) = ⎜ (0, 0) ⎟. ⎝ (−0.6, 0.2) (0, 0) (−0.5, 0.7) (1.1, −0.9) (0, 0) ⎠ (0, 0) (0, 0) (−0.4, 0.4) (−0.3, 0.1) (0.7, −0.5)

− → Here, L Spec p ( G ) = {0, 1.1410, 0.7385 + 0.4071i, 0.7385 − 0.4071i, 0.5820}, − → L Specn ( G ) = {0, −0.9724 + 0.0499i, −0.9724 − 0.0499i, −0.6276 + 0.3270i, − → −0.6276 − 0.3270i}, S L Spec( G ) = {(0, 0), (1.1410, −0.9724 + 0.0499i), (0.7385 + 0.4071i, −0.9724 − 0.0499i), (0.7385 − 0.4071i, −0.6276 + 0.3270i), n − → (0.5820, −0.6276 − 0.3270i)}, L E( G ) = (1.3960, 1.3296), i=1 θi = 0, and n γ = 0. i i=1

8.4 Signless Laplacian Energy of Bipolar Fuzzy Graphs In this section, the signless Laplacian energy of bipolar fuzzy graphs and bipolar fuzzy digraphs is discussed with certain important properties. Definition 8.15 The signless Laplacian matrix of a bipolar fuzzy graph G = (A, B) is defined as S L(G) = D(G) + A(G), where D(G) is a degree matrix of G and A(G) is the adjacency matrix of G. The signless Laplacian matrices correspondp ing to positive and negative membership values can be written as S L p (G) = [si j ]n×n n n and S L (G) = [si j ]n×n , respectively. Definition 8.16 The spectrum S L Spec(G) of signless Laplacian matrix can be p defined using the eigenvalues of S L p (G) = [si j ]n×n and S L n (G) = [sinj ]n×n as explained in Definition 8.6. Example 8.8 Consider a bipolar fuzzy graph G = (A, B) on X = {v1 , v2 , v3 , v4 , v5 , v6 } as shown in Fig. 8.5. The adjacency matrix, degree matrix, and signless Laplacian matrix of G are given as follows: ⎛

(0, 0) ⎜ (0.2, −0.5) ⎜ ⎜ (0, 0) A(G) = ⎜ ⎜ (0, 0) ⎜ ⎝ (0, 0) (0.2, −0.4)

(0.2, −0.5) (0, 0) (0, 0) ((0.1, −0.6) (0.1, −0.6) (0, 0) (0.3, −0.2) (0.2, −0.2) (0.2, −0.3) (0, 0) (0.3, −0.4) (0, 0)

(0, 0) (0, 0) (0.3, −0.2) (0.2, −0.3) (0.2, −0.2) (0, 0) (0, 0) (0.7, −0.2) (0.7, −0.2) (0, 0) (0.1, −0.2) (0.3, −0.4)

⎞ (0.2, −0.4) (0.3, −0.4) ⎟ ⎟ (0, 0) ⎟ ⎟ (0.1, −0.2) ⎟ ⎟ (0.3, −0.4) ⎠ (0, 0)

8.4 Signless Laplacian Energy of Bipolar Fuzzy Graphs

325 v3 (0.2, −0.9)

2) 0. ,− .2 (0

) −0.6 (0.1,

v2 (0.4, −0.7)

(0. 2,

v1 (0.3, −0.6) 4) 0. ,− .2 (0

v6 (0.4,

v4 (0.9, −0.3)

−0 .3)

(0.7, −0.2)

(0.3, −0.4)

(0 .2 ,− 0. 5)

(0.3, −0 .2)

.2) , −0 (0.1

(0.3,

0.5)

0.4)

v5 (0.8,

0.5)

Fig. 8.5 Bipolar fuzzy graph

⎞ (0.4, −0.9) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) ⎜ (0, 0) (1.1, −2.0) (0, 0) (0, 0) (0, 0) (0, 0) ⎟ ⎟ ⎜ ⎜ (0, 0) (0, 0) (0.3, −0.8) (0, 0) (0, 0) (0, 0) ⎟ ⎟ D(G) = ⎜ ⎜ (0, 0) (0, 0) (0, 0) (1.3, −0.8) (0, 0) (0, 0) ⎟ ⎟ ⎜ ⎝ (0, 0) (0, 0) (0, 0) (0, 0) (1.2, −0.9) (0, 0) ⎠ (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0.9, −1.4) ⎛

⎛

(0.4, −0.9) ⎜ (0.2, −0.5) ⎜ ⎜ (0, 0) S L(G) = ⎜ ⎜ (0, 0) ⎜ ⎝ (0, 0) (0.2, −0.4)

(0.2, −0.5) (0, 0) (0, 0) (0, 0) (1.1, −2.0) (0.1, −0.6) (0.3, −0.2) (0.2, −0.3) (0.1, −0.6) (0.3, −0.8) (0.2, −0.2) (0, 0) (0.3, −0.2) (0.2, −0.2) (1.3, −0.8) (0.7, −0.2) (0, 0) (0.7, −0.2) (1.2, −0.9) (0.2, −0.3) (0.3, −0.4) (0, 0) (0.1, −0.2) (0.3, −0.4)

⎞ (0.2, −0.4) (0.3, −0.4) ⎟ ⎟ (0, 0) ⎟ ⎟. (0.1, −0.2) ⎟ ⎟ (0.3, −0.4) ⎠ (0.9, −1.4)

By usual calculations, S L Spec p (G) = {0.2229, 0.3069, 0.4790, 0.8174, 1.1639, 2.2100}, S L Specn (G) = {−2.7434, −1.4384, −1.0168, −0.6964, −0.5432, −0.3618}, S L Spec(G) = {(0.2229, −2.7434), (0.3069, −1.4384), (0.4790, −1.0168), (0.8174, −0.6964), (1.1639, −0.5432), (2.2100, −0.3618)}. Theorem 8.10 Let G = (A, B) be a bipolar fuzzy graph on X with S L p (G) and S L n (G) as the signless Laplacian matrices corresponding to positive and negative + + + + + membership values of edges of G. If λ+ 1 ≥ λ2 ≥ · · · ≥ λn and μ1 ≤ μ2 ≤ · · · ≤ μn are the eigenvalues of S L p (G) and S L n (G), respectively, then 1. 2.

n i=1 n  i=1

λi+ =

(λi+ )2 =

n  i=1 n  i=1

deg p (vi ), p

(dii )2 +

n

n  n  i=1 j=1 i= j

i=1 p

μi+ =

(si j )2 ,

n 

deg n (vi ), vi ∈ X .

i=1

n 

i=1

(μi+ )2 =

n  i=1

(diin )2 +

n  n  i=1 j=1 i= j

(sinj )2 .

326

8 Energy of Bipolar Fuzzy Graphs

Definition 8.17 Let & G = (A, B) be a bipolar fuzzy graph on X and S L(G) = % p + n + [si j ]n×n , [si j ]n×n be the signless Laplacian matrix of G. Let λ+ 1 ≥ λ2 ≥ · · · ≥ λn + + p n and μ+ 1 ≤ μ2 ≤ · · · ≤ μn be the eigenvalues of S L (G) and S L (G), respectively. If 2  p 2  n si j and ϑi+ = μi+ − s , πi+ = λi+ − n 1≤i< j≤n n 1≤i< j≤n i j

then the signless Laplacian energy of G is defined as an ordered pair L E(G) = n n   (S L E p (G), S L E n (G)), where S L E p (G) = |πi+ | and S L E n (G) = |ϑi+ |. i=1

i=1

Example 8.9 The signless Laplacian energy of the bipolar fuzzy graph G = (A, B) shown in Fig. 8.5 is S L E(G) = (3.2811, 3.5964). Corollary 8.3 Let& G = (A, B) be a bipolar fuzzy graph on X and S L(G) = % p + + [si j ]n×n , [sinj ]n×n be the signless Laplacian matrix of G. Let λ+ 1 ≥ λ2 ≥ · · · ≥ λn + and μ+ · ≤ μ+ L p (G) and S L n (G), respectively. n be the eigenvalues of S 1 ≤ μ2 ≤ · ·  p + + si j and ϑi+ = μi+ − n2 sinj , then If πi = λi − n2 1≤i< j≤n

n 

1. 2. 3.

i=1 n  i=1 n  i=1

πi+ = 0 , (πi+ )2 (ϑi+ )2

= =

n  i=1 n  i=1

n  i=1

1≤i< j≤n

ϑi+ = 0,

p

(dii −

2 n

(diin

2 n

−

 1≤i< j≤n



1≤i< j≤n



p

si j )2 + 2 p si j )2

+2

1≤i< j≤n n  1≤i< j≤n

p

p

(si j )(s ji ), (sinj )(s nji ).

Example 8.10 Consider a bipolar fuzzy graph G = (A, B) on X  = {v1 , v2 , v3 , + , v , v } as given in Fig. 8.5. Here, S L E(G) = (3.2811, 3.5964), v 4 5 6 i=1 πi = 0, + i=1 ϑi = 0, 6 6    p p p p + 2 (dii − 26 si j )2 + 2 (si j )(s ji ), and i=1 (πi ) = 2.7733 = 6 i=1

(ϑi+ )2 = 4.3182 =

i=1 6 

i=1

1≤i< j≤6

(diin −

2 6

 1≤i< j≤6

1≤i< j≤6

sinj )2 + 2

 1≤i< j≤6

(sinj )(s nji ).

− → Definition 8.18 The signless Laplacian matrix of a bipolar fuzzy digraph G = − → − → − → − → p → → (A, B ) is defined as S L( G ) = (S L p ( G ), S L n ( G )) = ([− s i j ]n×n , [− s inj ]n×n ) = − → − → − → − → D out ( G ) − A( G ), where D out ( G ) and A( G ) are the out-degree matrix and adja− → cency matrix of bipolar fuzzy digraph G . Definition 8.19 The spectrum of a signless Laplacian matrix of G is defined as − → − → − → − → S L Spec( G ) = (S L Spec p ( G ), S L Specn ( G )), where S L Spec p ( G ) and S L Specn − → − → − → ( G ) can be computed using the eigenvalues of S L p ( G ) and S L n ( G ), respectively, as explained in Definition 8.6.

8.4 Signless Laplacian Energy of Bipolar Fuzzy Graphs

327

v4 (0.8, −0.3)

(0 .2,

) 0.1 ,− 5 . (0

.3) −0

−0 .3)

v3 (0.3, −0.7)

.3) −0

(0. 3,

(0.2, −0.4)

2, (0.

v1 (0.5, −0.4)

4, (0.

v6 (0.6,

−0 .1)

(0.4, −0.2)

(0.5, −0.1)

(0. 4,

−0 .2)

(0.1, −0.3)

v5 (0.7, −0.2)

(0.4, −0.2)

) 0.2 6, − . 0 (

(0. 1,

−0 .2)

v2 (0.4,

0.8)

0.5)

Fig. 8.6 Signless Laplacian energy of bipolar fuzzy digraph

− → − → − → p → Theorem 8.11 Let G = (A, B ) be a bipolar fuzzy digraph and S L p ( G ) = ([− s i j ]n×n − → − → → and S L n ( G ) = [− s inj ]n×n be the signless Laplacian matrices of G corresponding to '+ '+ μ+ μ+ positive and negative membership values. If ' λ+ 1 ≥ λ2 ≥ · · · ≥ λn and ' 1 ≤' 2 ≤ − → − → + p n ··· ≤ ' μn are the eigenvalues of S L ( G ) and S L ( G ), respectively, then n  i=1

Re(' λi+ ) =

n  −  −  p → → s i j and s inj . Re(' μi+ ) = 1≤i, j≤n

i=1

1≤i, j≤n

− → Definition 8.20 The signless Laplacian energy of a bipolar fuzzy digraph G = − → − → − → − → (A, defined )as an ordered pair L E( G ) = (L E p ( G ), L E n ( G )) = (n B ) +is  n + i=1 |θi |, i=1 |γi | such that 1  − 1  − p → → λi+ ) − μi+ ) − s i j and γi+ = Re(' s n. θi+ = Re(' n 1≤i, j≤n n 1≤i, j≤n i j n − → − → Corollary Let G = (A, B ) be a bipolar fuzzy digraph on X then i=1 θi+ = 0 n 8.4 + and i=1 γi = 0. The proof of Corollary 8.4 is a direct consequence of Definition 8.20. − → − → Example 8.11 Consider a bipolar fuzzy digraph G = (A, B ) on X = {v1 , v2 , v3 , v4 , v5 , v6 } as shown in Fig. 8.6. − → The adjacency matrix, out-degree matrix, and the signless Laplacian matrix of G are calculated as follows:

328

8 Energy of Bipolar Fuzzy Graphs

⎞ (0, 0) (0, 0) (0.2, −0.3) (0, 0) (0.4, −0.1) (0.4, −0.3) ⎜ (0.3, −0.2) (0, 0) (0.2, −0.4) (0, 0) (0, 0) (0, 0) ⎟ ⎟ ⎜ ⎜ (0, 0) − → (0.1, −0.3) (0, 0) (0.2, −0.2) (0, 0) (0, 0) ⎟ ⎟ A( G ) = ⎜ ⎜ (0.4, −0.2) (0, 0) (0.1, −0.3) (0, 0) (0.5, −0.1) (0, 0) ⎟ ⎟ ⎜ ⎝ (0, 0) (0, 0) (0, 0) (0.6, −0.2) (0, 0) (0.5, −0.1) ⎠ (0, 0) (0, 0) (0, 0) (0.4, −0.2) (0, 0) (0, 0) ⎛

⎛

⎞ (1.0, −0.7) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) ⎜ (0, 0) (0.5, −0.6) (0, 0) (0, 0) (0, 0) (0, 0) ⎟ ⎜ ⎟ ⎜ (0, 0) − → (0, 0) (0.3, −0.5) (0, 0) (0, 0) (0, 0) ⎟ out ⎟ D (G) = ⎜ ⎜ (0, 0) (0, 0) (0, 0) (1.0, −0.6) (0, 0) (0, 0) ⎟ ⎜ ⎟ ⎝ (0, 0) (0, 0) (0, 0) (0, 0) (1.1, −0.3) (0, 0) ⎠ (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0.4, −0.2) ⎛

⎞ (1.0, −0.7) (0, 0) (0.2, −0.3) (0, 0) (0.4, −0.1) (0.4, −0.3) ⎜ (0.3, −0.2) (0.5, −0.6) (0.2, −0.4) (0, 0) (0, 0) (0, 0) ⎟ ⎜ ⎟ ⎜ (0, 0) − → (0.1, −0.3) (0.3, −0.5) (0.2, −0.2) (0, 0) (0, 0) ⎟ ⎟. S L( G ) = ⎜ ⎜ (0.4, −0.2) (0, 0) (0.1 − 0.3) (1.0, −0.6) (0.5, −0.1) (0, 0) ⎟ ⎜ ⎟ ⎝ (0, 0) (0, 0) (0, 0) (0.6, −0.2) (1.1, −0.3) (0.5, −0.1) ⎠ (0, 0) (0, 0) (0, 0) (0, 0) (0.4, −0.2) (0.4, −0.2)

− → Here, S L Spec p ( G ) = {1.8150, 0.7995 + 0.2749i, 0.7995 − 0.2749i, 0.0889, − → 0.2520, 0.5451}, S L Spec p ( G ) = {−1.0790, −0.0509, −0.6616, −0.4613, − → −0.2834, −0.3639}, S L Spec(G) = {(1.8150, −1.0790), (0.7995 + 0.2749i, −0.0509), (0.7995 − 0.2749i, −0.6616), (0.0889, −0.4613), (0.2520, −0.2834), n − → + (0.5451, −0.3639)}, S L E( G ) = (1.9630, 1.5477), and i=1 θi = 0 n + δ = 0. i=1 i

8.5 Dominating and Double Dominating Energy of Bipolar Fuzzy Graphs In this section, the notions of dominating energy, double dominating energy, outdominating energy, and double out-dominating of bipolar fuzzy graphs and bipolar fuzzy digraphs are studied with certain lower and upper bounds. Definition 8.21 Let G = (A, B) be an bipolar fuzzy graph on a non-empty set X . For p p p any two vertices u and w in G, u dominates w if μ B (uw) = min{μ A (u), μ A (w)} and μnB (uw) = max{μnA (u), μnA (w)}. A subset D ⊆ X is a dominating set if for every vertex w ∈ / D, there is a vertex u ∈ D such that u dominates w. D is a minimal dominating set of G if, for any w ∈ X , D \ {w} is not a dominating set. The dominating set with minimum cardinality among all minimal dominating sets is called the minimum dominating set of G.

8.5 Dominating and Double Dominating Energy of Bipolar Fuzzy Graphs

329

v3 (0.6, −0.4)

(0 .5 ,− 0. 3)

2) 0. − 1, . (0 (0.2, −0.3) v1 (0.5,

(0.3, −0.4) v2 (0.3,

0.7)

(0.6, −0.1) v4 (0.7,

0.4)

0.5)

v5 (0.8,

0.3)

Fig. 8.7 Bipolar fuzzy graph G

Definition 8.22 Let D be a minimum dominating set of a bipolar fuzzy graph G = (A, B) then the dominating adjacency matrix of G is n × n matrix A D (G) = p p [ξi j ]n×n = [(ξi j , ξinj )]n×n = (A D (G), AnD (G)), where ⎧ ⎪ ⎨ B(u i u j ), u i = u j , ξi j = (0, 0), / D ui = u j , ui ∈ ⎪ ⎩ u i = u j , u i ∈ D. A(u i ), p

If μ1 ≥ μ2 ≥ · · · ≥ μn are the eigenvalues of A D (G) in nonincreasing order and ν1 ≤ ν2 ≤ · · · ≤ νn are the eigenvalues of AnD (G) in nondecreasing order, then the p p p μ p -spectrum of A D (G), denoted by Spec D (G), is defined as the set Spec D = n n {μ1 , μ2 , . . . , μn }. Similarly, the μn -spectrum of A D (G) is denoted by Spec D (G) and defined as the set SpecnD = {ν1 , ν2 , . . . , νn }. The spectrum of A D (G) is defined as a set Spec D (G) = {(μi , νi )|i = 1, 2, . . . , n}. Definition 8.23 Let D be a minimum dominating set of a bipolar fuzzy graph G then the minimum dominating energy corresponding to D is defined as an ordered n n   p p pair E D (G) = (E D (G), E nD (G)), where E D (G) = |μi | and E nD (G) = |νi |. If i=1

i=1

D is any minimal dominating set of G then E D (G) is called the dominating energy of G. Example 8.12 Consider the bipolar fuzzy graph G = (A, B) on X = {v1 , v2 , v3 , v4 , v5 } as shown in Fig. 8.7. Clearly, by calculations, D = {v2 , v4 } is a dominating set because every vertex in X − D is dominated by at least one vertex in D. The adjacency matrix of G is computed in Eq. (8.10). ⎛

⎞ (0, 0) (0, 0) (0.3, −0.4) (0, 0) ⎟ ⎟ (0.5, −0.3) (0, 0) ⎟ ⎟. (0.7, −0.5) (0.6, −0.1) ⎠ (0.6, −0.1) (0, 0) (8.10) Thus, Spec D (G) = {(−0.4334, −1.5156), (−0.0456, −0.6803), (0, 0), (0.8949, 0.0761), (1.5841, 0.1199)}, E D (G) = (2.958, 2.3919). (0, 0) ⎜ (0.2, −0.3) ⎜ A D (G) = ⎜ ⎜ (0, 0) ⎝ (0, 0) (0, 0)

(0.2, −0.3) (0, 0) (0.3, −0.4) (0.1, −0.2) (0.1, −0.2) (0, 0) (0.3, −0.4) (0.5, −0.3) (0, 0) (0, 0)

330

8 Energy of Bipolar Fuzzy Graphs

Theorem 8.12 Let G = (A, B) be a bipolar fuzzy graph with n vertices and minimum dominating set D = {w1 , w2 , . . . , wk }, then & % n n   1. A(wi ) = μi , νi , wi ∈D

2.

n 

(μi

i=1

)2

=

n  i=1

i=1 p (ξii )2

i=1

+

n  n 

p

p

(ξi j )(ξ ji )

n 

and

i=1 j=1 i= j

n 

(νi )2 =

i=1

i=1

(ξiin )2 +

n  n 

(ξinj )(ξ nji ).

i=1 j=1 i= j

Proof 1. As the sum of eigenvalues of a matrix is equal to trace of the matrix,

n 

μi ,

i=1

n 

νi

=

i=1

n 



ξii =

A(wi ).

wi ∈D

i=1

p

2. As the sum of squares of eigenvalues of a matrix A D (G) is equal to trace of p (A D (G))2 , n   p p p p p p p (μi )2 = (ξ11 )2 + ξ12 × ξ21 + ξ13 × ξ31 + · · · + ξ1n × ξn1 ] i=1

 p p p p p p p + ξ21 × ξ12 + (ξ22 )2 + ξ23 × ξ32 + · · · + ξ2n × ξn2 ] .. .  p p p p p p p 2 ) ] + ξn1 × ξ1n + ξn2 × ξ2n + ξn3 × ξ3n + · · · + (ξnn =

n n n    p p p (ξii )2 + (ξi j )(ξ ji ) i=1

=



i=1 j=1 i= j

(A(wi ))2 + 2

wi ∈D



p

(ξi j )2 .

1≤i< j≤n

The part of the theorem corresponding to negative membership values can be obtained similarly.  Theorem 8.13 Let G = (A, B) be a bipolar fuzzy graph with n vertices. If D is p p a minimum dominating set, Q = det(A D (G)) = det ([ξi j ]n×n ), and R = det(AnD n (G)) = det ([ξi j ]n×n ), then  n n    p p 2 p 1. (ξii )2 + (ξi j )(ξ ji ) + n(n − 1)Q n ≤ |μi |, i= j

i=1

2.

n  i=1



3.



 n   p p p |μi | ≤ n (ξii )2 + (ξi j )(ξ ji ) ,

n 

i=1

i=1

(ξiin )2 +

 i= j

i=1

i= j

2

(ξinj )(ξ nji ) + n(n − 1)Q n ≤

n  i=1

|νi |,

8.5 Dominating and Double Dominating Energy of Bipolar Fuzzy Graphs

4.

n  i=1

331



 n   |νi | ≤ n (ξiin )2 + (ξinj )(ξ nji ) . i= j

i=1

Proof The Cauchy Schwarz inequality is

n 

2 u i vi

≤

i=1

n  i=1



u i2

n  i=1

 vi2 .

2. In Cauchy Schwarz inequality, substituting u i = 1, vi = |μi | and using Theorem 8.12, we obtain

 n

2 |μi |

≤





   n n n  p p  p 1 (μi )2 = n (ξii )2 + (ξi j )(ξ ji )

i=1

i=1

i=1

i= j

i=1

   n n   p p   p ⇒ |μi | ≤ n (ξii )2 + (ξi j )(ξ ji ) i=1

i= j

i=1

n 

2 |μi |

i=1

=

n 

|μi |2 +



|μi ||μkj |.

(8.11)

i= j

i=1

Consider the set of n(n − 1) values |μi ||μkj |, i = j, 1 ≤ i, j ≤ n. Clearly, the arith 1 metic mean of these values is A.M = n(n−1) |μi ||μkj | and the geometric mean is G.M =

i= j

1

n(n−1)



i= j

|μi ||μkj |

. As arithmetic mean is always greater, than or equal to

geometric mean, ⎛ ⎞ 1 n n

2

1 n(n−1) n n(n−1)     2 1 k k ⎠ 2(n−1) ⎝ |μi ||μ j | ≥ |μi ||μ j | = |μi | = |μi | = Qn. n(n − 1) i= j

i= j

i=1

i=1

(8.12) Equation (8.11) takes the form as n 

2 |μi |

i=1

⇒

n  i=1

≥

n 

n   p p 2 p |μi | + n(n − 1)Q = (ξii )2 + (ξi j )(ξ ji ) + n(n − 1)Q n 2 n

2

i=1

i=1

i= j

   p p  n p 2 |μi | ≥  (ξii )2 + (ξi j )(ξ ji ) + n(n − 1)Q n . i=1

i= j

 Theorem 8.14 Let G = (A, B) be an bipolar fuzzy graph with adjacency matrix p A(G) = [u i j ]n×n = [(u i j , u inj )]n×n . Let λi , 1 ≤ i ≤ n, be the eigenvalues of adja-

332

8 Energy of Bipolar Fuzzy Graphs

cency matrix A p (G) and γi , 1 ≤ i ≤ n, be the eigenvalues of adjacency matrix An (G), then n 

2 |μi |

≤n

i=1

n n   p (ξii )2 + ( |λi |)2 i=1

i=1

and

n 

2 |νi |

≤n

i=1

n 

n  (ξiin )2 + ( |γi |)2 .

i=1

i=1

Proof Since u ii = (0, 0) and for i = j, u i j = ξi j , using Theorem 8.13, the following relation can be obtained

 n

2 |λi |

≥

 p p  p p 2 p (ξi j )(ξ ji ) + n(n − 1)[det Ak (G)] n ≥ (ξi j )(ξ ji ). i= j

i=1

(8.13)

i= j

From condition 2 of Theorem 8.13,

 n i=1

2 |μi |

≤n





 n n n  p p  ( )2 p p (ξii )2 + (ξi j )(ξ ji ) ≤ n (ξii )2 + |λi | . i=1

i= j

i=1

The remaining part can be proved on similar arguments as above.

i=1



− → Definition 8.24 An out-dominating set of a bipolar fuzzy digraph G = (A, B) is out out a subset of vertices D ⊆ X such that for each w ∈ / D , there is a vertex u ∈ D out p p p n n n which satisfies μ− → (uw) = min{μ A (u), μ A (w)} and μ− → (uw) = max{μ A (u), μ A B B − → (w)}. D out is a minimal out-dominating set of G if for every w ∈ D out , D out \ {w} is not an out-dominating set. The out-dominating set with minimum cardinality among all minimal out-dominating sets is called the minimum out-dominating set of G. Definition 8.25 Let D out be a minimum out-dominating set of an bipolar fuzzy − → − → − → digraph G = (A, B ), then the out-dominating adjacency matrix of G is an n × n − → − →p − →n − → matrix A Dout ( G ) = [ ξ i j ]n×n = [( ξ i j , ξ i j )]n×n , where ⎧− → ⎪ B (u i u j ), u i = u j , ⎨ − → ξ i j = (0, 0), / D out ui = u j , ui ∈ ⎪ ⎩ A(u i ), u i = u j , u i ∈ D out . − → p If τ1 ≥ τ2 ≥ · · · ≥ τn are the eigenvalues of A Dout ( G ) in nonincreasing order and − → ρ1 ≤ ρ2 ≤ · · · ≤ ρn are the eigenvalues of AnDout ( G ) in nondecreasing order, then the − → − → p p spectrum of A Dout ( G ), denoted by Spec Dout ( G ), is defined as the set Spec Dout = − → − → p {τ1 , τ2 , . . . , τn }. The spectrum of A Dout ( G ), denoted by Spec Dout ( G ), is defined n as the set Spec Dout = {ρ1 , ρ2 , . . . , ρn }. The spectrum of G is the set Spec Dout = {(τi , ρi )|i = 1, 2, . . . , n}.

8.5 Dominating and Double Dominating Energy of Bipolar Fuzzy Graphs

333

− → Definition 8.26 The out-dominating energy of a bipolar fuzzy digraph G corren  − → sponding to a minimum out-dominating set D out is defined as E Dout ( G ) = ( |τi |, n 

i=1

|ρi |).

i=1

Definition 8.27 Let G = (A, B) be a bipolar fuzzy graph on a non-empty set X . A + ⊆ X is a double dominating set of G if every vertex w ∈ + is dominated subset D / D + + by at least two vertices in D. D is a minimal double dominating set of G if, for + \ {w} is not a double dominating set. The double dominating set any w ∈ X , D with minimum cardinality among all minimal double dominating sets is called the minimum double dominating set of G. + be a minimum double dominating set of a bipolar fuzzy Definition 8.28 Let D graph G, then the double dominating adjacency matrix of G is an n × n matrix p p A D+ (G) = (A D+ (G), AnD+ (G)) = [(ξi j , ξinj )]n×n , where ⎧ ⎪ ⎨ B(u i u j ), u i = u j , + ξi j = (0, 0), / D ui = u j , ui ∈ ⎪ ⎩ + u i = u j , u i ∈ D. A(u i ), p

If φ1 ≥ φ2 ≥ · · · ≥ φn are the eigenvalues of A D+ (G) in nonincreasing order, then the p p spectrum of A D+ (G) is defined as the set Spec D+ = {φ1 , φ2 , . . . , φn }. If ψ1 ≤ ψ2 ≤ · · · ≤ ψn are the eigenvalues of AnD+ (G), then the spectrum of AnD+ (G) is defined as the set SpecnD+ = {ψ1 , ψ2 , . . . , ψn }. + be a minimum double dominating set of a bipolar fuzzy graph Definition 8.29 Let D + is defined as an ordered G, then the double dominating energy corresponding to D n n   p n pair E D+ (G) = (E D+ (G), E D+ (G)) such that E D+ (G) = ( |φi |, |ψi |). i=1

i=1

Example 8.13 Consider a bipolar fuzzy graph G = (A, B) on X = {v1 , v2 , v3 , v4 , v5 , v6 } as shown in Fig. 8.8. +1 = {v1 , v3 , v5 } and D +2 = By usual calculations, it is easy to check that D {v2 , v4 , v6 } are double dominating sets of G. The adjacency matrix corresponding to +1 is calculated as follows: D ⎛

(1, −1) (0.4, −0.5) (0, 0) (0, 0) (0, 0) ⎜ (0.4, −0.5) (0, 0) (0.3, −0.1) (0, 0) (0, 0) ⎜ ⎜ (0, 0) (0.3, −0.1) (1, −1) (0.2, −0.1) (0, 0) ⎜ AD +1 (G) = ⎜ (0, 0) (0.2, −0.1) (0, 0) (0.1, −0.3) ⎜ (0, 0) ⎝ (0, 0) (0, 0) (0, 0) (0.1, −0.3) (1, −1) (0, 0) (0.2, −0.2) (0.6, −0.1) (0.3, −0.4) (0.7, −0.3)

⎞ (0.3, −0.4) (0.7, −0.3) ⎟ ⎟ (0, 0) ⎟ ⎟. (0.2, −0.2) ⎟ ⎟ (0.6, −0.1) ⎠ (0, 0)

It can be written in the form of two different adjacency matrices as follows:

334

8 Energy of Bipolar Fuzzy Graphs v3 (0.4, −0.3)

(0 .2 ,− 0. 1)

1) 0. − , .3 (0 v2 (0.9, −0.6)

v4 (0.3, −0.6)

(0 .7 ,− 0. 3)

5) 0. − , .4 (0

(0.3, −0.4) v1 (0.5,

(0 .1 ,− 0. 3)

2) 0. ,− 2 . (0

(0.6, −0.1)

0.7)

v6 (0.8,

v5 (0.7,

0.5)

0.4)

Fig. 8.8 Bipolar fuzzy graph G

⎛

1 ⎜ 0.4 ⎜ ⎜ 0 p ⎜ AD (G) = +1 ⎜ 0 ⎜ ⎝ 0 0.3

0.4 0 0.3 0 0 0.7

0 0.3 1 0.2 0 0

0 0 0.2 0 0.1 0.2

0 0 0 0.1 1 0.6

⎞ 0.3 0.7 ⎟ ⎟ 0 ⎟ ⎟, 0.2 ⎟ ⎟ 0.6 ⎠ 0

⎛

⎞ −1 −0.5 0 0 0 −0.4 ⎜ −0.5 0 −0.1 0 0 −0.3 ⎟ ⎜ ⎟ ⎜ 0 −0.1 −1 −0.1 0 0 ⎟ n ⎜ ⎟. AD (G) = +1 ⎜ 0 0 −0.1 0 −0.3 −0.2 ⎟ ⎜ ⎟ ⎝ 0 0 0 −0.3 −1 −0.1 ⎠ −0.4 −0.3 0 −0.2 −0.1 0

Thus Spec D+1 (G) = {(−0.8523, −1.3948), (−0.0326, −1.0948), (0.1578, −1.0050), (1.0303, −0.0403), (1.1141, 0.1648), (1.5827, 0.3701)} and E D+1 (G) = (4.7698, 4.0698). +2 = {v2 , v4 , v6 }, For D ⎛

(0, 0) (0.4, −0.5) (0, 0) (0, 0) (0, 0) ⎜ (0.4, −0.5) (1, −1) (0.3, −0.1) (0, 0) (0, 0) ⎜ ⎜ (0, 0) (0.3, −0.1) (0, 0) (0.2, −0.1) (0, 0) ⎜ AD +2 (G) = ⎜ (0, 0) (0.2, −0.1) (1, −1) (0.1, −0.3) ⎜ (0, 0) ⎝ (0, 0) (0, 0) (0, 0) (0.1, −0.3) (0, 0) (0, 0) (0.2, −0.2) (0.6, −0.1) (0.3, −0.4) (0.7, −0.3)

⎞ (0.3, −0.4) (0.7, −0.3) ⎟ ⎟ (0, 0) ⎟ ⎟. (0.2, −0.2) ⎟ ⎟ (0.6, −0.1) ⎠ (1, −1)

Hence Spec D+2 (G) = {(−0.3855, −1.6054), (−0.2163, −1.1099), (−0.0307, −0.6556), (0.6225, 0.0101), (1.0327, 0.0911), (1.9773, 0.2797)} and E D+2 (G) = (4.2650, 3.7418). − → Definition 8.30 Let G = (A, B ) be a bipolar fuzzy digraph on a non-empty set X . − → A subset Dbout ⊆ X is a double out-dominating set of G if every vertex w ∈ / Dbout out out is out-dominated by at least two vertices in Db . Db is a minimal double out− → dominating set of G if, for any w ∈ X , Dbout \ {w} is not a double out-dominating set. The double out-dominating set with minimum cardinality among all minimal double out-dominating sets is called a minimum double out-dominating set of

8.5 Dominating and Double Dominating Energy of Bipolar Fuzzy Graphs

335

− → − → p G . If γ1 ≥ γ2 ≥ · · · ≥ γn are the eigenvalues of A Dout ( G ) and δ1 ≤ δ2 ≤ · · · ≤ δn b − → − → are the eigenvalues of AnDout ( G ), then the double out-dominating energy of G is b (n ) n − → E Dbout ( G ) = i=1 |γi |, i=1 |δi | .

8.6 Applications of Energy to Decision-Making Problems Definition 8.31 A bipolar fuzzy preference relation R on the set of choices W = p p {w1 , w2 , . . . , wn } is defined as a matrix R = [ξi j ]n×n , where ξi j = (ξi j , ξinj ), ξi j is n a positive certainty degree to which wi is positively preferred to w j and ξi j is a positive certainty degree to which wi is negatively preferred to w j . Moreover, ξii = (0.5, −0.5).

8.6.1 Smooth Communication Problem In modern warfare, it is very important to maintain smooth communication among individuals. The performance of communication equipments play a key role in campaign victory and defeat of an army. It is necessary for the communication units to keep regular communication drills. Suppose that the headquarters are drawing up a plan of communication drill for the next round. According to consultants with different simulation environments, there are four possible training venues (alternatives), xi (i = 1, 2, 3, 4), to choose from. The leaders of the communication unit invite a decision group which contains six experts ek (k = 1, 2, . . . , 6) to evaluate all venues so as to make the most reasonable choice. Based on their experiences, the experts compare each pair of alternatives and give individual judgments using the following p(k) bipolar fuzzy preference relations Rk = (ri j , rin(k) j )4×4 (k = 1, 2, . . . , 6). ⎡

(0.50, −0.50) (0.23, −0.17) (0.57, −0.67) ⎢ (0.77, −0.83) (0.50, −0.50) (0.17, −1) R1 = ⎢ ⎣ (0.43, −0.33) (0.83, 0) (0.50, −0.50) (0.37, −0.50) (0.16, −0.33) (0.58, −0.83) ⎡

(0.50, −0.50) ⎢ (0.62, −0.62) R2 = ⎢ ⎣ (0.49, −0.42) (0.73, −0.16) ⎡

(0.50, −0.50) ⎢ (0.43, −0.90) ⎢ R3 = ⎣ (0.60, −0.40) (0.54, −0.30)

⎤ (0.63, −0.5) (0.84, −0.67) ⎥ ⎥ (0.42, −0.17) ⎦ (0.50, −0.50)

(0.38, −0.38) (0.50, −0.50) (0.40, −0.31) (0.25, −0.10)

(0.51, −0.58) (0.60, −0.69) (0.50, −0.50) (0.86, −0.20)

⎤ (0.27, −0.84) (0.75, −0.90) ⎥ ⎥ (0.14, −0.80) ⎦ (0.50, −0.50)

(0.57, −0.10) (0.50, −0.50) (0.39, −0.20) (0.81, −0.60)

(0.40, −0.60) (0.61, −0.80) (0.50, −0.50) (0.20, −0.10)

⎤ (0.46, −0.70) (0.19, −0.40) ⎥ ⎥ (0.80, −0.90) ⎦ (0.50, −0.50)

336

8 Energy of Bipolar Fuzzy Graphs

⎡

(0.50, −0.50) ⎢ (0.70, −0.67) R4 = ⎢ ⎣ (0.58, −0.83) (0.74, −0.33) ⎡

(0.50, −0.50) ⎢ (0.30, −0.66) R5 = ⎢ ⎣ (0.84, −0.80) (0.59, −0.04) ⎡

(0.50, −0.50) ⎢ (0.77, −0.50) R6 = ⎢ ⎣ (0, −0.30) (0.70, 0)

⎤ (0.30, −0.33) (0.42, −0.17) (0.26, −0.67) (0.50, −0.50) (0.90, −0.33) (0.72, −0.17) ⎥ ⎥ (0.10, −0.67) (0.50, −0.50) (0.81, −1) ⎦ (0.28, −0.83) (0.19, 0) (0.50, −0.50) (0.70, −0.34) (0.50, −0.50) (0.20, −0.67) (0.71, −0.02)

(0.16, −0.20) (0.80, −0.33) (0.50, −0.50) (0.37, −0.01)

⎤ (0.41, −0.96) (0.29, −0.98) ⎥ ⎥ (0.63, −0.99) ⎦ (0.50, −0.50)

(0.23, −0.50) (0.50, −0.50) (0.40, −0.20) (0.64, −0.40)

(1.0, −0.70) (0.60, −0.80) (0.50, −0.50) (0.28, −0.20)

⎤ (0.30, −1.0) (0.36, −0.60) ⎥ ⎥. (0.72, −0.80) ⎦ (0.50, −0.50)

− → The bipolar fuzzy digraphs G i corresponding to bipolar fuzzy preference relations given in matrices Ri are shown in Fig. 8.9. The energy of each bipolar fuzzy digraph is − → − → − → E( G 1 ) = (2.9357, 2.3961), E( G 2 ) = (2.8289, 2.5249), E( G 3 ) = (2.9602, 2.84 − → − → − → 95), E( G 4 ) = (2.7304, 2.6413), E( G 5 ) = (2.9699, 1.9252), E( G 6 ) = (2.9510, 2.5897). Then the weights can be calculated as: ⎛

⎞

→ − → ⎜ E p (− G k) En( G k) ⎟ ⎜ ⎟ p n wk = (wk , wk ) = ⎜ 6 , 6 ⎟, ⎝ p −  n − → → ⎠ E ( G j) E ( G j) j=1

k = 1, 2, . . . , 6.

j=1

Here, w1 = (0.1690, 0.1605), w2 = (0.1628, 0.1692), w3 = (0.1704, 0.1909), w4 = (0.1571, 0.177), w5 = (0.1709, 0.1290), w6 = (0.1698, 0.1735). The collective bipolar fuzzy preference relation aggregated from the six bipolar fuzzy preference relations is determined as: ⎤ (0.5, −0.5) (0.4037, −0.2997) (0.5106, −0.4976) (0.3907, −0.7719) ⎥ ⎢ (0.5963, −0.7004) (0.5, −0.5) (0.6103, −0.6697) (0.5202, −0.5968) ⎥. =⎢ ⎣ (0.4894, −0.5025) (0.3897, −0.3304) (0.5873, −0.7780) ⎦ (0.5, −0.5) (0.6093, −0.2282) (0.4798, −0.4033) (0.4127, −0.2221) (0.5, −0.5) ⎡

R=

6  k=1

wk Rk = [(ηi+j , ηi−j )]6×6

Using the score function si j = ηi−j + ηi+j , the score values can be calculated in matrix S. ⎡ ⎤ 0 0.1040 0.0130 −0.3812 ⎢ −0.1041 0 −0.0594 −0.0766 ⎥ ⎥. S = [si j ]6×6 = ⎢ ⎣ −0.0131 0.0593 0 −0.1907 ⎦ 0.3811 0.0765 0.1906 0

8.6 Applications of Energy to Decision-Making Problems

337

Fig. 8.9 Bipolar fuzzy digraphs

The net flow of xi , i.e., the net degree of preference of xi over the other alternatives can be calculated as ⎞ ⎛ 5 5   ψ(ai ) = wj ⎝ (s jk − sk j )⎠, i = 1, 2, 3, 4, 5, 6. j=1

k=1 j=k

So, the net flows of the four alternatives are φ(x1 ) = −0.5281, φ(x2 ) = −0.4799, φ(x3 ) = −0.2887, φ(x4 ) = 1.2967, which give the ranking x4 x3 x2 x1 . Thus, the best choice is x4 .

338

8 Energy of Bipolar Fuzzy Graphs

8.6.2 Selection of a Business Partner International Business Machines Corporation (IBM) is an American multinational technology company. It offers a prosperous portfolio of businesses, industry solutions including product engineering solutions and platform services. It is committed to becoming a universally superior IT solution and maintenance by way of regular development of organization and process, adequacy improvement, leadership, agreement, and open change of employees. To improve the competitiveness capability and operations in the universal market, IBM plans to set up a strategic agreement with a transnational corporation. After various meetings, four corporations would like to set up a strategic agreement with IBM; these are HPQ (a1 ), VZ (a2 ), DELL (a3 ), and NT (a4 ). To select the attractive strategic agreement partner, five experts e j ( j = 1, 2, 3, 4, 5) are requested to cooperate in the decision analysis who come from various departments of IBM. The experts compare given choices and give personal judgments using ( j) bipolar fuzzy preference relations R j = [rik ]4×4 ( j = 1, 2, 3, 4, 5) given in the fol− → lowing matrices with corresponding bipolar fuzzy digraphs G i , i = 1, 2, 3, 4, 5 (Fig. 8.10). ⎡

(0.5, −0.5) ⎢ (0.7, −0.4) R1 = ⎢ ⎣ (0.3, −0.7) (0.2, −0.3) ⎡

(0.5, −0.5) ⎢ (0.7, −0.4) R2 = ⎢ ⎣ (0.2, −0.8) (0.2, −0.9) ⎡

(0.5, −0.5) ⎢ (0.2, −0.2) R3 = ⎢ ⎣ (0.2, −0.7) (0.3, −0.8) ⎡

(0.5, −0.5) ⎢ (0.7, −0.4) R4 = ⎢ ⎣ (0.4, −0.3) (0.5, −0.9) ⎡

(0.5, −0.5) ⎢ (0.1, −0.8) ⎢ R5 = ⎣ (0.6, −0.9) (0.1, −0.7)

(0.3, −0.6) (0.5, −0.5) (0.4, −0.9) (0.9, −0.5)

(0.7, −0.3) (0.6, −0.1) (0.5, −0.5) (0.8, −0.3)

⎤ (0.8, −0.7) (0.1, −0.5) ⎥ ⎥ (0.2, −0.7) ⎦ (0.5, −0.5)

(0.3, −0.6) (0.5, −0.5) (0.3, −0.8) (0.2, −0.7)

(0.8, −0.2) (0.7, −0.2) (0.5, −0.5) (0.1, −0.6)

⎤ (0.8, −0.1) (0.8, −0.3) ⎥ ⎥ (0.9, −0.4) ⎦ (0.5, −0.5)

(0.8, −0.8) (0.5, −0.5) (0.2, −0.9) (0.2, −0.8)

(0.8, −0.3) (0.8, −0.1) (0.5, −0.5) (0.7, −0.4)

⎤ (0.7, −0.2) (0.8, −0.2) ⎥ ⎥ (0.3, −0.6) ⎦ (0.5, −0.5)

(0.3, −0.6) (0.5, −0.5) (0.7, −0.3) (0.8, −0.8)

(0.6, −0.7) (0.3, −0.7) (0.5, −0.5) (0.8, −0.3)

⎤ (0.5, −0.1) (0.2, −0.2) ⎥ ⎥ (0.2, −0.7) ⎦ (0.5, −0.5)

(0.9, −0.2) (0.5, −0.5) (0.2, −0.4) (0.7, −0.7)

(0.4, −0.1) (0.8, −0.6) (0.5, −0.5) (0.4, −0.3)

⎤ (0.9, −0.3) (0.3, −0.3) ⎥ ⎥. (0.6, −0.7) ⎦ (0.5, −0.5)

8.6 Applications of Energy to Decision-Making Problems

Fig. 8.10 Bipolar fuzzy digraphs

339

340

8 Energy of Bipolar Fuzzy Graphs

− → The Laplacian matrix of each bipolar fuzzy digraph G j is denoted by R Lj ( j = 1, 2, 3, 4, 5) and calculated as follows: ⎡

(1.8, −1.6) ⎢ (−0.7, 0.4) R1L = ⎢ ⎣ (−0.3, 0.7) (−0.2, 0.3) ⎡

(1.9, −0.9) ⎢ (−0.7, 0.4) R2L = ⎢ ⎣ (−0.2, 0.8) (−0.2, 0.9) ⎡

(2.4, −0.9) ⎢ (−0.2, 0.2) L ⎢ R3 = ⎣ (−0.2, 0.7) (−0.3, 0.8) ⎡

(1.4, −1.4) ⎢ (−0.7, 0.4) L R4 = ⎢ ⎣ (−0.4, 0.3) (−0.5, 0.9) ⎡

(2.2, −0.6) ⎢ (−0.1, 0.8) L R5 = ⎢ ⎣ (−0.6, 0.9) (−0.1, 0.7)

(−0.3, 0.6) (1.4, −1.0) (−0.4, 0.9) (−0.9, 0.5)

(−0.7, 0.3) (−0.6, 0.1) (0.9, −2.3) (−0.8, 0.3)

⎤ (−0.8, 0.7) (−0.1, 0.5) ⎥ ⎥ (−0.2, 0.7) ⎦ (1.9, −1.1)

(−0.3, 0.6) (2.2, −0.9) (−0.3, 0.8) (−0.2, 0.7)

(−0.8, 0.2) (−0.7, 0.2) (1.4, −2.0) (−0.1, 0.6)

⎤ (−0.8, 0.1) (−0.8, 0.3) ⎥ ⎥ (−0.9, 0.4) ⎦ (0.5, −2.2)

(−0.8, 0.8) (1.7, −0.9) (−0.2, 0.9) (−0.2, 0.8)

(−0.8, 0.3) (−0.8, 0.1) (0.7, −2.2) (−0.7, 0.4)

⎤ (−0.7, 0.2) (−0.8, 0.2) ⎥ ⎥ (−0.3, 0.6) ⎦ (1.2, −2.0)

(−0.3, 0.6) (1.2, −1.3) (−0.7, 0.3) (−0.8, 0.8)

(−0.6, 0.7) (−0.3, 0.7) (1.3, −1.3) (−0.8, 0.3)

⎤ (−0.5, 0.1) (−0.2, 0.2) ⎥ ⎥ (−0.2, 0.7) ⎦ (2.1, −2.0)

(−0.9, 0.2) (1.2, −1.7) (−0.2, 0.4) (−0.7, 0.7)

(−0.4, 0.1) (−0.8, 0.6) (1.4, −2.0) (−0.4, 0.3)

⎤ (−0.9, 0.3) (−0.3, 0.3) ⎥ ⎥. (−0.6, 0.7) ⎦ (1.2, −1.7)

− → The Laplacian energy of each bipolar fuzzy digraph is computed as L E( G 1 ) = − → − → (4.6500, 4.5500), L E( G 2 ) = (3.8501, 5.0999), L E( G 3 ) = (3.9091, 47438), − → − → L E( G 4 ) = (4.9500, 4.5000), L E( G 5 ) = (4.0501, 4.8999). The weight of each expert can be calculated using Formula (8.14). ⎞

⎛

→ − → ⎟ ⎜ L E p (− L En( G j ) G j) ⎟ ⎜ p , w j = (w j , w nj ) = ⎜ ⎟ , j = 1, 2, 3, 4, 5. 5 5 ⎠ ⎝  − → − → p n LE ( G i) L E(L E ( G i ) i=1

i=1

(8.14) By calculations, w1 = (0.2172, 0.1912), w2 = (0.1798, 0.2143), w3 = (0.1826, 0.1891), w4 = (0.2312, 0.1891), w5 = (0.1892, 0.2059). The collective bipolar fuzzy preference relation combined from five bipolar fuzzy preference relations and weights are computed in Eq. (8.15).

8.6 Applications of Energy to Decision-Making Problems ⎡

341

(0.5000, −0.5000) (0.5048, −0.5574) (0.6564, −0.3130) ⎢ (0.4952, −0.4426) (0.5000, −0.5000) (0.6230, −0.3378) p n L ⎢ R = [(ri j , ri j )]5×5 = wj Rj = ⎣ (0.3436, −0.6870) (0.3770, −0.6622) (0.5000, −0.5000) j=1 (0.2687, −0.7242) (0.5854, −0.7006) (0.5802, −0.3843) 5 

⎤ (0.7313, −0.2758) (0.4146, −0.2994) ⎥ ⎥. (0.4198, −0.6157) ⎦ (0.5000, −0.5000)

(8.15)

p

Using score function si j = ri j + rinj , the scores are given in Eq. (8.16). ⎡

S = [si j ]5×5

⎤ 0 −0.0526 0.3434 0.4555 ⎢ 0.0526 0 0.2852 0.1152 ⎥ ⎥ =⎢ ⎣ −0.3434 −0.2852 0 −0.1959 ⎦ . −0.4555 −0.1152 0.1959 0

(8.16)

The net degree of preference of each alternative ai over the other choices can be calculated using Formula (8.17). ψ(ai ) =

5  j=1

wj

 5

 sk j − s jk , i = 1, 2, 3, 4, 5.

(8.17)

k=1 i= j

The net degrees of the four choices are ψ(a1 ) = 1.4926, ψ(a2 ) = 0.906, ψ(a3 ) = −1.649, ψ(a4 ) = −0.7496 which give the ranking a1 a2 a4 a3 . Thus the best choice is a1 . − → The signless Laplacian matrices of the bipolar fuzzy digraphs G j , j = 1, 2, 3, 4, 5 are computed as follows: ⎡

R1L

+

(1.8, −1.6) ⎢ (0.7, −0.4) =⎢ ⎣ (0.3, −0.7) (0.2, −0.3) ⎡

R2L

+

(1.9, −0.9) ⎢ (0.7, −0.4) =⎢ ⎣ (0.2, −0.8) (0.2, −0.9) ⎡

R3L

+

(2.4, −0.9) ⎢ (0.2, −0.2) =⎢ ⎣ (0.2, −0.7) (0.3, −0.8) ⎡

R4L

+

(1.4, −1.4) ⎢ (0.7, −0.4) =⎢ ⎣ (0.4, −0.3) (0.5, −0.9)

(0.3, −0.6) (1.4, −1.0) (0.4, −0.9) (0.9, −0.5)

(0.7, −0.3) (0.6, −0.1) (0.9, −2.3) (0.8, −0.3)

⎤ (0.8, −0.7) (0.1, −0.5) ⎥ ⎥ (0.2, −0.7) ⎦ (1.9, −1.1)

(0.3, −0.6) (2.2, −0.9) (0.3, −0.8) (0.2, −0.7)

(0.8, −0.2) (0.7, −0.2) (1.4, −2.0) (0.1, −0.6)

⎤ (0.8, −0.1) (0.8, −0.3) ⎥ ⎥ (0.9, −0.4) ⎦ (0.5, −2.2)

(0.8, −0.8) (1.7, −0.9) (0.2, −0.9) (0.2, −0.8)

(0.8, −0.3) (0.8, −0.1) (0.7, −2.2) (0.7, −0.4)

⎤ (0.7, −0.2) (0.8, −0.2) ⎥ ⎥ (0.3, −0.6) ⎦ (1.2, −2.0)

(0.3, −0.6) (1.2, −1.3) (0.7, −0.3) (0.8, −0.8)

(0.6, −0.7) (0.3, −0.7) (1.3, −1.3) (0.8, −0.3)

⎤ (0.5, −0.1) (0.2, −0.2) ⎥ ⎥ (0.2, −0.7) ⎦ (2.1, −2.0)

342

8 Energy of Bipolar Fuzzy Graphs

⎡

R5L

+

(2.2, −0.6) ⎢ (0.1, −0.8) =⎢ ⎣ (0.6, −0.9) (0.1, −0.7)

(0.9, −0.2) (1.2, −1.7) (0.2, −0.4) (0.7, −0.7)

(0.4, −0.1) (0.8, −0.6) (1.4, −2.0) (0.4, −0.3)

⎤ (0.9, −0.3) (0.3, −0.3) ⎥ ⎥. (0.6, −0.7) ⎦ (1.2, −1.7)

The signless Laplacian energy of each bipolar fuzzy digraph is computed as − → − → − → S L E( G 1 ) = (3.4884, 3.4080), S L E( G 2 ) = (3.0805, 4.2173), S L E( G 3 ) = − → − → (3.0688, 3.8348), S L E( G 4 ) = (3.9000, 3.0000), S L E( G 5 ) = (3.3673, 4.0769). The weights of each expert can be computed using Formula (8.18). ⎛

⎞

→ − → ⎜ S L E p (− SL En( G j ) ⎟ G j) ⎜ ⎟ p , w j = (w j , w nj ) = ⎜ ⎟ , j = 1, 2, 3, 4, 5. (8.18) 5 5 ⎝ ⎠  − → − → p n SL E ( G i) SL E ( G i) i=1

i=1

Here, w1 = (0.2064, 0.1838), w2 = (0.1822, 0.2275), w3 = (0.1815, 0.2069), w4 =(0.2307, 0.1618), w5 = (0.1992, 0.2199). The collective bipolar fuzzy preference relation combined from the five bipolar fuzzy preference relations and corresponding weights are computed as follows: ⎡

(0.5000, −0.5000) (0.5103, −0.5534) (0.6535, −0.2980) (0.7326, −0.2749)

⎤

5  ⎢ (0.4897, −0.4466) (0.5000, −0.5000) (0.6252, −0.3298) (0.4175, −0.2999) ⎥ p ⎥ R = [(ri j , rinj )]n×n = wj Rj = ⎢ ⎣ (0.3465, −0.7020) (0.3748, −0.6702) (0.5000, −0.5000) (0.4254, −0.6110) ⎦ . j=1

(0.2674, −0.7251) (0.5825, −0.7001) (0.5746, −0.3890) (0.5000, −0.5000)

p

Using the score function si j = ri j + rinj , the score values are calculated in matrix S. ⎡

S = [si j ]5×5

⎤ 0 −0.0431 0.3555 0.4577 ⎢ 0.0431 0 0.2954 0.1176 ⎥ ⎥ =⎢ ⎣ −0.3555 −0.2954 0 −0.1856 ⎦ . −0.4577 −0.1176 0.1856 0

The net degree of preference of ai over the other choices can be computed using Formula (8.19). ψ(ai ) =

5  j=1

wj

 5

 (s jk − sk j ) , i = 1, 2, 3, 4, 5.

(8.19)

k=1 j=k

Hence the net degrees of the four choices are ψ(a1 ) = 1.5402, ψ(a2 ) = 0.9122, ψ(a3 ) = −1.6730, ψ(a4 ) = −0.7794 which gives the ranking of a1 a2 a4 a3 . Thus the best choice is a1 .

8.7 Conclusions

343

8.7 Conclusions In connection with energy, the properties of graphical models can be described using various matrices associated with graphs such as Laplacian matrix to study diffusion over networks, dominating, out-dominating, and double dominating matrices for characterizing domination parameters in network models. The energy-like quantities related to these matrices have a clear connection to chemical problems, and in newer times attracted much attention of mathematicians and mathematical chemists. To deal with bipolar fuzzy networks, in this chapter, we have studied the concept of energy in bipolar fuzzy graphs and presented certain formulae, lower and upper bounds of Laplacian energy, signless Laplacian energy, dominating energy, outdominating energy, double dominating energy, and double out-dominating energy of bipolar fuzzy graphs and bipolar fuzzy digraphs. Using bipolar fuzzy preference relations, we have studied multi-criteria decision-making models based on the energy of bipolar fuzzy graphs in business partnerships and smooth communication problems.

Exercises 8 1. Let G = (A, B) be a complete bipolar fuzzy graph such that B(x y) = (c p , cn ), for all x, y ∈ X . Calculate the energy of G. (v, a)(0.2, −0.3)

(u, a)(0.2, −0.3)

(0.1, −0.2) (0.1, −0.3)

(u, b)(0.3, −0.4)

(x, a)(0.2, −0.3)

(0.1, −0.3)

(0.1, −0.3)

(0.1, −0.3)

(0.1, −0.3)

(v, b)(0.3, −0.5)

(w, b)(0.3, −0.5)

(x, b)(0.3, −0.5) (0.1, −0.3)

(0.1, −0.2)

(0.2, −0.3)

(0.2, −0.1)

(0.2, −0.1)

(0.2, −0.1)

(v, c)(0.4, −0.5)

(w, c)(0.4, −0.5)

(x, c)(0.4, −0.5)

(0.2, −0.1)

(u, c)(0.0, −0.4)

(w, a)(0.2, −0.3)

(0.2, −0.3)

(0.1, −0.2)

(0.3, −0.1)

(0.2, −0.3)

(u, d)(0.3, −0.4)

− → Fig. 8.11 Cartesian product G

(0.3, −0.1)

(0.3, −0.1)

(0.3, −0.1) (0.1, −0.2)

(0.1, −0.3)

(0.2, −0.3)

(v, d)(0.3, −0.4)

(0.1, −0.3)

(w, d)(0.3, −0.4)

(x, d)(0.3, −0.4)

344

8 Energy of Bipolar Fuzzy Graphs

2. Calculate the double out-dominating set and double out-dominating energy of the − → Cartesian product G of two bipolar fuzzy digraphs as shown in Fig. 8.11. 3. Let G = (A, B) be a bipolar fuzzy graph with n vertices and minimum double dominating set D = {w1 , w2 , . . . , wk }, then

.8) −0

(0 .3,

.3) −0

−0 .2)

.4, (0

.7, (0 w2

(0.7, −0.4)

w1 (0.9, −0.4)

(0. 6,

.8) −0 , 4 (0.

−0 .7)

(0.2, −0.6)

w4 (0.4, −0.9)

w3 (0.6, −0.7)

− → Fig. 8.12 Bipolar fuzzy digraph G u3 ( 0.5

)

)

−

0.6

, .4 (0

(0 .1 ,

−

0. 3) u6 (0.1, −0.3)

Fig. 8.13 Bipolar fuzzy digraph G

(0.3 ,

,−

u7 (0.5, −0.6)

, (0.2

−0. 1) u5 (0.3, −0.5)

4) −0.

(0 .7

)

(0.4, −0.1)

)

u4

0.1

0.2

)

0.2

,−

3, − (0.

0.3

,−

(0.2

3) 0.

,−

(0. 3

0. 4) − (0 .2 , u1

(0.1, −0.2)

) , −0.7 u2(0.8

(0.1, −0.3)

, (0.3

2) −0.

8.7 Conclusions



(a) (b)

+ wi ∈ D n 

B(wi ) = (

i=1 n 

n 

φi ,

i=1

(φi )2 =

(c)

345

n  i=1

(ψi )2 =

i=1

p

(ξii )2 +

n  i=1

n 

ψi ), i=1 n  n 

(ξiin )2 +

i=1 j=1 i= j n  n 

p

p

(ξi j )(ξ ji ),

i=1 j=1 i= j

(ξinj )(ξ nji ).

+ is a minimum dou4. Let G = (A, B) be a bipolar fuzzy graph with n vertices. If D p p ble dominating set, R = det(A D+ (G)) = det ([ξi j ]n×n ), and T = det(AnD+ (G)) = det ([ξinj ]n×n ), then 

(a) (b)



n 



n n 2    p  p p p p p (ξii )2 + (ξi j )(ξ ji ) + n(n − 1)R n ≤ |φi | ≤ n (ξii )2 + (ξi j )(ξ ji ) . i=1 i = j i=1 i=1 i = j 

  n n n 2     n  n n n n 2 2 (ξii ) + (ξi j )(ξ ji ) + n(n − 1)T n ≤ |ψi | ≤ n (ξii ) + (ξi j )(ξ nji ) . i=1 i = j i=1 i=1 i = j

5. Let G = (A, B) be a bipolar fuzzy graph with adjacency matrix A(G) = [u i j ]n×n . Let λi , 1 ≤ i ≤ n, be the eigenvalues of adjacency matrix A p (G) and ωi , 1 ≤ i ≤ n, be the eigenvalues of adjacency matrix An (G), then

n 

2 |φi |

n

n

2 n n n  p     2 2 n 2 2 and ≤n (ξii ) + ( |λi |) |ψi | ≤ n (ξii ) + ( |ωi |) .

i=1

i=1

i=1

i=1

i=1

i=1

6. If G = (A, B) is a null bipolar fuzzy graph, then prove that L E(G) = (0, 0). 7. If G is a regular and totally regular bipolar fuzzy graph, then prove that the sum of eigenvalues of G is zero. 8. Calculate the out-dominating set and out-dominating energy of bipolar fuzzy − → − → digraph G = (A, B ) on X = {w1 , w2 , w3 , w4 } shown in Fig. 8.12. 9. Let A(G), L(G) and L + (G) be the adjacency, the Laplacian and the signless Laplacian matrices, respectively, of a bipolar fuzzy graph G, as given in Fig. , , 8.13. Then show that , L E + (G) − L E(G), ≤ 2E(G). 10. Prove that E(G) = L E(G) = L E + (G), if the bipolar fuzzy graph G is a regular bipolar fuzzy graph. 11. Let G = (A, B) be a bipolar fuzzy graph on n vertices and let L(G) and L + (G) be the Laplacian and the signless Laplacian matrices of G, respectively. Then prove that , , , + ≥ max LE ,deglipj (vi ) − , , ,  , L E + (linj ) + L E(linj ) ≥ max 2E(linj ), 2 nj=1 ,deglinj (vi ) − , -

+

p (li j )

p L E(li j )

 p 2E(li j ), 2 nj=1

,. , , , , n ,  n ,. 2 li j , , 1≤i< j≤n , . n , 2



1≤i< j≤n

p

li j

346

8 Energy of Bipolar Fuzzy Graphs

12. Let G = (A, B) be a bipolar fuzzy graph on n vertices and let L(G) and L + (G) be the Laplacian and the signless Laplacian matrices of G, respectively. Then show that and rank r . 4r L E + (li j ) + L E(li j ) ≥ 4E(li j ) − p

p

p

4r L E + (linj ) + L E(linj ) ≥ 4E(linj ) −

 1≤i< j≤n

n  1≤i< j≤n

p

li j , linj

n

,

where r is the number of non-zero eigenvalues of a bipolar fuzzy graph G. p 13. Let G = (A, B) be a bipolar fuzzy graph on n vertices and let L(G) = (L(li j ), L(linj )) be the Laplacian matrix of G. Then prove that  , p, , 2 li j ,, n ,  1≤i< j≤n , , p p L E(li j ) ≤ E(li j ) + ,degl p (vi ) − ,, ij , , n j=1 , ,  , , , 2 linj ,, n ,  1≤i< j≤n , , L E(linj ) ≤ E(linj ) + ,deglinj (vi ) − ,. , , n j=1 , ,

14. Let G = (A, B) be a bipolar fuzzy graph on n vertices and let L + (G) = p (L + (li j ), L + (linj )) be the signless Laplacian matrix of G. Then show that p L E + (li j )

≤

p E(li j ) +

 , p, , 2 li j ,, n ,  1≤i< j≤n , , ,degl p (vi ) − ,, ij , , n j=1 , ,

 , , , 2 linj ,, n ,  1≤i< j≤n , , L E + (linj ) ≤ E(linj ) + ,. ,deglinj (vi ) − , , n j=1 , ,

15. Let G be a connected bipolar fuzzy graph on n vertices and let L + (G) = p (L + (li j ), L + (linj )) be the signless Laplacian matrix of G. Then show that  ⎛    n   p + p L E (li j ) ≤ E(li j ) + n degl2p (vi ) − 4 ⎝ j=1

ij

ij

,

1≤i< j≤n

 ⎛    n   degl2n (vi ) − 4 ⎝ L E + (linj ) ≤ E(linj ) + n j=1

⎞2 p li j ⎠

1≤i< j≤n

⎞2 linj ⎠ .

References

347

References 1. Akram, M.: Bipolar fuzzy graphs. Inf. Sci. 181(24), 5548–5564 (2011) 2. Akram, M., Dudek, W.A.: Regular bipolar fuzzy graphs. Neural Comput. Appl. 21(supp 1), 197–205 (2012) 3. Akram, M., Saleem, D., Rashmanlou, H.: Signless Laplacian energy of bipolar fuzzy graphs with application. Int. J. Adv. Intell. Paradigms. In press 2019 4. Akram, M., Saleem, D., Davvaz, D.: Energy of double dominating bipolar fuzzy graphs. J. Appl. Math. Comput. 61(1–2), 219–234 (2019) 5. Anjali, N., Mathew, S.: Energy of a fuzzy graph. Ann. Fuzzy Math. Inf. 6(3), 455–465 (2013) 6. Borzooei, R.A., Rashmanlou, H.: New concepts of vague graphs. Int. J. Mach. Learn. Cybern. 8(4), 1081–1092 (2017) 7. Gutman, I.: The energy of a graph. Ber. Math-Statist. Sekt. Forschungsz. Graz 103, 1–22 (1978) 8. Gutman, I.: The energy of a graph: Old and New Results. In: Betten, A., Kohnert, A., Laue, R., Wassermann, A. (eds.) Algebraic Combinatorics and Applications. Springer, Berlin (2001) 9. Gutman, I., Zhou, B.: Laplacian energy of a graph. Linear Algebra Appl. 414(1), 29–37 (2014) 10. Kahraman, C., Otay, I. (eds.): Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets. Springer, Berlin (2019) 11. Kaladevi, V., Devi, G.S.: Double dominating energy of some graphs. Int. J. Fuzzy Math. Archive 4(1), 1–7 (2014) 12. Kanna, M.R., Dharmendra, B.N., Sridhara, G.: The minimum dominating energy of a graph. Int. J. Pure Appl. Math. 85(4), 707–718 (2013) 13. Li, X., Shi, Y., Gutman, I.: Graph Energy. Springer Science and Business Media, Berlin (2012) 14. Naz, S., Ashraf, S., Karaaslan, F.: Energy of a bipolar fuzzy graph and its application in decision making. Italian J. Pure Appl. Math. 40, 339–352 (2018) 15. Praba, B., Chandrasekaran, V.M., Deepa, G.: Energy of an intuitionistic fuzzy graph. Italian J. Pure Appl. Math. 32, 431–444 (2014) 16. Rahimi Sharbaf, S., Fayazi, F.: Laplacian energy of a fuzzy graph. Iranian J. Math. Chem. 5(1), 1–10 (2014) 17. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 18. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998) 19. Zhang, W.-R.: A communication on bipolarity. A Publication of Society for Mathematics of Uncertainty, vol. III, pp. 27–35 (2009)

Chapter 9

Bipolar Neutrosophic Competition Graphs

In this chapter, we present a concise review of bipolar neutrosophic sets and apply this concept to graphs and digraphs. We discuss the notion of bipolar neutrosophic competition graphs and present certain characterizations of bipolar neutrosophic outneighborhoods, bipolar neutrosophic in-neighborhoods, bipolar neutrosophic open neighborhood graphs, bipolar neutrosophic closed neighborhood graphs, bipolar neutrosophic p-competition graphs, m-step bipolar neutrosophic competition graphs, and strong preys and strong independent predator–prey relations. We describe various methods for the construction of bipolar neutrosophic competition graphs of certain products of bipolar neutrosophic digraphs. We elaborate certain algorithms to compute the strength of competition with a number of real-world applications in different fields including economics, business marketing, and organizational designations. The basic results of this chapter are from [1, 5].

9.1 Introduction The first successful attempt toward incorporating non-probabilistic uncertainty, i.e. uncertainty which is not caused by randomness of an event, into mathematical modeling was made in 1965 by Zadeh [19] through his remarkable theory on fuzzy sets. A fuzzy set is a generalization of a set in which each element of the universe belongs to it but with some grade or degree of belongingness which lies in [0, 1]. This gradation concept is very well suited for applications involving imprecise data such as natural language processing, artificial intelligence, handwriting, and speech recognition. Although fuzzy set theory is very successful in handling uncertainties arising from vagueness or partial belongingness of elements, it cannot model all sorts of uncertainties prevailing in various physical problems. In some applications such as expert system, belief system, and information fusion, we should consider not only the truth membership supported by the evidence but also the falsity membership against the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_9

349

350

9 Bipolar Neutrosophic Competition Graphs

evidence. That is beyond the scope of fuzzy sets and interval-valued fuzzy sets. In 1983, Atanassov [11] introduced the notion of intuitionistic fuzzy sets as a generalization of fuzzy sets. In an intuitionistic fuzzy set A, both truth membership value (belongingness) T A (x) and falsity membership value (non-belongingness) FA (x) are attached with each element x with the restriction that the sum of these two membership values is less or equal to 1. A fuzzy set can be considered as a special case of an intuitionistic fuzzy set, where the degree of non-belongingness of an element is exactly equal to 1 minus the degree of belongingness. In real-world problems, there are many situations containing tricomponents of uncertainty, e.g., games (winning, defeating, or tie scores), voting (pro, contra, null/black votes), positive/negative/zero numbers, yes/no/NA, decision-making and control theory (making a decision, not making, or hesitating), and accepted/rejected/pending. Smarandache [16], in 1995, proposed the term “neutrosophic” because “neutrosophic” etymologically comes from neutro-sophy means knowledge of neutral thought and this neutral represents the main distinction between fuzzy/intuitionistic fuzzy logic/set and neutrosophic logic/set. Neutrosophic logic is a logic in which each proposition is estimated to have a degree of truth (T), a degree of indeterminacy (I), and a degree of falsity (F). A neutrosophic set A is a set in which each element (x) of the universe has a degree of truth T (x), indeterminacy I (x), and falsity F(x), where T (x), I (x), and F(x) are real standard or real nonstandard subsets of ]0− , 1+ [, the nonstandard unit interval. In contrast to intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets, the indeterminacy is characterized explicitly in a neutrosophic set which is independent of truth and falsity membership values. Neutrosophic sets are indeed more general than intuitionistic fuzzy sets as there are no constraints between truth, indeterminacy, and falsity membership values. All these degrees can individually vary within ]0− , 1+ [. Since, for software engineering proposals, the classical unit interval [0, 1] is used, a neutrosophic set will be more difficult to apply in real scientific and engineering fields. To overcome this limitation, Wang et al. [18] proposed the concept of single-valued neutrosophic sets in which the degree of truth T (x), indeterminacy I (x), and falsity F(x) belong to [0, 1]. Single-valued neutrosophic sets/graphs have been an active domain of research in the last few years. A lot of work has been done by researchers on neutrosophic set/graph theory [2, 3, 7, 9, 13, 15]. The concept of bipolar neutrosophic set was introduced in [14] and the bipolar single-valued neutrosophic graph in [6]. Other concepts of bipolar single-valued neutrosophic graphs can be seen in [1, 5, 8, 12]. For further terminologies and studies on single-valued neutrosophic theory, readers are referred to [4, 10, 17, 20–22].

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs A neutrosophic set A in a universe X is defined by a truth-membership function T A (x), an indeterminacy-membership function I A (x), and a falsity-membership

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs

351

function FA (x), where T A (x), I A (x), and FA (x) are real standard or real nonstandard subsets of ]0− , 1+ [, the nonstandard unit interval, that is, T A : X →]0− , 1+ [, I A : X →]0− , 1+ [, and FA : X →]0− , 1+ [ are the mappings. There is no restriction on the sum of T A (x), I A (x), and FA (x), so 0− ≤ sup T A (x) + sup I A (x) + sup FA (x) ≤ 3+ . Definition 9.1 A single-valued neutrosophic set A on a non-empty set X is characterized by a truth-membership function T A : X → [0, 1], indeterminacy-membership function I A : X → [0, 1], and a falsity-membership function FA : X → [0, 1]. There is no restriction on the sum of T A (x), T A (x), and FA (x), for all x ∈ X . Definition 9.2 A bipolar single-valued neutrosophic set A on a non-empty set X is an object of the form A= p



p

  p p p x, T A (x), I A (x), FA (x), T An (x), I An (x), FAn (x) | x ∈ X , p

where T A , I A , FA : X → [0, 1] and T An , I An , FAn : X → [−1, 0] are mappings. The p p p positive membership values T A (x), I A (x), FA (x) denote, respectively, the truth, indeterminacy, and falsity membership values of an element x ∈ X , whereas the negative membership values T An (x), I An (x), FAn (x) denote the implicit counter properties of truth, indeterminacy, and falsity membership degrees of an element x ∈ X corresponding to the bipolar neutrosophic set A. p

Definition 9.3 The height of a bipolar single-valued neutrosophic set A = (T A (x), p p I A (x), FA (x), T An (x), I An (x), FAn (x)) in a universe of discourse X is defined as h(A) = (h 1 (A), h 2 (A), h 3 (A), h 4 (A), h 5 (A), h 6 (A)) p

p

p

p

p

p

= (sup T A (x), sup I A (x), inf FA (x), sup T A (x), sup I A (x), inf FA (x)). x∈X

x∈X

x∈X

x∈X

x∈X

x∈X

Example 9.1 Consider a bipolar single-valued neutrosophic set A = {(a, 0.5, 0.7, 0.2, −0.8, −0.9, −0.3), (b, 0.1, 0.2, 1, −0.5, −0.7, −0.6), (c, 0.3, 0.5, 0.3, −0.8, −0.6, −0.4)} on X = {a, b, c}, then h(A) = (0.5, 0.7, 0.2, 0.5, 0.7, 0.2). p

p

p

Definition 9.4 Let A1 = (T A1 (x), I A1 (x), FA1 (x), T An1 (x), I An 1 (x), FAn1 (x)) and A2 = p p p (T A2 (x), I A2 (x), FA2 (x), T An2 (x), I An 2 (x), FAn2 (x)) be two bipolar single-valued neutrosophic sets on X . The union of A1 and A2 is denoted by A1 ∪ A2 and defined as  p p p p p p (A1 ∪ A2 )(x) = T A1 (x) ∨ T A2 (x), I A1 (x) ∨ I A2 (x), FA1 (x) ∧ FA2 (x), T An1 (x) ∧ T An2 (x),  I An 1 (x) ∧ I An 2 (x), FAn 1 (x) ∨ FAn 2 (x) , for all x ∈ X. p

p

p

Definition 9.5 Let A1 = (T A1 (x), I A1 (x), FA1 (x), T An1 (x), I An 1 (x), FAn1 (x)) and A2 = p p p (T A2 (x), I A2 (x), FA2 (x), T An2 (x), I An 2 (x), FAn2 (x)) be two bipolar single-valued neutrosophic sets on X . The intersection of A1 and A2 is denoted by A1 ∩ A2 and defined as

352

9 Bipolar Neutrosophic Competition Graphs

Table 9.1 Bipolar neutrosophic set A A x p TA p IA p FA n TA I An FAn

0.3 0.4 0.5 −0.6 −0.5 −0.2

y

z

0.5 0.4 0.2 −0.1 −0.8 −0.2

0.4 0.3 0.2 −0.5 −0.5 −0.5

 p p p p p p (A1 ∩ A2 )(x) = T A (x) ∧ T A (x), I A (x) ∧ I A (x), F A (x) ∨ F A (x), T An1 (x) ∨ T An2 (x), 1 2 1 2 1 2  I An 1 (x) ∨ I An 2 (x), F An 1 (x) ∧ F An 2 (x) , for all x ∈ X. p

p

p

Definition 9.6 Let A = (T A (x), I A (x), FA (x), T An (x), I An (x), FAn (x)) be a bipolar single-valued neutrosophic set on X . The complement of A is denoted by Ac = p p p (T Ac (x), I Ac (x), FAc (x), T Anc (x), I An c (x), FAnc (x)) and defined as   p p p Ac (x) = 1 − T A (x), 1 − I A (x), 1 − FA (x), −1 − T An (x), −1 − I An 1 (x), −1 − FAn 1 (x) , for all x ∈ X.

Definition 9.7 A bipolar single-valued neutrosophic relation on a non-empty p set X is a mapping B : X × X → [0, 1]3 × [−1, 0]3 , that is, B = {(yz, TB (yz), p p I B (yz), FB (yz), TBn (yz), I Bn (yz), FBn (yz)) | yz ∈ X × X }. Definition 9.8 A bipolar single-valued neutrosophic graph on a non-empty set X is a pair G = (A, B), where A is a bipolar single-valued neutrosophic set on X and B is a bipolar single-valued neutrosophic relation in X such that p

p

p

p

p

p

TB (yz) ≤ T A (y) ∧ T A (z),

I B (yz) ≤ I A (y) ∧ I A (z),

TBn (yz)

I Bn (yz)

≥

T An (y) ∨

T An (z),

≥

I An (y) ∨ I An (z),

p

p

p

FB (yz) ≤ FA (y) ∨ FA (z), FBn (yz)

≥

FAn (y) ∧

FAn (z),

for all y, z ∈ X.

Note that B(yz) = (0, 0, 1, 0, 0, −1), for all yz ∈ X × X \ E, where E is the set of edges in G. Example 9.2 Consider a bipolar single-valued neutrosophic graph G = (A, B) on X = {x, y, z}, where A is a bipolar single-valued neutrosophic set on X , given in Table 9.1, and B is a bipolar single-valued neutrosophic relation in X given in Table 9.2. The bipolar single-valued neutrosophic graph G is shown in Fig. 9.1. Definition 9.9 A bipolar single-valued neutrosophic digraph on a non-empty set − → − → X is a pair G = (A, B ), where A is a bipolar single-valued neutrosophic set on X − → and B is a bipolar single-valued neutrosophic relation in X such that p

p

p

T− → (yz) ≤ T A (y) ∧ T A (z),

B n T− → (yz) B

≥ T An (y) ∨ T An (z),

p

p

p

I− → (yz) ≤ I A (y) ∧ I A (z),

B n I− → (yz) B

≥ I An (y) ∨ I An (z),

p

p

p

F− → (yz) ≤ F A (y) ∨ F A (z),

B n F− → (yz) B

≥ FAn (y) ∧ FAn (z),

for all y, z ∈ X.

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs Table 9.2 Bipolar neutrosophic relation A B xy 0.3 0.4 0.5 −0.1 −0.8 −0.2

xz

0.3 0.4 0.2 −0.1 −0.8 −0.5

0.3 0.4 0.5 −0.5 −0.5 −0.5

z(0.4, 0.3, 0.2,

0.5,

−0 .1,

−0 .8,

−0 .2)

) 0.2 ,− 5 . −0 .6, 0 − (0.3, 0.4, 0.5, −0.1, −0.5, −0.2) .5, ,0 0.4

5) 0. − 5, 0. − 1, 0. − , 2 0. 3, . ,0 .3 (0 0.5,

0.4 ,

0.2 ,

(0.3, 0.4, 0.5, −0.5, −0.5, −0.5)

, 0.3 x(

yz

y (0 .5,

p TB p IB p FB n TB I Bn FBn

353

0.5)

Fig. 9.1 Bipolar neutrosophic graph G

− → Note that B is a nonsymmetric bipolar single-valued neutrosophic relation in X . Example 9.3 A bipolar single-valued neutrosophic digraph on X = {a, b, c} is shown in Fig. 9.2. − → Definition 9.10 Let G be a bipolar single-valued neutrosophic digraph on X . A bipolar single-valued neutrosophic out-neighborhood of a vertex x ∈ X is a bipop p p lar single-valued neutrosophic set N + (x) = (X x+ , Tx , Ix , Fx , Txn , Ixn , Fxn ) such that − → p p p X x+ = {y | B (x y) = (0, 0, 1, 0, 0, −1)} and Tx , Ix , Fx : X × X → [0, 1], p p p p Txn , Ixn , f xn : X × X → [−1, 0] are defined by Tx (y) = T− → (x y), I x (y) = I− → (x y), B B p p n n n n n n Fx (y) = F− → (x y), Tx (y) = T− → (x y), I x (y) = I− → (x y), Fx (y) = F− → (x y). B B B B A bipolar single-valued neutrosophic in-neighborhood of a vertex x ∈ X is a p p p bipolar single-valued neutrosophic set N − (x) = (X x− , Tx , Ix , Fx , Txn , Ixn , Fxn ) such

354

9 Bipolar Neutrosophic Competition Graphs

a(0.6, 0.7, 0.8, −0.1, −0.3, −0.7)

(0.3, 0.6, 0.7, −0.1, −0.2, −0.6)

(0.1 , 0.2 , 0.6 , −0 .1, − 0.2, −0.8 )

, 0.2 1, (0.

.5, −0 .4, 0 ,− 0.1

c(0.1, 0.3, 0.5, −0.5, −0.6, −0.7)

.2) −0

b(0.8, 0.9, 0.1, −0.5, −0.6, −0.1)

Fig. 9.2 Bipolar neutrosophic digraph

− → p p p that X x− = {y | B (yx) = (0, 0, 1, 0, 0, −1)} and Tx , Ix , Fx : X × X → [0, 1], p p p p Txn , Ixn , f xn : X × X → [−1, 0] are defined by Tx (y) = T− → (yx), I x (y) = I− → (yx), B B p p n n n n n n Fx (y) = F− → (yx), Tx (y) = T− → (yx), I x (y) = I− → (yx), Fx (y) = F− → (yx). B

B

B

B

Definition 9.11 For any two n-tuples (a1 , a2 , . . . , an ) and (b1 , b2 , . . . , bn ), where ai , bi , for each 1 ≤ i ≤ n, are real numbers, then 1. (a1 , a2 , . . . , an ) = (b1 , b2 , . . . , bn ) ⇔ ai = bi , for each 1 ≤ i ≤ n 2. (a1 , a2 , . . . , an ) ≤ (b1 , b2 , . . . , bn ) ⇔ ai ≤ bi , for each 1 ≤ i ≤ n 3. (a1 , a2 , . . . , an ) ≥ (b1 , b2 , . . . , bn ) ⇔ ai ≥ bi , for each 1 ≤ i ≤ n 4. (a1 , a2 , . . . , an ) × (b1 , b2 , . . . , bn ) = (a1 b1 , a2 b2 , . . . , an bn ). Definition 9.12 A bipolar single-valued neutrosophic competition graph of a − → − → bipolar single-valued neutrosophic digraph G = (A, B ) is a undirected bipolar − → single-valued neutrosophic graph C( G ) = (A, R) which has the same vertex set as − → in G , and there is an edge between two vertices x and y if and only if N + (x) ∩ N + (y) is non-empty. The positive truth-membership, indeterminacy-membership, falsitymembership, and negative truth-membership, indeterminacy-membership, falsitymembership values of the edge x y are defined as 1. 2. 3. 4. 5. 6.

TR (x y) = (T A (x) ∧ T A (y))h 1 (N + (x) ∩ N + (y)), p p p I R (x y) = (I A (x) ∧ I A (y))h 2 (N + (x) ∩ N + (y)), p p p FR (x y) = (FA (x) ∨ FA (y))h 3 (N + (x) ∩ N + (y)), n n n TR (x y) = (T A (x) ∨ T A (y))h 4 (N + (x) ∩ N + (y)), I Rn (x y) = (I An (x) ∨ I An (y))h 5 (N + (x) ∩ N + (y)), FRn (x y) = (FAn (x) ∧ FAn (y))h 6 (N + (x) ∩ N + (y)), for all x, y ∈ X . p

p

p

(0.2, 0.1, 0.1, −0.4, −0.1, −0.2)

355

, 04 − 5, 0. − 1, 0. 3, 0. 8, 0. b( 2) 0. −

a( 0. 3, 0. 8, 0. 2, − 0. 5, − 0. 2, − 0. 1)

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs

(0.3, 0.5, 0.6, −0.2, −0.2, −0.1)

(0.6, 0.2, 0.2, −0.1, −0.2, −0.3) c(0 .4,

0.5 ,

0.6 ,

−0 .2,

(0.2, 0.2, 0.2, −0.2, −0.3, −0.5) −0 .3,

−0 .5)

.7, d(0

, 0.3

.5) −0 .3, 0 − .2, −0 , 4 0.

− → Fig. 9.3 Bipolar neutrosophic digraph G Table 9.3 Bipolar neutrosophic out-neighborhoods of vertices x N + (x) a b c d

{(b, 0.2, 0.1, 0.1, –0.4, –0.1, –0.2), (c, 0.3, 0.5, 0.6, –0.2, –0.2, –0.1)} {(d, 0.6, 0.2, 0.2, –0.1, –0.2, –0.3)} ∅ {(c, 0.2, 0.2, 0.2, –0.2, –0.3, –0.5)}

Example 9.4 Consider a bipolar single-valued neutrosophic digraph as shown in Fig. 9.3. The bipolar single-valued neutrosophic out-neighborhoods are calculated in Table 9.3. The bipolar single-valued neutrosophic competition graph is shown in Fig. 9.4. Definition 9.13 Let A be a bipolar single-valued neutrosophic set on X , then the support of A, denoted by supp(A), is defined as the set of those elements x ∈ X such that A(x) = (0, 0, 1, 0, 0, −1). Example 9.5 The support of a bipolar single-valued neutrosophic set A = {(a, 0.5, 0.7, 0.2, −0.8, −0.9, −0.3), (b, 0.1, 0.2, 1, −0.5, −0.7, −0.6), (c, 0.3, 0.5, 0.3, −0.8, −0.6, −0.4), (d, 0, 0, 1, −1, −1, 0)} in X = {a, b, c, d} is supp(A) = {a, b, c}. Definition 9.14 Let p be a positive integer. Then the bipolar single-valued neu− → trosophic p-competition graph C p ( G ) of the bipolar single-valued neutrosophic

9 Bipolar Neutrosophic Competition Graphs

0.5 ,

0.6 ,

−0 .2,

−0 .3,

−0 .5)

d

2) 0. −

c(0 .4,

(0 .06 ,0 .06 ,0 .24 ,− 0.0 4, −0 .04 ,− 0.3 0)

, 04 − 5, 0. − 1, 0. 3, 0. 8, 0. b(

a( 0. 3, 0. 8, 0. 2, − 0. 5, − 0. 2, − 0. 1)

356

5) 0. − , 3 0. ,− 2 0. − 4, . ,0 .3 ,0 7 . (0

Fig. 9.4 Bipolar neutrosophic competition graph

− → − → digraph G = (A, B ) is a undirected bipolar single-valued neutrosophic graph G = (A, R ) which has the same bipolar single-valued neutrosophic set of vertices − → as in G and has a bipolar single-valued neutrosophic edge between two vertices x, − → y ∈ X in C p ( G ) if and only if |supp(N + (x) ∩ N + (y))| ≥ p. The positive truthmembership, indeterminacy-membership, falsity-membership, and negative truthmembership, indeterminacy-membership, falsity-membership values of the edge x y are defined as 1. 2. 3. 4. 5. 6.

(T A (x) ∧ T A (y))h 1 (N + (x) ∩ N + (y)), TR (x y) = (i− p)+1 i p p p (i− p)+1 (I A (x) ∧ I A (y))h 2 (N + (x) ∩ N + (y)), I R (x y) = i p p p (i− p)+1 (FA (x) ∨ FA (y))h 3 (N + (x) ∩ N + (y)), FR (x y) = i (T An (x) ∨ T An (y))h 4 (N + (x) ∩ N + (y)), TRn (x y) = (i− p)+1 i (i− p)+1 (I An (x) ∨ I An (y))h 5 (N + (x) ∩ N + (y)), I Rn (x y) = i (i− p)+1 n (FAn (x) ∧ FAn (y))h 6 (N + (x) ∩ N + (y)), for all x, y ∈ X , FR (x y) = i p

p

p

where i = |supp(N + (x) ∩ N + (y))|. The bipolar single-valued neutrosophic 3-competition graph is illustrated in Example 9.6. − → − → Example 9.6 Consider a bipolar single-valued neutrosophic digraph G = (A, B ) on X = {x, y, z, a, b, c} as shown in Fig. 9.5. The bipolar single-valued neutrosophic 3-competition graph is shown in Fig. 9.6.

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs b(0.5, 0.6, 0.7, −0.9, −0.8, −0.7)

− 0.

.6 0. −

,0

2,

.5

,0

.4

0. ,− 2,

(0 .4

c(0.5, 0.6, 0.3, −0.1, −0.2, −0.4)

(0.4

0. − 6)

y(0.6, 0.7, 0.5, 0.3, 0.2, 0.7)

5, , 0.

0.

0. 1 4, −

(0 .4

.4 (0

,0

.5

,0

,0

.

.3

0. 1 ,−

0 5,

.4

,− 0.

0. 6 ,−

0. ,−

1,

1

− 0.

)

0. ,−

2,

2,

− 0.

0. −

3)

(0.4, 0.5, 0.2, −0.1, −0.2, −0.3)

,0

,− 0.

.5

(0.3, 0.4, 0.6, −0.4, −0.5, −0.7)

4,

,0

− 0.

5,

.4 (0 x(0.7, 0.8, 0.5, 0.5, 0.6, 0.7)

(0.4, 0.4, 0.6, −0.2, −0.2, −0.6)

5)

a(0.5, 0.6, 0.7, −0.5, −0.6, −0.8)

357

6)

z(0.6, 0.7, 0.3, 0.2, 0.3, 0.4)

Fig. 9.5 Bipolar neutrosophic digraph a(0.5, 0.6, 0.7, −0.5, −0.6, −0.8)

b(0.5, 0.6, 0.7, −0.9, −0.8, −0.7)

c(0.5, 0.6, 0.3, −0.1, −0.2, −0.4)

(0.08, 0.1166, 0.066, −0.04, −0.033, −0.09933) x(0.7, 0.8, 0.5,

.5, 0.6, 0.7)

y(0.6, 0.7, 0.5, 0.3, 0.2, 0.7)

z(0.6, 0.7, 0.3, 0.2, 0.3, 0.4)

Fig. 9.6 Bipolar neutrosophic 3-competition graph

Definition 9.15 Let G = (A, B) be a bipolar single-valued neutrosophic graph on X , then an edge x y, x, y ∈ X , is called independent strong if the following conditions are satisfied. 1 n 1 p 1 p p p p p [T (x) ∧ T A (y)] < TB (x y), [T (x) ∨ T An (y)] > TBn (x y), [I (x) ∧ I A (y)] < I B (x y), 2 A 2 A 2 A 1 p 1 n 1 n p p [I (x) ∨ I An (y)] > I Bn (x y), [F (x) ∨ FA (y)] > FB (x y), [F (x) ∧ FAn (y)] < FBn (x y). 2 A 2 A 2 A

Otherwise, it is called a weak edge. − → i Theorem 9.1 Let C [ 2 ] ( G ) be a bipolar single-valued neutrosophic [ 2i ]-competition − → − → graph corresponding to G = (A, B ). If for any two vertices x, y ∈ X , h 1 (N + (x) ∩ N + (y)) = 1, h 2 (N + (x) ∩ N + (y)) = 1, h 3 (N + (x) ∩ N + (y)) = 0, h 4 (N + (x) ∩ N + (y)) = 1, h 5 (N + (x) ∩ N + (y)) = 1, h 6 (N + (x) ∩ N + (y)) = 0,

then the edge x y is strong, where i = |supp(N + (x) ∩ N + (y))|. Note that for any real number x, [x] = greatest integer not exceeding x.

358

9 Bipolar Neutrosophic Competition Graphs

− → i Proof Let C [ 2 ] ( G ) = (A, R ) be a bipolar single-valued neutrosophic [ 2i ]− → − → competition graph corresponding to G = (A, B ), then by the given statement, for all x, y ∈ X , (i − [ 2i ]) + 1 p (i − [ 2i ]) + 1 n p [T A (x) ∧ T A (y)] × 1, T Rn (x y) = [T A (x) ∨ T An (y)] × 1, i i (i − [ 2i ]) + 1 p (i − [ 2i ]) + 1 n p p I R (x y) = [I A (x) ∧ I A (y)] × 1, I Rn (x y) = [I A (x) ∨ I An (y)] × 1, i i (i − [ 2i ]) + 1 p (i − [ 2i ]) + 1 n p p FR (x y) = [F A (x) ∨ F A (y)] × 0, FRn (x y) = [F A (x) ∧ F An (y)] × 0. i i p

T R (x y) =

It clearly follows that p TR (x y) TRn (x y) (i − [ 2i ]) + 1 (i − [ 2i ]) + 1 = > 0.5, = < 0.5, p p n n i [T A (x) ∨ T A (y)] i [T A (x) ∧ T A (y)] p I Rn (x y) I R (x y) (i − [ 2i ]) + 1 (i − [ 2i ]) + 1 > 0.5, = < 0.5, = p p n n i [I A (x) ∨ I A (y)] i [I A (x) ∧ I A (y)] p FRn (x y) FR (x y) (i − [ 2i ]) + 1 (i − [ 2i ]) + 1 < 0.5, = < 0.5. = p p i [F An (x) ∧ F An (y)] i [F A (x) ∨ F A (y)]

− → i Hence, the edge x y is strong in C [ 2 ] ( G ).  − → − → Definition 9.16 Let G = (A, B ) be a bipolar single-valued neutrosophic digraph − → on X . The m-step bipolar single-valued neutrosophic digraph of G is denoted − → − → by G m = (A, R ), where the bipolar single-valued neutrosophic set of vertices of − → − → G m is same as the bipolar single-valued neutrosophic set of vertices of G , and − → there is a directed edge between x and y in G m if and only if there exists a bipolar single-valued neutrosophic directed path P : x = x1 , e1 , x2 , e2 , . . . , em−1 , xm , y of p p p p p m vertices between x and y and TR (y) = min T− → (e), I R (y) = min I− → (e), FR (y) = B

e∈P

p

e∈P

B

n n n n n n max F− → (e), I R (y) = max I− → (e), TR (y) = max T− → (e), FR (y) = min F− → (e). e∈P

B

e∈P

B

e∈P

B

B

e∈P

Definition 9.17 The m-step bipolar single-valued neutrosophic out-neighbor− → − → hood of a vertex x in a bipolar single-valued neutrosophic digraph G = (A, B ) p p p + + n n is a bipolar single-valued neutrosophic set Nm (x) = (X x , Tx , Ix , Fx , Tx , Ix , Fxn ) such that y ∈ X x+ if there exists a bipolar single-valued neutrosophic directed path P : x = x1 , e1 , x2 , e2 , . . . , em−1 , xm = y of m vertices between x and y. The membership p p p p p values of Nm+ (x) are defined as Tx (y) = min T− → (e), I x (y) = min I− → (e), Fx (y) = e∈P

p

B

e∈P

B

n n n n n n max F− → (e), I x (y) = max I− → (e), Tx (y) = max T− → (e), Fx (y) = min F− → (e). e∈P

B

e∈P

B

e∈P

B

e∈P

B

Definition 9.18 The m-step bipolar single-valued neutrosophic in-neighbor− → − → hood of a vertex x in a bipolar single-valued neutrosophic digraph G = (A, B ) p p p is bipolar single-valued neutrosophic set Nm− (x) = (X x+ , Tx , Ix , Fx , Txn , Ixn , Fxn )

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs x(0.5, 0.2, 0.7, −0.5, −0.3, −0.2)

359

a(0.4, 0.3, 0.5, −0.3, −0.7, −0.8)

(0.2, 0.2, 0.5, −0.3, −0.3, −0.3)

(0 .2 ,0 .3 ,0 .3 ,− 0. 2, − 0. 5, − 0. 2)

(0.3

, 0.2

(0.3, 0.5, 0.2, −0.3, −0.4, −0.4)

(0 .1 ,0 .3 ,0 .1 ,− 0. 2, − 0. 3, − 0. 6)

b(0.2, 0.4, 0.3, −0.7, −0.6, −0.3)

(0.4

, 0.4

, −0

.2, − 0.6,

−0. 2)

d(0.5, 0.6, 0.7, −0.6, −0.7, −0.2)

, 0.5

3, −0. 0.2, − , , 0.6

5) −0.

c(0.7, 0.6, 0.2, 0.3, 0.4, 0.7)

y(0.4, 0.6, 0.7, 0.3, 0.4, 0.5)

− → Fig. 9.7 Bipolar neutrosophic digraph G

such that y ∈ X x+ if there exists a bipolar single-valued neutrosophic directed path P : y = x1 , e1 , x2 , e2 , . . . , em−1 , xm = x of m vertices from y to x. The membership p p p p p values of Nm− (x) are defined as Tx (y) = min T− → (e), I x (y) = min I− → (e), Fx (y) = e∈P

p

B

e∈P

B

n n n n n n max F− → (e), I x (y) = max I− → (e), Tx (y) = max T− → (e), Fx (y) = min F− → (e). B

e∈P

B

e∈P

e∈P

B

e∈P

B

− → − → Example 9.7 Consider a bipolar single-valued neutrosophic digraph G = (A, B ) on X = {x, y, a, b, c, d} as shown in Fig. 9.7. The 2-step bipolar single-valued neutrosophic out-neighborhoods of vertices x and y are calculated as N2+ (x) = {(b, 0.2, 0.2, 0.5, −0.2, −0.3, −0.2), (d, 0.2, 0.2, 0.5, −0.2, −0.3, −0.3)}, N2+ (y) = {(b, 0.1, 0.3, 0.2, −0.2, −0.3, −0.6), (d, 0.3, 0.5, 0.6, −0.2, −0.3, −0.5)}. − → − → Example 9.8 Consider a bipolar single-valued neutrosophic digraph G = (A, B ) on X = {a, b, c, d, e, f } as shown in Fig. 9.8. The 2-step bipolar single-valued inneighborhood of vertices a and b are calculated as N2− (a) = {( f , 0.1, 0.1, 0.5, −0.1, −0.2, −0.6), (e, 0.3, 0.1, 0.7, −0.1, −0.2, −0.4)}, N2− (b) = {( f , 0.1, 0.3, 0.6, −0.3, −0.4, −0.7), (e, 0.4, 0.3, 0.6, −0.3, −0.4, −0.5)}. Definition 9.19 An m-step bipolar single-valued neutrosophic competition − → − → graph of a bipolar single-valued neutrosophic digraph G = (A, B ) is a undirected − → bipolar single-valued neutrosophic graph Cm ( G ) = (A, Q) which has the same ver− → tex set as in G , and there is an edge between two vertices x and y if and only if Nm+ (x) ∩ Nm+ (y) is non-empty. The positive truth-membership, indeterminacymembership, falsity-membership, and negative truth-membership, indeterminacymembership, falsity-membership values of the edge x y are defined as 1. 2. 3. 4.

TQ (x y) = (T A (x) ∧ T A (y))h 1 (Nm+ (x) ∩ Nm+ (y)), p p p I Q (x y) = (I A (x) ∧ I A (y))h 2 (Nm+ (x) ∩ Nm+ (y)), p p p FQ (x y) = (FA (x) ∨ FA (y))h 3 (Nm+ (x) ∩ Nm+ (y)), TQn (x y) = (T An (x) ∨ T An (y))h 4 (Nm+ (x) ∩ Nm+ (y)), p

p

p

360

9 Bipolar Neutrosophic Competition Graphs

b(0.5, 0.4, 0.6, −0.4, −0.5, −0.6)

d(0.6, 0.7, 0.8, −0.5, −0.6, −0.8)

(0.4, 0.3, 0.6, −0.3, −0.4, −0.5)

(0 (0.1 .4 , 0.3 ,0 .5 , 0.4 ,0 , −0 .6 ,− .3, − 0. 0.5, 4, −0. − 0. 7) 5, − 0. 5)

(0.5, 0.1, 0.5, −0.1, −0.2, −0.4) a(0.6, 0.7, 0.8, 0.2, 0.3, 0.6)

(0 .3 ,0 .1 ,0 .7 ,− 0. 3, − 0. 5, − 0. 4)

e(0.5, 0.6, 0.7, −0.5, −0.6, −0.7)

f (0.2, 0.4, 0.5, −0.6, −0.7, −0.8)

,0 (0.1

.1, 0

.4,

0.5, 5, − −0.

6) −0.

c(0.8, 0.2, 0.7, 0.8, 0.6, 0.5)

− → Fig. 9.8 Bipolar neutrosophic digraph G x(0.5, 0.2, 0.7, −0.5, −0.3, −0.2)

a(0.4, 0.3, 0.5, −0.3, −0.7, −0.8)

(0.2, 0.2, 0.5, −0.3, −0.3, −0.3)

(0 (0.3 .2 , 0.2 ,0 , 0.4 .3 ,0 , −0 .3 .2, − ,− 0. 0.6, 2, −0. − 0. 2) 5, − 0. 2)

(0.3, 0.5, 0.2, −0.3, −0.4, −0.4) y(0.4, 0.6, 0.7, 0.3,

.4, 0.5)

(0 .1 ,0 .3 ,0 .1 ,− 0. 2, − 0. 3, − 0. 6)

b(0.2, 0.4, 0.3, −0.7, −0.6, −0.3)

, (0.4

d(0.5, 0.6, 0.7, −0.6, −0.7, −0.2)

0.5,

0.6,

0.3, 2, − −0.

5) −0.

c(0.7, 0.6, 0.2, 0.3, 0.4, 0.7)

− → Fig. 9.9 Bipolar neutrosophic digraph G

5. I Qn (x y) = (I An (x) ∨ I An (y))h 5 (Nm+ (x) ∩ Nm+ (y)), 6. FQn (x y) = (FAn (x) ∧ FAn (y))h 6 (Nm+ (x) ∩ Nm+ (y)),

for all x, y ∈ X .

− → − → Example 9.9 Consider a bipolar single-valued neutrosophic digraph G = (A, B ) as shown in Fig. 9.9. The corresponding 2-step bipolar single-valued neutrosophic competition graph is shown in Fig. 9.10. → If a predator x attacks one prey y, then the linkage is shown by an edge − x y in a bipolar single-valued neutrosophic digraph. But, if the predator needs help of many other mediators x1 , x2 , …, xm−1 , then the linkage among them is shown by a bipolar single− → valued neutrosophic directed path P mx,y with a vertex set {x, x1 , x2 , …, xm−1 , y}. So

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs

361

a(0.4, 0.3, 0.5, −0.3, −0.7, −0.8)

x(0.5, 0.2, 0.7, −0.5, −0.3, −0.2)

(0.08, 0.04, 0.35, −0.06, −0.0

b(0.2, 0.4, 0.3, −0.7, −0.6, −0.3) d(0.5, 0.6, 0.7, −0.6, −0.7, −0.2)

6, −0.25) y(0.4, 0.6, 0.7, 0.3, 0.4, 0.5)

c(0.7, 0.6, 0.2, 0.3, 0.4, 0.7)

Fig. 9.10 2-step bipolar single-valued neutrosophic competition graph

the m-step prey in a bipolar single-valued neutrosophic digraph is represented by a vertex which is in the m-step bipolar single-valued neutrosophic out-neighborhood of the other vertex. − → − → Definition 9.20 Let G = (A, B ) be a bipolar single-valued neutrosophic digraph. Let w be a common vertex of m-step out-neighborhoods of vertices x1 , x2 ,…, xl . Also, p p p let T− → (u 1 v1 ), T− → (u 2 v2 ),…, T− → (u l vl ) be the minimum positive truth-membership B B B p p p values, I− → (u 1 v1 ), I− → (u 2 v2 ),…, I− → (u l vl ) be the minimum positive indeterminacyB B B p p p membership values, F− → (u 1 v1 ), F− → (u 2 v2 ),…, F− → (u l vl ) be the maximum positive B B B n n n false-membership values, T− → (u 1 v1 ), T− → (u 2 v2 ),…, T− → (u l vl ) be the maximum negaB B B n n tive truth-membership values, I− I− → (u 1 v1 ), → (u 2 v2 ), B B n …, I− → (u l vl ) be the maximum negative indeterminacy-membership values, and B n n n F− → (u 1 v1 ), F− → (u 2 v2 ),…, F− → (u l , vl ) be the minimum negative false-membership B B B − → − → values of edges of bipolar single-valued neutrosophic directed paths P mx1 ,w , P mx2 ,w , − → …, P mxl ,w . The m-step prey w ∈ X is called a strong prey if the following conditions are satisfied. p p p B B B n n n T− → (u i vi ) < −0.5, I− → (u i vi ) < −0.5, F− → (u i vi ) > −0.5, B B B

T− → (u i vi ) > 0.5, I− → (u i vi ) > 0.5, F− → (u i vi ) < 0.5,

for all i = 1, 2, . . . , l.

The strength of the prey w can be measured by the mapping S : X → [0, 1] such that S(w) =

1 l

 l  i=1

p

T− → (u i vi ) + B

l  i=1

p

I− → (u i vi ) + B

l  i=1

p

F− → (u i vi ) − B

l  i=1

n T− → (u i vi ) − B

l  i=1

n I− → (u i vi ) − B

l  i=1

 n F− → (u i vi ) . B

− → Example 9.10 Consider the bipolar single-valued neutrosophic digraph G = − → (A, B ) as shown in Fig. 9.9. The strength of the prey b is equal to

362

9 Bipolar Neutrosophic Competition Graphs

[0.2 + 0.2] + [0.2 + 0.3] + [0.5 + 0.3] − [−0.3 − 0.2] − [−0.3 − 0.5] − [−0.3 − 0.2] = 1.75 > 0.5. 2

Hence, b is a strong 2-step prey. − → − → Theorem 9.2 If a prey w of G = (A, B ) is strong, then the strength of w, S(w) > 0.5. − → − → Proof Let G = (A, B ) be a bipolar single-valued neutrosophic digraph on X . Let w be a common vertex of m-step out-neighborhoods of vertices x1 , x2 ,…, xl , i.e., there − → − → − → exist bipolar single-valued neutrosophic directed paths P mx1 ,w , P mx2 ,w ,…, P mxl ,w , − → p p p in G . Let T− → (u 1 v1 ), T− → (u 2 v2 ),…, T− → (u l vl ) be the minimum positive truthB B B p p p membership values, I− → (u 1 v1 ), I− → (u 2 v2 ), . . ., I− → (u l vl ) be the minimum positive B B B p p p indeterminacy-membership values, F− → (u 1 v1 ), F− → (u 2 v2 ),…, F− → (u l vl ) be the maxB B B n n n imum positive false-membership values, T− → (u 1 v1 ), T− → (u 2 v2 ),…, T− → (u l vl ) be the B B B n n maximum negative truth-membership values, I− I− → (u 1 v1 ), → (u 2 v2 ), B B n …, I− → (u l vl ) be the maximum negative indeterminacy-membership values, and B n n n F− → (u 1 v1 ), F− → (u 2 v2 ),…, F− → (u l vl ) be the minimum negative false-membership B B B − → values of bipolar single-valued neutrosophic edges of the directed paths P mx1 ,w , − → − →m P x2 ,w ,…, P mxl ,w , respectively. If w is strong, then each edge u i vi , i = 1, 2, . . . , l is strong and so, p

p

p

B

B

T− → (u i vi ) > 0.5, I− → (u i vi ) > 0.5, F− → (u i vi ) < 0.5, B n T− → (u i vi ) B

n n < −0.5, I− → (u i vi ) < −0.5, F− → (u i vi ) > −0.5, for all i = 1, 2, . . . , l. B

B

0.5 + 0.5 + · · · (l times) + 0.5 > 0.5. S(w) > l



It completes the proof.

Remark 9.1 The converse of Theorem 9.2 is not true in general, i.e., if S(w) > 0.5, then all preys may not be strong. It can be explained as follows: Let S(w) > 0.5 for − → a prey w in G , then S(w) =

1 l

 l  i=1

p

T− → (u i vi ) + B

l  i=1

p

I− → (u i vi ) + B

l  i=1

p

F− → (u i vi ) − B

l  i=1

n T− → (u i vi ) − B

l  i=1

n I− → (u i vi ) − B

l  i=1

 n F− → (u i vi ) . B

Hence, l  i=1

p

T− → (u i vi ) + B

l  i=1

p

I− → (u i vi ) + B

l  i=1

p

F− → (u i vi ) − B

l  i=1

n T− → (u i vi ) − B

l  i=1

n I− → (u i vi ) − B

l  i=1

n F− → (u i vi ) > B

l . 2

This result does not necessarily imply that p p p B B B n n n T− → (u i vi ) < −0.5, I− → (u i vi ) < −0.5, F− → (u i vi ) > −0.5, B B B

T− → (u i vi ) > 0.5, I− → (u i vi ) > 0.5, F− → (u i vi ) < 0.5,

for all i = 1, 2, . . . , l.

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs

363

− → Since all edges of the bipolar single-valued neutrosophic directed paths P mx1 ,w , − → − →m P x2 ,w ,…, P mxl ,w are not strong, the converse Theorem 9.2 is not true, i.e., if S(w) > − → 0.5, the prey w of G may not be strong. − → − → − → Theorem 9.3 If all preys of G = (A, B ) are strong, then all edges of Cm ( G ) = (A, B) are strong. − → − → Proof Let G = (A, B ) be a bipolar single-valued neutrosophic digraph and all − → preys of it are strong. Let Cm ( G ) = (A, R), then TR (x y) = [T A (x) ∧ T A (z)]h 1 (Nm+ (x) ∩ Nm+ (y)), p

p

p I R (x y) p FR (x y) TRn (x y) I Rn (x y) FRn (x y)

= = = = =

p

p p [I A (x) ∧ I A (y)]h 2 (Nm+ (x) ∩ Nm+ (y)), p p [FA (x) ∨ FA (y)]h 3 (Nm+ (x) ∩ Nm+ (y)), [T An (x) ∨ T An (z)]h 4 (Nm+ (x) ∩ Nm+ (y)), [I An (x) ∨ I An (y)]h 5 (Nm+ (x) ∩ Nm+ (y)), [FAn (x) ∧ FAn (y)]h 6 (Nm+ (x) ∩ Nm+ (y)),

− → for all edges x y in Cm ( G ) = (A, R). Then there are two cases. Case 1: If Nm+ (x) ∩ Nm+ (y) = ∅, then there does not exist any edge between x − → and y in Cm ( G ). + Case 2: If Nm (x) ∩ Nm+ (y) = ∅, then h 1 (Nm+ (x) ∩ Nm+ (y)) > 0.5, h 2 (Nm+ (x) ∩ Nm+ (y)) > 0.5, h 3 (Nm+ (x) ∩ Nm+ (y)) < 0.5, h 4 (Nm+ (x) ∩ Nm+ (y)) > 0.5, h 5 (Nm+ (x) ∩ Nm+ (y)) > 0.5, h 6 (Nm+ (x) ∩ Nm+ (y)) < 0.5,

− → − → in G as all preys are strong. So the edge x y in Cm ( G ) has the memberships values, TR (x y) = [T A (x) ∧ T A (z)]h 1 (Nm+ (x) ∩ Nm+ (y)), I R (x y) = [I A (x) ∧ I A (y)]h 2 (Nm+ (x) ∩ Nm+ (y)), p

p FR (x y) I Rn (x y)

p

= =

p

p

p

p

p p [FA (x) ∨ FA (y)]h 3 (Nm+ (x) ∩ Nm+ (y)), TRn (x y) = [T An (x) ∨ T An (z)]h 4 (Nm+ (x) ∩ Nm+ (y)), [I An (x) ∨ I An (y)]h 5 (Nm+ (x) ∩ Nm+ (y)), FRn (x y) = [FAn (x) ∧ FAn (y)]h 6 (Nm+ (x) ∩ Nm+ (y)).

Hence all the edges are strong.



A relation is established between the m-step bipolar single-valued neutrosophic competition graph of a bipolar single-valued neutrosophic digraph and a bipolar single-valued neutrosophic competition graph of the m-step bipolar single-valued neutrosophic digraph. − → − → Theorem 9.4 Let G = (A, B ) be a bipolar single-valued neutrosophic digraph − → − → on X , then C( G m ) = Cm ( G ). − → − → − → Proof Let G = (A, B ) be a bipolar single-valued neutrosophic digraph and G m = − → − → (A, J ) be the m-step bipolar single-valued neutrosophic digraph of G . Also, let

364

9 Bipolar Neutrosophic Competition Graphs

− → − → C( G m ) = (A, J ) (say) and Cm ( G ) = (A, B). It can be easily observed that bipolar − → − → − → single-valued neutrosophic vertex sets of G m , Cm ( G ), and C( G m ) are the same. So, − → we have to show that the bipolar single-valued neutrosophic edge sets of C( G m ) and − → − → Cm ( G ) are equal. Let x y be an edge in C( G m ), then there exist bipolar single-valued → −→ −→ − → − → →, − neutrosophic directed edges − xa 1 ya1 ; xa2 , ya2 ,…, xal , yal , for some positive integer − → l. Assume that for any two vertices x and y of G m , N + (x) ∩ N + (y) = {ai | i = 1, 2, . . . , l}, where − → − → p p n n si = T− → (xai ) ∨ J (y, ai ), → (xai ) ∧ J (y, ai ), si = T− J J − → − → p p n n qi = I − → (xai ) ∨ J (y, ai ), → (xai ) ∧ J (y, ai ), qi = I− J J − → − → p p n n ri = F− → (xai ) ∧ J (y, ai ). → (xai ) ∨ J (y, ai ), ri = F− J

J

p

p

Let S p = max{si | i = 1, 2, . . . , l}, Q p = max{qi | i = 1, 2, . . . , l}, and R p = p min{ri | i = 1, 2, . . . , l}, then TJ (x y) = (T A (x) ∧ T A (y))h 1 (N + (x) ∩ N + (y)) = S p × T A (x) ∧ T A (y), p p p p p I J (x y) = (I A (x) ∧ I A (y))h 2 (N + (x) ∩ N + (y)) = Q p × I A (x) ∧ I A (y), p

p

p

p

p

FJ (x y) = (FA (x) ∨ FA (y))h 3 (N + (x) ∩ N + (y)) = R p × FA (x) ∨ FA (y), TJn (x y) = (T An (x) ∨ T An (y))h 4 (N + (x) ∩ N + (y)) = S p × T An (x) ∨ T An (y), p

p

p

p

p

I Jn (x y) = (I An (x) ∨ I An (y))h 5 (N + (x) ∩ N + (y)) = Q p × I An (x) ∨ I An (y), FJn (x y) = (FAn (x) ∧ FAn (y))h 6 (N + (x) ∩ N + (y)) = R p × FAn (x) ∧ FAn (y). → → that exists in − An edge − xa G m implies that there exists a bipolar single-valued i − →m neutrosophic directed path P x,ai from x to ai with m vertices such that − →m p p T− → (xai ) = min{T− → (uv)|uv is an edge in P x,ai }, J B − →m p p I− → (xai ) = min{I− → (uv)|uv is an edge in P x,ai }, J B − →m p p F− → (xai ) = max{F− → (uv)|uv is an edge in P x,ai }, J B − →m n n T− → (xai ) = max{T− → (uv)|uv is an edge in P x,ai }, J B − →m n n I− → (xai ) = max{I− → (uv)|uv is an edge in P x,ai }, J B − →m n n F− → (xai ) = min{F− → (uv)|uv is an edge in P x,ai }. J

B

− → Thus, the edge x y is also available in Cm ( G ), then

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs

365

h 1 (Nm+ (x) ∩ Nm+ (y)) = S p , h 4 (Nm+ (x) ∩ Nm+ (y)) = S p , h 2 (Nm+ (x) ∩ Nm+ (y)) = Q p , h 5 (Nm+ (x) ∩ Nm+ (y)) = Q p , h 3 (Nm+ (x) ∩ Nm+ (y)) = R p , h 6 (Nm+ (x) ∩ Nm+ (y)) = R p . It clearly follows that TB (x y) = [T A (x) ∧ T A (z)]h 1 (Nm+ (x) ∩ Nm+ (y)) = S p × T A (x) ∧ T A (z), p

p

p

p

p

I B (x y) = [I A (x) ∧ I A (y)]h 2 (Nm+ (x) ∩ Nm+ (y)) = Q p × I A (x) ∧ I A (y), p p p p p FB (x y) = [FA (x) ∨ FA (y)]h 3 (Nm+ (x) ∩ Nm+ (y)) = R p × FA (x) ∨ FA (y), p

p

p

p

p

TBn (x y) = [T An (x) ∨ T An (z)]h 4 (Nm+ (x) ∩ Nm+ (y)) = S p × T An (x) ∨ T An (z), I Bn (x y) = [I An (x) ∨ I An (y)]h 5 (Nm+ (x) ∩ Nm+ (y)) = Q p × I An (x) ∨ I An (y),

FBn (x y) = [FAn (x) ∧ FAn (y)]h 6 (Nm+ (x) ∩ Nm+ (y)) = R p × FAn (x) ∧ FAn (y). − → − → It shows that there exists an edge in Cm ( G ) for every edge in C( G m ). Similarly, for − → − → every edge in Cm ( G ) there exists an edge in C( G m ).  − → − → Theorem 9.5 Let G = (A, B ) be a bipolar single-valued neutrosophic digraph. − → If m > |X |, then Cm ( G ) = (A, B) has no edge. Proof If m > |X |, then there does not exist any bipolar single-valued neutrosophic − → directed path with m vertices in G . So Nm+ (x) ∩ Nm+ (y) is an empty set, for all − → x, y ∈ X . Hence, there does not exist any edge in Cm ( G ).  Theorem 9.6 If all the edges of the bipolar single-valued neutrosophic digraph − → − → − → G = (A, B ) are independent strong, then all the edges of Cm ( G ) are independent strong. − → − → Proof Suppose G = (A, B ) is a bipolar single-valued neutrosophic digraph and − → Cm ( G ) = (A, B) is the corresponding m-step bipolar single-valued neutrosophic − → competition graph. Since all the edges of G are independent strong, h 1 (Nm+ (x) ∩ Nm+ (y)) > 0.5, h 2 (Nm+ (x) ∩ Nm+ (y)) > 0.5, h 3 (Nm+ (w) ∩ Nm+ (z)) < 0.5, + h 4 (Nm+ (x) ∩ Nm+ (y)) < 0.5, h 5 (Nm+ (x) ∩ Nm+ (y)) < 0.5, h 6 (N+ m (w) ∩ Nm (z)) < 0.5.

TB (x y) = (T A (x) ∧ T A (y))h 1 (Nm+ (x) ∩ Nm+ (y)) p

p

p

or,

p

TB (x y) or, (T p (x)∧T > 0.5, p A A (y)) p p p I B (x y) = (I A (x) ∧ I A (y))h 2 (Nm+ (x) p I B (x y) > 0.5, or, (I p (x)∧I p A A (y))

or,

p

p FB (x y) p p (FA (x)∨FA (y))

p

< 0.5,

p

p

∩ Nm+ (y)) or, I B (x y) > 0.5(I A (x) ∧ I A (y))

FB (x y) = (F A (x) ∨ F A (y))h 3 (Nm+ (x) ∩ Nm+ (y)) p

p

TB (x y) > 0.5(T A (x) ∧ T A (y)) p

or,

p

p

p

p

p

FB (x y) < 0.5(F A (x) ∨ F A (y))

366

9 Bipolar Neutrosophic Competition Graphs

TBn (x y) = (T An (x) ∨ T An (y))h 4 (Nm+ (x) ∩ Nm+ (y)) or, TBn (x y) < 0.5(T An (x) ∨ T An (y)) TBn (x y) or, (T n (x)∨T < 0.5, I Bn (x y) = (I An (x) ∨ I An (y))h 5 (Nm+ (x) ∩ Nm+ (y)) or, n (y)) A

A

I Bn (x y) < 0.5(I An (x) ∨ I An (y)) or,

I Bn (x y) (I An (x)∨I An (y))

FBn (x y) < 0.5(FAn (x) ∧ FAn (y)) or,

< 0.5, FBn (x y) = (FAn (x) ∧ FAn (y))h 6 (Nm+ (x) ∩ Nm+ (y))

or,

FBn (x y) (FAn (x)∧FAn (y))

< 0.5. − → Hence, the edge x y is independent strong in Cm ( G ). Since x y is taken to be an − → − → arbitrary edge of Cm ( G ), all the edges of Cm ( G ) are independent strong.  Definition 9.21 An m-step bipolar single-valued neutrosophic neighborhood of a vertex x in a bipolar single-valued neutrosophic graph G = (A, B) is a p p p bipolar single-valued neutrosophic set Nm (x) = (X x , Tx , Ix , Fx , Txn , Ixn , Fxn ) such that y ∈ X x+ if there exists a bipolar single-valued neutrosophic directed path p P : x = x1 , e1 , x2 , e2 , . . . , em−1 , xm = y of m vertices between x and y and Tx (y) = p p p p p n min TB (e), Ix (y) = min I B (e), Fx (y) = max FB (e), Ixn (y) = max I B (e), Txn (y) = e∈P

e∈P

max TBn (e), Fxn (y) = min FBn (e). e∈P

e∈P

e∈P

e∈P

Definition 9.22 An m-step bipolar single-valued neutrosophic neighborhood graph of a bipolar single-valued neutrosophic graph G = (A, B) is a bipolar single− → valued neutrosophic graph Cm ( G ) = (A, Q) which has the same vertex set as in G, and there is an edge between two vertices x and y if and only if Nm (x) ∩ Nm (y) is non-empty. The positive truth-membership, indeterminacy-membership, falsitymembership, and negative truth-membership, indeterminacy-membership, falsitymembership values of the edge x y are defined as 1. 2. 3. 4. 5. 6.

p

p

p

TQ (x y) = (T A (x) ∧ T A (y))h 1 (Nm (x) ∩ Nm (y)), p p p I Q (x y) = (I A (x) ∧ I A (y))h 2 (Nm (x) ∩ Nm (y)), p p p FQ (x y) = (FA (x) ∨ FA (y))h 3 (Nm (x) ∩ Nm (y)), n n n TQ (x y) = (T A (x) ∨ T A (y))h 4 (Nm (x) ∩ Nm (y)), I Qn (x y) = (I An (x) ∨ I An (y))h 5 (Nm (x) ∩ Nm (y)), FQn (x y) = (FAn (x) ∧ FAn (y))h 6 (Nm (x) ∩ Nm+ (y)),

for all x, y ∈ X .

− → Theorem 9.7 Suppose G is a bipolar single-valued neutrosophic digraph. If N + (x) − → − → ∩ N + (y) contains only one element of G , then the edge x y of C( G ) is independent strong if and only if |[N + (x) ∩ N + (y)]|T p > 0.5, |[N + (x) ∩ N + (y)]| I p > 0.5, |[N + (x) ∩ N + (y)]| F p < 0.5.

− → Proof Suppose, G is a bipolar single-valued neutrosophic digraph. Suppose N + (x) + ∩ N (y) = (a, p˘p , q p , r p , p˘n , q n , r n ), where p˘p , q p , r p , p˘n , q n , r n are the positive truth-membership, indeterminacy-membership, falsity-membership, and negative truth-membership, indeterminacy-membership, falsity-membership values of either the edge (x, a) or the edge (y, a), respectively, such that

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs

367

|[N + (x) ∩ N + (y)]|T p = p˘p = h 1 (N + (x) ∩ N + (y)) = h 4 (N + (x) ∩ N + (y)), |[N + (x) ∩ N + (y)]| I p = q p = h 2 (N + (x) ∩ N + (y)) = h 5 (N + (x) ∩ N + (y)), |[N + (x) ∩ N + (y)]| F p = r p = h 3 (N + (x) ∩ N + (y)) = h 6 (N + (x) ∩ N + (y)). It then follows that p p p TB (x y) = p˘p × [T A (x) ∧ T A (y)], TBn (x y) = p˘p × [T An (x) ∨ T An (y)], p

p

p

I B (x y) = q p × [I A (x) ∧ I A (y)], I Bn (x y) = q p × [I An (x) ∨ I An (y)], p p p FB (x y) = r p × [FA (x) ∨ FA (y)], FBn (x y) = r p × [FAn (x) ∧ FAn (y)]. − → Therefore, the edge x y in C( G ) is independent strong if and only if p˘p > 0.5, − → q p > 0.5, and r p < 0.5. Hence, the edge x y of C( G ) is independent strong if and only if |[N + (x) ∩ N + (y)]|T p > 0.5, |[N + (x) ∩ N + (y)]| I p > 0.5, |[N + (x) ∩ N + (y)]| F p < 0.5.



The proof is complete.

− → Theorem 9.8 If all the edges of a bipolar single-valued neutrosophic digraph G = − → − → (A, B ) are independent strong, then for all edges x y in C( G ) = (A, R), p

TR (x y) > 0.5, p p (T A (x) ∧ T A (y))2

TRn (x y) < 0.5, (T An (x) ∨ T An (y))2

I Rn (x y) < 0.5, n (I A (x) ∨ I An (y))2

FR (x y) < 0.5, p p (FA (x) ∨ FA (y))2

p

p

I R (x y) > 0.5, p p (I A (x) ∧ I A (y))2 FRn (x y) n (FA (x) ∧ FAn (y))2

< 0.5.

− → Proof Suppose all the edges of the bipolar single-valued neutrosophic digraph G = − → − → (A, B ) are independent strong, then for all the edges x y in G , 1 p p (T (x) ∧ T A (y)) < 2 A 1 n (I (x) ∨ I An (y)) > 2 A

p B

T− → (x y), n I− → (x y), B

1 n 1 p p p n (T (x) ∨ T An (y)) > T− (I (x) ∧ I A (y)) < I− → (x y), → (x y), B 2 A 2 A B 1 p 1 n p p n (F (x) ∨ F A (y)) > F− (F (x) ∧ F An (y)) < F− → (x y). → (x y), B 2 A 2 A B

Let the corresponding bipolar single-valued neutrosophic competition graph be − → C( G ) = (A, R), then there are two cases: Case 1: When N + (x) ∩ N + (y) = ∅ for all x, y ∈ X , then there exists no edge in − → C( G ) between x and y. Thus, there is nothing to prove in this case. p p p Case 2: When N + (x) ∩ N + (y) = ∅. Let N + (x) ∩ N + (y) = {(a1 , m 1 , r1 , p˘ 1 , m n1 , p p p p p p p n n n n n n n n r1 , p˘ 1 ), (a2 , m 2 , r2 , p˘ 2 , m 2 , r2 , p˘ 2 ), …, (al , m l , rl , p˘l , m l , rl , p˘l )}, where m i , p p n n n ri , p˘ i , m i , ri , p˘ i are the positive and negative truth-membership, indeterminacy→ for i = 1, 2, . . . , l, → or − ya membership, and falsity-membership values of either − xa i i respectively. Thus,

368

9 Bipolar Neutrosophic Competition Graphs

p

p

p

p

p

p

n n n m i = [T− → (xai ) ∨ T− → (yai )], ri = [I− → (xai ) ∧ T− → (yai )], m i = [T− → (xai ) ∧ I− → (yai )], B B n n [I− → (xai ) ∨ I− → (yai )], B B

rin =

p

p˘ i =

B p [F− → (xai ) ∨ B

B p F− → (yai )], B

p˘ in =

B B n n [F− → (xai ) ∨ F− → (yai )], B B

i = 1, 2, . . . , l.

Suppose that p h 1 (N + (x) ∩ N + (y)) = max{m i , i = 1, 2, . . . , l} = m max = h 4 (N + (x) ∩ N + (y)), p

+

+

p max{ri , p min{ p˘ i ,

h 2 (N (x) ∩ N (y)) = h 3 (N + (x) ∩ N + (y)) = p

p

i = 1, 2, . . . , l} = p˘ min = h 6 (N + (x) ∩ N + (y)). p

p

p

p

p

p

n m max > T− ˘ min < F− → (x y), r max > I− → (x y), p → (x y), m max > T− → (x y), B B B B p − → n n > I− ˘ min > F− → (x y), p → (x y) for all edges x y shows that

Obviously, rmax

p

p i = 1, 2, . . . , l} = rmax = h 5 (N + (x) ∩ N + (y)),

B

B

p

p

T A (x) ∧ T A (y)

>

p B

T− → (x y)

p

m max

p

p

T A (x) ∧ T A (y)

n p T− → (x y) m max B < < 0.5, T An (x) ∨ T An (y) T An (x) ∨ T An (y)

> 0.5,

p n p p I− I− → (x y) → (x y) rmax rmax B < n B < 0.5, > p > 0.5, p p p n n I A (x) ∨ I A (y) I A (x) ∨ I An (y) I A (x) ∧ I A (y) I A (x) ∧ I A (y) p n p p F− F− → (x y) → (x y) p˘ min p˘ min B B < < 0.5, < < 0.5. p p p p F An (x) ∧ F An (y) F An (x) ∧ F An (y) F A (x) ∨ F A (y) F A (x) ∨ F A (y)

Thus, TR (x y) = (T A (x) ∧ T A (y))h 1 (N + (x) ∩ N + (y)) ⇒ TR (x y) = [T A (x) ∧ p p T A (y)] × m max , p p p TR (x y) TR (x y) p m max = (T p (x)∧T > 0.5, or (T p (x)∧T p (y)) = m max ⇒ (T p (x)∧T p p (y))2 (y)) p

A

p

A

p

A

p

A

A

A

p

I R (x y) = (I A (x) ∧ I A (y))h 2 (N + (x) ∩ N + (y)) ⇒ I R (x y) = [I A (x) ∧ I A (y)] × rmax p

or

p

p I R (x y) p p (I A (x)∧I A (y))

p FR (x y)

or

=

=

=

=

(T An (x) ∨

=

=

p p˘ min

⇒

p

⇒

p

= rmax ⇒

FR (x y) p p (FA (x)∨FA (y))2

=

TRn (x y) (T An (x)∨T An (y))2

=

(FAn (x) ∧

FRn (x y) (FAn (x)∧FAn (y))

=

I Rn (x y) (I An (x)∨I An (y))2

=

⇒

FRn (x y) (FAn (x)∧FAn (y))2

p

⇒

p

p

p

=

⇒

FRn (x y) p

p

p

< 0.5, p

= [T An (x) ∨ T An (y)] × m max ,

m max (T An (x)∨T An (y))

I Rn (x y)

p

= [FA (x) ∨ FA (y)] × p˘ min ,

p˘ min p p (FA (x)∨FA (y))

TRn (x y)

p

> 0.5,

p

rmax (FAn (x)∨I An (y))

=

p

p FR (x y)

⇒

⇒

FAn (y))h 6 (N + (x) ∩ N + (y)) p p˘ min

p rmax

p

T An (y))h 4 (N + (x) ∩ N + (y)) p m max

p

(I A (x)∧I A (y))

p FA (y))h 3 (N + (x) ∩ N + (y))

(I An (x) ∨ I An (y))h 5 (N + (x) ∩ N + (y))

I Rn (x y) (I An (x)∨I An (y))

FRn (x y)

or

p

p

p (FA (x) ∨

TRn (x y) (T An (x)∨T An (y))

I Rn (x y)

or

=

p I R (x y) p p (I A (x)∧I A (y))2

= rmax , ⇒

FR (x y) p p (FA (x)∨FA (y))

TRn (x y)

or

p

< 0.5,

p [I An (x) ∨ I An (y)] × rmax ,

< 0.5, p

= [FAn (x) ∧ FAn (y)] × p˘ min ,

p˘ min (FAn (x)∧FAn (y))

< 0.5.

9.2 Certain Types of Bipolar Neutrosophic Competition Graphs

p

TR (x y) > 0.5, p p (T A (x) ∧ T A (y))2

Hence,

p I R (x y) p p (I A (x) ∧ I A (y))2 p FR (x y) p p (FA (x) ∨ FA (y))2

> 0.5, < 0.5,

369

TRn (x y) < 0.5, n (T A (x) ∨ T An (y))2 I Rn (x y) < 0.5, n (I A (x) ∨ I An (y))2 FRn (x y) < 0.5, (FAn (x) ∧ FAn (y))2

− → for all edges x y in C( G ).



9.3 Bipolar Neutrosophic Neighborhood Graphs In this section, various types of bipolar single-valued neutrosophic open neighborhood graphs are discussed in relation with m-step bipolar single-valued neutrosophic competition graphs. Definition 9.23 Let G = (A, B) be a bipolar single-valued neutrosophic graph on X . A bipolar single-valued neutrosophic open neighborhood of a vertex x ∈ X is p p p a bipolar single-valued neutrosophic set N (x) = (X x , Tx , Ix , Fx , Txn , Ixn , Fxn ) such p p p that X x = {y | B(x y) = (0, 0, 1, 0, 0, −1)} and Tx , Ix , Fx : X × X → [0, 1], Txn , p p p p p Ixn , f xn : X × X → [−1, 0] are defined by Tx (y) = TB (x y), Ix (y) = I B (x y), Fx (y) p n n n n n n = FB (x y), Tx (y) = TB (x y), Ix (y) = I B (x y), Fx (y) = FB (x y). The bipolar singlevalued neutrosophic closed neighborhood of a vertex x is defined as N [x] = N (x) ∪ A(x). Definition 9.24 A bipolar single-valued neutrosophic open neighborhood graph of a bipolar single-valued neutrosophic graph G = (A, B) is a bipolar singlevalued neutrosophic graph C(G) = (A, R) which has the same vertex set as in G, and there is an edge between two vertices x and y if and only if N (x) ∩ N (y) is non-empty. The positive truth-membership, indeterminacy-membership, falsitymembership, and negative truth-membership, indeterminacy-membership, falsitymembership values of the edge x y are defined as 1. 2. 3. 4. 5. 6.

p

p

p

TR (x y) = (T A (x) ∧ T A (y))h 1 (N (x) ∩ N (y)), p p p I R (x y) = (I A (x) ∧ I A (y))h 2 (N (x) ∩ N (y)), p p p FR (x y) = (FA (x) ∨ FA (y))h 3 (N (x) ∩ N (y)), TRn (x y) = (T An (x) ∨ T An (y))h 4 (N (x) ∩ N (y)), I Rn (x y) = (I An (x) ∨ I An (y))h 5 (N (x) ∩ N (y)), FRn (x y) = (FAn (x) ∧ FAn (y))h 6 (N (x) ∩ N + (y)),

for all x, y ∈ X .

Definition 9.25 A bipolar single-valued neutrosophic closed neighborhood graph of a bipolar single-valued neutrosophic graph G = (A, B) is a bipolar singlevalued neutrosophic graph C[G] = (A, R ) which has the same vertex set as in G,

370

9 Bipolar Neutrosophic Competition Graphs a(0.7, 0.8, 0.6, −0.5, −0.3, −0.7)

b(0.3, 0.5, 0.7, −0.7, −0.8, −0.9)

(0.6, 0.6, 0.5, −0.2, −0.5, −0.7)

c(0.8, 0.9, 0.7, 0.5, 0.6, 0.7)

(0.2, 0.4, 0.6, −0.2, −0.6, −0.8)

(0.6, 0.7, 0.6, −0.5, −0.2, −0.6)

(0.2, 0.4, 0.6, −0.4, −0.2, −0.8)

d(0.8, 0.7, 0.6, 0.3, 0.7, 0.8)

Fig. 9.11 Bipolar neutrosophic graph G

and there is an edge between two vertices x and y if and only if N [x] ∩ N [y] is non-empty. The positive truth-membership, indeterminacy-membership, falsitymembership, and negative truth-membership, indeterminacy-membership, falsitymembership values of the edge x y are defined as p

p

p

TR (x y) = (T A (x) ∧ T A (y))h 1 (N [x] ∩ N [y]), p p p I R (x y) = (I A (x) ∧ I A (y))h 2 (N [x] ∩ N [y]), p p p FR (x y) = (FA (x) ∨ FA (y))h 3 (N [x] ∩ N [y]), TRn (x y) = (T An (x) ∨ T An (y))h 4 (N [x] ∩ N [y]), I Rn (x y) = (I An (x) ∨ I An (y))h 5 (N [x] ∩ N [y]), FRn (x y) = (FAn (x) ∧ FAn (y))h 6 (N [x] ∩ N + [y]),

1. 2. 3. 4. 5. 6.

for all x, y ∈ X .

Example 9.11 Consider a bipolar single-valued neutrosophic graph G = (A, B) on X = {a, b, c, d} as shown in Fig. 9.11. The corresponding bipolar single-valued neutrosophic open and closed neighborhood graphs are shown in Figs. 9.12 and 9.13, respectively. Theorem 9.9 For each edge of a bipolar single-valued neutrosophic graph G, there exists an edge in N [G]. Proof Suppose x y is an edge of a bipolar single-valued neutrosophic graph G = (A, B). Suppose N [G] = (A, B ) is the corresponding bipolar single-valued neutrosophic closed neighborhood graph. Suppose x, y ∈ supp(N [x] ∩ N [y]), then h(N [x] ∩ N [y]) = (0, 0, 1, 0, 0, 1). p

p

p

TB (x y) = [T A (x) ∧ T A (y)]h 1 (N [x] ∩ N [y]) = 0, TBn (x y) = [T An (x) ∨ T An (y)]h 4 (N [x] ∩ N [y]) = 0, p I B (x y) p FB (x y)

= =

p p [I A (x) ∧ I A (y)]h 2 (N [x] ∩ N [y]) = 1, p p [FA (x) ∨ FA (y)]h 3 (N [x] ∩ N [y]) = 0,

I Bn (x y) = [I An (x) ∨ I An (y)]h 5 (N [x] ∩ N [y]) = 0, FBn (x y) = [FAn (x) ∧ FAn (y)]h 6 (N [x] ∩ N [y]) = −1.

Hence, for each edge x y in bipolar single-valued neutrosophic graph G, there exists an edge x y in N [G]. 

9.3 Bipolar Neutrosophic Neighborhood Graphs

371

a(0.7, 0.8, 0.6, −0.5, −0.3, −0.7)

(0.42

, 0.42

, 0.36

, −0.

18, − 0.18,

0. 2 6, 0 −0.4 . 8) (0

0 0,

.4

b(0.3, 0.5, 0.7, −0.7, −0.8, −0.9) ) . 54 −0 , . 24 −0 , 1 . −0 2,

d(0.8, 0.7, 0.6, 0.3, 0.7, 0.8)

c(0.8, 0.9, 0.7, 0.5, 0.6, 0.7)

Fig. 9.12 Bipolar neutrosophic open neighborhood graph N (G)

c(0.8, 0.9, 0.7, 0.5, 0.6,

(0.06, 0.20, 0.42, −0.06, −0.28, −0.54)

(0.42, 0.56, 0.42, −0.30, −0.21, −0.42)

a(0.7, 0.8, 0.6, −0.5, −0.3, −0.7) (0.06, 0.20, 0.42, −0.1, −0.12, −0.54) b(0.3, 0.5, 0.7, −0.7, −0.8, −0.9) −0.54) −0.24, , −0.1, 2 .4 0 , .20 (0.06, 0

(0.42, 0.42, 0.36, −0.18, −0.18, −0.48)

.7) (0.48, 0.42, 0.42, 0.18, 0.36, 0.48) d(0.8, 0.7, 0.6, 0.3, 0.7, 0.8)

Fig. 9.13 Bipolar neutrosophic closed neighborhood graph N [G]

Definition 9.26 The Cartesian product of two bipolar single-valued neutrosophic graphs G 1 and G 2 is denoted by the pair G 1 G 2 = (A1 A2 , B1 B2 ) and defined as p

TA

p

1 A2

T An

1 B2

p

p

p

p

p

(y) = T A1 (y) ∧ TB2 (y),

IA

1 A2

(y) = I A1 (y) ∧ I A2 (y),

FA

(y) = T An1 (y) ∨ T An2 (y),

I An

1 A2

(y) = I An 1 (y) ∨ I An 2 (y),

FAn

p

1 A2 1 A2

p

(y) = FA1 (y) ∨ FA2 (y), (y) = FAn 1 (y) ∧ FAn 2 (y).

for all y ∈ X 1 × X 2 . The membership values of the edges in G 1 G 2 can be calculated as p

p

p

1. TB1 B2 ((y1 , y2 )(y1 , z 2 )) = T A1 (y1 ) ∧ TB2 (y2 z 2 ), TBn1 B2 ((y1 , y2 )(y1 , z 2 )) = T An1 (y1 ) ∨ TBn2 (y2 z 2 ), for all y1 ∈ X 1 , y2 z 2 ∈ E 2 , p p p 2. TB1 B2 ((y1 , y2 )(z 1 , y2 )) = TB1 (y1 z 1 ) ∧ T A2 (y2 ), TBn1 B2 ((y1 , y2 )(z 1 , y2 )) = n n TB1 (y1 z 1 ) ∨ T A2 (y2 ), for all y1 z 1 ∈ E 1 , y2 ∈ X 2 , p p p 3. I B1 B2 ((y1 , y2 )(y1 , z 2 )) = I A1 (y1 ) ∧ I B2 (y2 z 2 ), I Bn1 B2 ((y1 , y2 )(y1 , z 2 )) = I An 1 (y1 ) ∨ I Bn2 (y2 z 2 ), for all y1 ∈ X 1 , y2 z 2 ∈ E 2 ,

372

9 Bipolar Neutrosophic Competition Graphs p

p

p

4. I B1 B2 ((y1 , y2 )(z 1 , y2 )) = I B1 (y1 z 1 ) ∧ I A2 (y2 ), I Bn1 B2 ((y1 , y2 )(z 1 , y2 )) = I Bn1 (y1 z 1 ) ∨ I An 2 (y2 ), for all y1 z 1 ∈ E 1 , y2 ∈ X 2 , p p p 5. FB1 B2 ((y1 , y2 )(y1 , z 2 )) = FA1 (y1 ) ∨ FB2 (y2 z 2 ), FBn1 B2 ((y1 , y2 )(y1 , z 2 )) = FAn1 (y1 ) ∧ FBn2 (y2 z 2 ), for all y1 ∈ X 1 , y2 z 2 ∈ E 2 , p p p 6. FB1 B2 ((y1 , y2 )(z 1 , y2 )) = FB1 (y1 z 1 ) ∨ FA2 (y2 ), FBn1 B2 ((y1 , y2 )(z 1 , y2 )) = FBn1 (y1 z 1 ) ∧ FAn2 (y2 ), for all y1 z 1 ∈ E 1 , y2 ∈ X 2 . Definition 9.27 The direct product of two bipolar single-valued neutrosophic graphs G 1 = (A1 , B1 ) and G 2 = (A2 , B2 ) is denoted by the pair G 1 × G 2 = (A1 × A2 , B1 × B2 ) such that p

p

p

p

p

p

p

p

p

T A1 ×A2 (y) = T A1 (y) ∧ TB2 (y),

I A1 ×A2 (y) = I A1 (y) ∧ I A2 (y),

FA1 ×A2 (y) = FA1 (y) ∨ FA2 (y),

T An1 ×A2 (y)

I An 1 ×A2 (y)

FAn 1 ×A2 (y)

=

T An1 (y) ∨

T An2 (y),

=

I An 1 (y) ∨ I An 2 (y),

=

FAn 1 (y) ∧

FAn 2 (y),

for all y ∈ X 1 × X 2 . p

p

p

1. TB1 ×B2 ((y1 , y2 )(z 1 , z 2 )) = TB1 (y1 z 1 ) ∧ TB2 (y2 z 2 ), TBn1 ×B2 ((y1 , y2 )(z 1 , z 2 )) = TBn1 (y1 z 1 ) ∨ TBn2 (y2 z 2 ), for all y1 z 1 ∈ E 1 , y2 z 2 ∈ E 2 , p p p 2. I B1 ×B2 ((y1 , y2 )(z 1 , z 2 )) = I B1 (y1 z 1 ) ∧ I B2 (y2 z 2 ), I Bn1 ×B2 ((y1 , y2 )(z 1 , z 2 )) = n n I B1 (y1 z 1 ) ∨ I B2 (y2 z 2 ), for all y1 z 1 ∈ E 1 , y2 z 2 ∈ E 2 , p p p 3. FB1 ×B2 ((y1 , y2 )(z 1 , z 2 )) = FB1 (y1 z 1 ) ∨ FB2 (y2 z 2 ), FBn1 ×B2 ((y1 , y2 )(z 1 , z 2 )) = n n FB1 (y1 z 1 ) ∧ FB2 (y2 z 2 ), for all y1 z 1 ∈ E 1 , y2 z 2 ∈ E 2 . The method of construction of a bipolar single-valued neutrosophic competition graph of the Cartesian product of bipolar single-valued neutrosophic digraphs in explained in Theorem 9.10. − → − → Theorem 9.10 Let C(G 1 ) = (A1 , B1 ) and C(G 2 ) = (A2 , B2 ) be two bipolar singlevalued neutrosophic competition graphs of bipolar single-valued neutrosophic − → − → − → − → − → − → digraphs G 1 =(A1 , l1 ) and G 2 =(A2 , l2 ), respectively. Then C(G 1 G 2 ) = → − → ∪ G  , where G − → − → is a bipolar single-valued neutrosophic G C(− G 1 )∗ C(G 2 )∗ C(G 1 )∗ C(G 2 )∗ − → − → → E − → ), C(G 1 )∗ and C(G 2 )∗ are the graph on the crisp graph (X 1 × X 2 , E C(− G 1 )∗ C(G 2 )∗ − → − → crisp competition graphs of G 1 and G 2 , respectively. G  is a bipolar single-valued  neutrosophic graph on (X 1 × X 2 , E ) such that the following conditions are satisfied. 1. E  = {(x1 , x2 )(y1 , y2 ) : y1 ∈ N − (x1 )∗ , y2 ∈ N + (x2 )∗ } → E − → = E C(− G 1 )∗ C(G 2 )∗ → } ∪ {(x 1 , x 2 )(y1 , x 2 ) : x 2 ∈ X 2 , x 1 y1 {(x1 , x2 )(x1 , y2 ) : x1 ∈ X 1 , x2 y2 ∈ E C(− G 2 )∗ → ∗ }, ∈ E C(− G1) p p p p p p p I A1 A2 = I A1 (x1 ) ∧ I A2 (x2 ), FA1 A2 = 2. T A1 A2 = T A1 (x1 ) ∧ T A2 (x2 ), p p FA1 (x1 ) ∨ FA2 (x2 ), I An 1 A2 = I An 1 (x1 ) ∨ I An 2 (x2 ), FAn1 A2 = T An1 A2 = T An1 (x1 ) ∨ T An2 (x2 ), n n FA1 (x1 ) ∧ FA2 (x2 ),

9.3 Bipolar Neutrosophic Neighborhood Graphs p

p

p

373 p

p

3. TB ((x1 , x2 )(x1 , y2 )) = [T A1 (x1 ) ∧ T A2 (x2 ) ∧ T A2 (y2 )] × ∨a2 {T A1 (x1 ) ∧ p p T− → (x 2 a2 ) ∧ T− → (y2 a2 )}, l2

l2

→ E − → , a2 ∈ (N + (x 2 ) ∩ N + (y2 ))∗ , (x1 , x2 )(x1 , y2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ p p p p p 4. I B ((x1 , x2 )(x1 , y2 )) = [I A1 (x1 ) ∧ I A2 (x2 ) ∧ I A2 (y2 )] × ∨a2 {I A1 (x1 ) ∧ p p I− → (x 2 a2 ) ∧ I− → (y2 a2 )}, l2

l2

→ E − → , a2 ∈ (N + (x 2 ) ∩ N + (y2 ))∗ , (x1 , x2 )(x1 , y2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ p p p p p 5. FB ((x1 , x2 )(x1 , y2 )) = [FA1 (x1 ) ∨ FA2 (x2 ) ∨ FA2 (y2 )] × ∨a2 {FA1 (x1 ) ∨ p p F− → (x 2 a2 ) ∨ F− → (y2 a2 )}, l2

l2

l2

l2

→ E − → , a2 ∈ (N + (x 2 ) ∩ N + (y2 ))∗ , (x1 , x2 )(x1 , y2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ n n 6. TB ((x1 , x2 )(x1 , y2 )) = [T A1 (x1 ) ∨ T An2 (x2 ) ∨ T An2 (y2 )] × ∨a2 {T An1 (x1 ) ∨ n n T− → (x 2 a2 ) ∨ T− → (y2 a2 )}, → E − → , a2 ∈ (N + (x 2 ) ∩ N + (y2 ))∗ , (x1 , x2 )(x1 , y2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ n n 7. I B ((x1 , x2 )(x1 , y2 )) = [I A1 (x1 ) ∨ I An 2 (x2 ) ∨ I An 2 (y2 )] × ∨a2 {I An 1 (x1 ) ∨ n n I− → (x 2 a2 ) ∨ I− → (y2 a2 )}, l2

l2

→ E − → , a2 ∈ (N + (x 2 ) ∩ N + (y2 ))∗ , (x1 , x2 )(x1 , y2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ n n 8. FB ((x1 , x2 )(x1 , y2 )) = [FA1 (x1 ) ∧ FAn2 (x2 ) ∧ FAn2 (y2 )] × ∨a2 {FAn1 (x1 ) ∧ n n F− → (x 2 a2 ) ∧ F− → (y2 a2 )}, l2

l2

l1

l1

→ E − → , a2 ∈ (N + (x 2 ) ∩ N + (y2 ))∗ , (x1 , x2 )(x1 , y2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ p p p p p 9. TB ((x1 , x2 )(y1 , x2 )) = [T A1 (x1 ) ∧ T A1 (y1 ) ∧ T A2 (x2 )] × ∨a1 {T A2 (x2 ) ∧ p p T− → (x 1 a1 ) ∧ T− → (y1 a1 )}, → E − → , a1 ∈ (N + (x 1 ) ∩ N + (y1 ))∗ , (x1 , x2 )(y1 , x2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ p p p p p 10. I B ((x1 , x2 )(y1 , x2 )) = [I A1 (x1 ) ∧ I A1 (y1 ) ∧ I A2 (x2 )] × ∨a1 {I A2 (x2 ) ∧ p p I− → (x 1 a1 ) ∧ I− → (y1 a1 )}, l1

l1

→ E − → , a1 ∈ (N + (x 1 ) ∩ N + (y1 ))∗ , (x1 , x2 )(y1 , x2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ p p p p p 11. FB ((x1 , x2 )(y1 , x2 )) = [FA1 (x1 ) ∨ FA1 (y1 ) ∨ FA2 (x2 )] × ∨a1 {FA2 (x2 ) ∨ p p F− → (x 1 a1 ) ∨ F− → (y1 a1 )}, l1

l1

l1

l1

→ E − → , a1 ∈ (N + (x 1 ) ∩ N + (y1 ))∗ , (x1 , x2 )(y1 , x2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ n n 12. TB ((x1 , x2 )(y1 , x2 )) = [T A1 (x1 ) ∨ T An1 (y1 ) ∨ T An2 (x2 )] × ∨a1 {T An2 (x2 ) ∨ n n T− → (x 1 a1 ) ∨ T− → (y1 a1 )}, → E − → , a1 ∈ (N + (x 1 ) ∩ N + (y1 ))∗ , (x1 , x2 )(y1 , x2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ n n 13. I B ((x1 , x2 )(y1 , x2 )) = [I A1 (x1 ) ∨ I An 1 (y1 ) ∨ I An 2 (x2 )] × ∨a1 {I An 2 (x2 ) ∨ n n I− → (x 1 a1 ) ∨ I− → (y1 a1 )}, l1

14.

l1

→ E − → , a1 ∈ (N + (x 1 ) ∩ N + (y1 ))∗ , (x1 , x2 )(y1 , x2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ FBn ((x1 , x2 )(y1 , x2 )) = [F An (x1 ) ∧ F An (y1 ) ∧ F An (x2 )] × ∨a1 {F An (x2 ) ∧ 1 1 2 2 n n F− → (x 1 a1 ) ∧ F− → (y1 a1 )}, l1

l1

→ E − → , a1 ∈ (N + (x 1 ) ∩ N + (y1 ))∗ , (x1 , x2 )(y1 , x2 ) ∈ E C(− G 1 )∗ C(G 2 )∗ p 15. TB ((x1 , x2 )(y1 , y2 )) = p p p p p [T A1 (x1 ) ∧ T A1 (y1 ) ∧ T A2 (x2 ) ∧ T A2 (y2 )] × [T A1 (x1 ) ∧ p p T− → (y1 x 1 ) ∧ T A2 (y2 ) ∧ l1

374

9 Bipolar Neutrosophic Competition Graphs

Fig. 9.14 Bipolar neutrosophic digraphs

p

T− → (x 2 y2 )], l2

(x1 , y1 )(x2 , y2 ) ∈ E  , 16.

p

p

p

p

p

p

p

p

p

I B ((x1 , x2 )(y1 , y2 )) = [I A1 (x1 ) ∧ I A1 (y1 ) ∧ I A2 (x2 ) ∧ I A2 (y2 )] × [I A1 (x1 ) ∧ I− → (y1 x 1 ) ∧ I A2 (y2 ) ∧ I− → (x 2 y2 )], l1



(x1 , y1 )(x2 , y2 ) ∈ E , p 17. FB ((x1 , x2 )(y1 , y2 )) = p

p

p

p

p

p l1

l2

p

p l2

[F A (x1 ) ∨ F A (y1 ) ∨ F A (x2 ) ∨ F A (y2 )] × [F A (x1 ) ∨ F− → (y1 x 1 ) ∨ F A (y2 ) ∨ F− → (x 2 y2 )], 1

1

2

(x1 , y1 )(x2 , y2 ) ∈ E  , 18. TBn ((x1 , x2 )(y1 , y2 )) =

2

1

2

n n n [T An (x1 ) ∨ T An (y1 ) ∨ T An (x2 ) ∨ T An (y2 )] × [T An (x1 ) ∨ T− → (y1 x 1 ) ∨ T A2 (y2 ) ∨ T− → (x 2 y2 )], 1 1 2 2 1

(x1 , y1 )(x2 , y2 ) ∈ E  , 19.

l1

l2

n n n I Bn ((x1 , x2 )(y1 , y2 )) = [I An 1 (x1 ) ∨ I An 1 (y1 ) ∨ I An 2 (x2 ) ∨ I An 2 (y2 )] × [I An 1 (x1 ) ∨ I− → (y1 x 1 ) ∨ I A2 (y2 ) ∨ I− → (x 2 y2 )], l1

(x1 , y1 )(x2 , y2 ) ∈ E  , 20. FBn ((x1 , x2 )(y1 , y2 )) =

l2

n n n [F An (x1 ) ∧ F An (y1 ) ∧ F An (x2 ) ∧ F An (y2 )] × [F An (x1 ) ∧ F− → (y1 x 1 ) ∧ F A2 (y2 ) ∧ F− → (x 2 y2 )], 1 1 2 2 1

(x1 , y1 )(x2 , y2 ) ∈ E  .

l1

l2

− → − → Example 9.12 Consider G 1 = (A1 , l1 ) and G 2 = (A2 , l2 ) as two bipolar singlevalued neutrosophic digraphs as shown in Fig. 9.14. The bipolar single-valued neu− → − → trosophic out- and in-neighborhoods of G 1 and G 2 are given in Tables 9.4 and 9.5. − → − → The bipolar single-valued neutrosophic competition graphs C( G 1 ) and C( G 2 ) are given in Fig. 9.15. → − → Construct the bipolar single-valued neutrosophic competition graph G C(− G 1 )∗ C(G 2 )∗ p p p p p p n n n  ∪ G = (W , B), where W = (TW , IW , FW , TW , IW , FW ) and B = (TB , I B , FB , TBn ,

9.3 Bipolar Neutrosophic Neighborhood Graphs

375

− → Table 9.4 Bipolar neutrosophic out- and in-neighborhoods of G 1 x ∈ X 1 N + (x) N − (x) a1 b1 c1 d1

{b1 (0.5, 0.1, 0.7, −0.1, −0.1, −0.3)} ∅ {b1 (0.1, 0.1, 0.7, −0.1, −0.2, −0.7)} {c1 (0.1, 0.2, 0.7, −0.1, −0.6, −0.6)}

∅ {a1 (0.5, 0.1, 0.7, −0.1, −0.1, −0.3)} {d1 (0.1, 0.2, 0.7, −0.1, −0.6, −0.6)} ∅

− → Table 9.5 Bipolar neutrosophic out- and in-neighborhoods of G 2 x ∈ X 2 N + (x) N − (x) a2 b2 c2

{c2 (0.1, 0.3, 0.6, −0.1, −0.1, −0.1)} {c2 (0.3, 0.4, 0.5, −0.1, −0.2, −0.4)} ∅

∅ ∅ {a2 (0.1, 0.3, 0.6, −0.1, −0.1, −0.1), b2 (0.3, 0.4, 0.5, −0.1, −0.2, −0.4)}

Fig. 9.15 Bipolar neutrosophic competition graphs

− → − → I Bn , FBn ), from C(G 1 )∗ and C(G 2 )∗ using Theorem 1.4. We obtain the following sets of edges using condition 1. → − → = {(a1 , a2 )(a1 , b2 ), (b1 , a2 )(b1 , b2 ), (c1 , a2 )(c1 , b2 ), (d1 , a2 )(d1 , b2 ), E C(− G 1 )∗ C(G 2 )∗ (a1 , a2 )(c1 , a2 ), (a1 , b2 )(c1 , b2 ), (a1 , c2 )(c1 , c2 )}, E  = {(b1 , a2 )(a1 , c2 ), (b1 , a2 ) (c1 , c2 ), (b1 , b2 )(a1 , c2 ), (b1 , b2 )(c1 , c2 ), (c1 , a2 )(d1 , c2 ), (c1 , b2 )(d1 , c2 )}. According to conditions 3–20, the degrees of positive truth-membership, indeterminacymembership, falsity-membership, negative truth-membership, indeterminacy→ − → membership, and falsity-membership values of the adjacent vertices of G C(− G 1 )∗ C(G 2 )∗  and G are given in Table 9.6.

376

9 Bipolar Neutrosophic Competition Graphs

→ − → ∪ G Table 9.6 Adjacent vertices of G C (− G )∗ C (G )∗ 1

2

(x1 , x2 )(y1 , y2 )

B(x1 , x2 )(y1 , y2 )

(a1 , a2 )(a1 , b2 ) (b1 , a2 )(b1 , b2 ) (c1 , a2 )(c1 , b2 ) (d1 , a2 )(d1 , b2 ) (a1 , a2 )(c1 , a2 ) (a1 , b2 )(c1 , b2 ) (a1 , c2 )(c1 , c2 ) (b1 , a2 )(a1 , c2 ) (b1 , a2 )(c1 , c2 ) (b1 , b2 )(a1 , c2 ) (b1 , b2 )(c1 , c2 ) (c1 , a2 )(d1 , c2 ) (c1 , b2 )(d1 , c2 )

(0.01, 0.09, 0.64, −0.01, −0.03, −0.32) (0.01, 0.04, 0.24, −0.01, −0.04, −0.24) (0.01, 0.09, 0.64, −0.01, −0.06, −0.64) (0.01, 0.09, 0.64, −0.01, −0.06, −0.56) (0.01, 0.03, 0.56, −0.01, −0.01, −0.56) (0.02, 0.03, 0.56, −0.02, −0.01, −0.56) (0.02, 0.03, 0.56, −0.02, −0.01, −0.56) (0.01, 0.02, 0.56, −0.01, −0.01, −0.35) (0.01, 0.02, 0.56, −0.01, −0.02, −0.56) (0.12, 0.02, 0.56, −0.06, −0.01, −0.35) (0.02, 0.02, 0.56, −0.02, −0.03, −0.56) (0.01, 0.06, 0.64, −0.01, −0.04, −0.64) (0.02, 0.06, 0.64, −0.02, −0.06, −0.64)

a2

a1

(0.1, 0.3, 0.8, −0.1, −0.1, −0.2)

c2

b2

(0.01, 0.09, 0.64, −0.01, −0.03, −0.32) (0.7, 0.5, 0.8, −0.2, −0.1, −0.4) (0.4, 0.5, 0.8, −0.2, −0.1, −0.5) (0.02, 0.03, 0.56, −0.02, −0.01, −0.56)

1

(0.0

2, , 0. 0

−0 .01, , −0 0. 5 6

. 0 1,

35 −0.

) ,0 . 02

(0.1, 0.2, 0.4, −0.1, −0.2, −0.4)

(0.0

(0.6, 0.2, 0.1, −0.2, −0.3, −0.4) 1, 0 . 0 2, 0. 5 6 , −0 . 0 1, −0. 0 2, −0. 56)

(0.01, 0.09, 0.64, −0.01, −0.06, −0.64)

c1

(0.1, 0.3, 0.8, −0.1, −0.2, −0.8)

(0.2, 0.3, 0.8, −0.2, −0.3, −0.8) (0.0

d1

1, 0 .06, 0

.64,

−0. 01

, −0

.04,

−0. 64

0 ,−

,0 . 12 (0 (0.4, 0.2, 0.7, −0.2, −0.3, −0.5) (0 .02 ,0 .02 ,0 .56 ,− 0. 0 2, −0 .03 ,− 0. 5 6)

(0.01, 0.04, 0.24, −0.01, −0.04, −0.24)

b1

. 56

5)

(0 . 02

,0

)

(0.02, 0.03, 0.56, −0.02, −0.01, −0.56)

(0.01, 0.03, 0.56, −0.01, −0.01, −0.56)

.3 −0 1, 0 . 0 ,− . 06

(0.2, 0.3, 0.8, −0.2, −0.3, −0.8) . 06

,0

. 64

,− 0. 0

2,

− 0. 0

6,

− 0. 6

4)

(0.01, 0.09, 0.64, −0.01, −0.06, −0.56) (0.5, 0.6, 0.8, 0.2, 0.3, 0.7) (0.4, 0.5, 0.8, 0.2, 0.3, 0.7)

(0.1, 0.3, 0.8, 0.1, 0.2, 0.7)

→ − → ∪ G Fig. 9.16 Bipolar neutrosophic graph G C (− G )∗ C (G )∗ 1

2

9.3 Bipolar Neutrosophic Neighborhood Graphs

a1

377

a2

b2

(0.1, 0.3, 0.8, −0.1, −0.1, −0.2)

(0.7, 0.5, 0.8, −0.2, −0.1, −0.4)

c2 (0.3, 0.4, 0.5, −0.1, −0.1, −0.4)

(0.4, 0.5, 0.8, −0.2, −0.1, −0.5)

(0.5, 0.1, 0.7, −0.1, −0.1, −0.4)

(0.1, 0.1, 0.7, −0.1, −0.1, −0.3)

b1

(0.1, 0.2, 0.4, −0.1, −0.2, −0.4)

(0.4, 0.1, 0.7, −0.1, −0.1, −0.5) (0.1, 0.3, 0.8, −0.1, −0.1, −0.2)

(0.6, 0.2, 0.1, −0.2, −0.3, −0.4)

(0.3, 0.2, 0.5, −0.1, −0.2, −0.4)

(0.4, 0.2, 0.7, −0.2, −0.3, −0.5)

(0.1, 0.1, 0.7, −0.1, −0.2, −0.7)

(0.1, 0.1, 0.7, −0.1, −0.2, −0.7)

(0.1, 0.1, 0.7, −0.1, −0.2, −0.7)

(0.1, 0.2, 0.6, −0.1, −0.1, −0.4)

c1

(0.1, 0.3, 0.8, −0.1, −0.2, −0.8)

(0.2, 0.3, 0.8, −0.2, −0.3, −0.8)

(0.2, 0.3, 0.8, −0.1, −0.2, −0.8)

(0.2, 0.3, 0.8, −0.2, −0.3, −0.8)

(0.1, 0.2, 0.7, −0.1, −0.3, −0.6) (0.1, 0.2, 0.7, −0.1, −0.2, −0.6)

d1

(0.1, 0.3, 0.8, −0.1, −0.2, −0.7)

(0.1, 0.2, 0.7, −0.1, −0.3, −0.6)

(0.1, 0.3, 0.8, −0.1, −0.1, −0.8)

(0.5, 0.6, 0.8, −0.2, −0.3, −0.7)

(0.1, 0.3, 0.8, −0.1,

(0.3, 0.4, 0.8, −0.1, −0.2, −0.7)

(0.4, 0.5, 0.8, −0.2, −0.3, −0.7)

−0.1, −0.7)

− → − → Fig. 9.17 Cartesian product G 1 G 2

The bipolar single-valued neutrosophic competition graph obtained using this method is given in Fig. 9.16, where the solid lines indicate the part of the bipo→ − → , the lar single-valued neutrosophic competition graph obtained from G C(− G 1 )∗ C(G 2 )∗  dotted lines represent the part G . − → − → The Cartesian product G 1 G 2 of bipolar single-valued neutrosophic digraphs − → − → G 1 and G 1 is shown in Fig. 9.17. The bipolar single-valued neutrosophic out− → − → neighborhoods of G 1 G 2 are calculated in Table 9.7. The bipolar single-valued − → − → neutrosophic competition graph of G 1 G 2 is shown in Fig. 9.18. → − → − → − → ∪ G ∼ It is clear from Figs. 9.16 and 9.18 that G C(− = C(G 1 G 2 ). G 1 )∗ C(G 2 )∗

9.4 Applications of Competition Graphs in Bipolar Neutrosophic Environment In this section, the applications of bipolar single-valued neutrosophic competition graphs are studied in organizational designations, economics, business marketing, and sports.

378

9 Bipolar Neutrosophic Competition Graphs

− → − → Table 9.7 Bipolar neutrosophic out-neighborhoods of G 1 G 2 (x, y) N + (x, y) (a1 , a2 ) (a1 , b2 ) (a1 , c2 ) (b1 , a2 ) (b1 , b2 ) (b1 , c2 ) (c1 , a2 ) (c1 , b2 ) (c1 , c2 ) (d1 , a2 ) (d1 , b2 ) (d1 , c2 )

{((a1 , c2 ), 0.1, 0.3, 0.8, −0.1, −0.1, −0.2), ((b1 , a2 ), 0.1, 0.1, 0.7, −0.1, −0.1, −0.3)} {((a1 , c2 ), 0.3, 0.4, 0.5, −0.1, −0.1, −0.4), ((b1 , b2 ), 0.5, 0.1, 0.7, −0.1, −0.1, −0.4)} {((b1 , c2 ), 0.4, 0.1, 0.7, −0.1, −0.1, −0.5)} {((b1 , c2 ), 0.1, 0.2, 0.6, −0.1, −0.1, −0.4)} {((b1 , c2 ), 0.3, 0.2, 0.5, −0.1, −0.2, −0.4)} ∅ {((c1 , c2 ), 0.1, 0.3, 0.8, −0.1, −0.1, −0.8), ((b1 , a2 ), 0.1, 0.1, 0.7, −0.1, −0.2, −0.7)} {((b1 , b2 ), 0.1, 0.1, 0.7, −0.1, −0.2, −0.7), ((c1 , c2 ), 0.2, 0.3, 0.8, −0.1, −0.2, −0.8)} {((b1 , c2 ), 0.1, 0.1, 0.7, −0.1, −0.2, −0.7)} {((d1 , c2 ), 0.1, 0.3, 0.8, −0.1, −0.1, −0.7), ((c1 , a2 ), 0.1, 0.2, 0.7, −0.1, −0.2, −0.6)} {((d1 , c2 ), 0.3, 0.4, 0.8, −0.1, −0.2, −0.7), (c1 , b2 ), 0.1, 0.2, 0.7, −0.1, −0.3, −0.6)} {((c1 , c2 ), 0.1, 0.2, 0.7, −0.1, −0.3, −0.6)}

a2

a1 (0.1, 0.3, 0.8, −0.1, −0.1, −0.2)

c2

b2

(0.01, 0.09, 0.64, −0.01, −0.03, −0.32) (0.7, 0.5, 0.8, −0.2, −0.1, −0.4) (0.4, 0.5, 0.8, −0.2, −0.1, −0.5) (0.02, 0.03, 0.56, −0.02, −0.01, −0.56)

(0.0

1,

, 0. 0 2

−0 .01, , −0 0. 5 6

.01,

35 −0.

) ,0 . 02

(0.1, 0.2, 0.4, −0.1, −0.2, −0.4)

(0.6, 0.2, 0.1, −0.2, −0.3, −0.4) (0.0 1, 0 .02, 0. 5 6 , −0 .01, −0. 02, −0. 56)

(0.01, 0.09, 0.64, −0.01, −0.06, −0.64)

c1

(0.1, 0.3, 0.8, −0.1, −0.2, −0.8)

(0.2, 0.3, 0.8, −0.2, −0.3, −0.8) (0.0

d1

1, 0

.06,

0.64

, −0

.01,

−0. 04

, −0

5)

,0 . 12 (0 (0.4, 0.2, 0.7, −0.2, −0.3, −0.5) (0 . 02 ,0 . 02 ,0 . 56 ,− 0. 0 2, −0 . 03 ,− 0. 5 6)

(0.01, 0.04, 0.24, −0.01, −0.04, −0.24)

b1

0 ,− . 56

0. 3 ,−

(0 . 02

,0

(0.02, 0.03, 0.56, −0.02, −0.01, −0.56)

(0.01, 0.03, 0.56, −0.01, −0.01, −0.56)

1 0. 0 ,− . 06

(0.2, 0.3, 0.8, −0.2, −0.3, −0.8) . 06

. 64)

,0

. 64

,− 0. 0

2,

− 0. 0

6,

− 0. 6

4)

(0.01, 0.09, 0.64, −0.01, −0.06, −0.56) (0.1, 0.3, 0.8, 0.1, 0.2, 0.7)

(0.5, 0.6, 0.8, 0.2, 0.3, 0.7) (0.4, 0.5, 0.8, 0.2, 0.3, 0.7)

− → − → Fig. 9.18 Bipolar neutrosophic competition graph C (G 1 G 2 )

9.4 Applications of Competition Graphs in Bipolar Neutrosophic Environment .5) −0

7) 0. − 5, 0. − 1, 0. ,− .5 ,0 .4 ,0 .6 (0

, 0.4 .6, (0

, 0.5 ,− 0.1 ,− 0.5

379

Angus (0.7, 0.5, 0.2, −0.1, −0.7, −0.8)

(0.6, 0.4, 0.2, −0.1, −0.5, −0.4)

BOD

CEO

(0.9, 0.6, 0.8, −0.1, −0.6, −0.5)

0.5, (0.5,

0.4) 0.5, − 0.4, − 0.5, −

(0.2 , 0.4 , 0.5 , −0 .5, − 0.6, −0.5 )

(0.5, 0.5, 0.6, −0.1, −0.4, −0.4)

(0.7, 0.7, 0.7, −0.5, −0.6, −0.6)

Alina (0.6, 0.7, 0.8, −0.7, −0.6, −0.5)

(0 .4,

0.4 ,

0.4 ,

−0 .3,

DOM −0 .4,

(0.7, 0.8, 0.6, −0.6, −0.7, −0.5)

Alma (0.4, 0.5, 0.7, −0.8, −0.7, −0.6)

(0.3, 0.4, 0.6, −0.4, −0.3, −0.5)

DOHR (0.6, 0.7, 0.5, −0.6, −0.5, −0.4)

−0 .2)

(0.4, 0.5, 0.5, −0. 2,

−0.3, −0.4) (0.2, 0.4, 0.6, −0.5, −0.4, −0.6)

Colin (0.5, 0.6, 0.8, −0.2, −0.5, −0.8)

, , 0.5 , 0.6 (0.4

0.4) .3, − , −0 −0.2

Alvin (0.3, 0.4, 0.6, −0.7, −0.8, −0.7)

Fig. 9.19 Bipolar neutrosophic organization model

9.4.1 Designation Competition in an Organization Consider a bipolar single-valued neutrosophic digraph as shown in Fig. 9.19 representing the competition between applicants for designations in an organization. Let {Angus, Alvin, Alina, Colin, Alma} be the set of applicants for the designations {Board of director(BOD), CEO, Director of marketing(DOM), Director of human resources(DOHR)}. The positive degree of membership T p (x) of each applicant represents the percentage of hard work toward the goals of organization, and I p (x) and F p (x) represent the indeterminacy and falsity in this percentage. The negative degree of membership T n (x) represents the percentage that the applicant is not effective in order to fulfill the goals of that organization, and I n (x) and F n (x) represent the indeterminacy and falsity in this percentage. The positive degree of membership T p (x) of each directed edge between applicants and designations represents the percentage of eligibility for that designation in the organization, and I p (x) and F p (x) represent the indeterminacy and falsity in this percentage. The negative degree of membership T n (x) of each directed edge between the applicants and the designation represents the percentage of non-eligibility for that designation in the organization, and I n (x) and F n (x) represent the indeterminacy and falsity in this percentage. The bipolar single-valued neutrosophic competition graph can be used in order to find the designations of the applicants. The bipolar neutrosophic out-neighborhoods are computed in Table 9.8.

380

9 Bipolar Neutrosophic Competition Graphs

Table 9.8 Bipolar neutrosophic out-neighborhoods x∈X N + (x) Angus

Alma Alvin Colin Alina

{(B O D, 0.6, 0.4, 0.5, −0.1, −0.5, −0.5), (C E O, 0.6, 0.4, 0.5, −0.1, −0.5, −0.7), (D O M, 0.6, 0.4, 0.2, −0.1, −0.5, −0.4)} {(D O H R, 0.3, 0.4, 0.6, −0.4, −0.3, −0.5)} {(D O H R, 0.2, 0.4, 0.6, −0.5, −0.4, −0.6), (C E O, 0.2, 0.4, 0.5, −0.5, −0.6, −0.5)} {(D O H R, 0.4, 0.6, 0.5, −0.2, −0.3, −0.4), (D O M, 0.4, 0.5, 0.5, −0.2, −0.3, −0.4)} {(B O D, 0.5, 0.5, 0.6, −0.1, −0.4, −0.4), (C E O, 0.5, 0.5, 0.5, −0.4, −0.5, −0.4), (D O M, 0.4, 0.4, 0.4, −0.3, −0.4, −0.2)}

Therefore, N + (Angus) ∩ N + (Alma) = ∅, N + (Alma) ∩ N + (Alina) = ∅, N + (Angus) ∩ N + (Alvin) = {(CEO, 0.2, 0.4, 0.5, −0.1, −0.5, −0.7)}, N + (Angus) ∩ N + (Colin) = {(DOM, 0.4, 0.4, 0.5, −0.1, −0.3, −0.4)}, N + (Alma) ∩ N + (Alvin)= {(DOHR, 0.2, 0.4, 0.6, −0.4, −0.3, −0.6)}, N + (Alma) ∩ N + (Colin) = {(DOHR, 0.3, 0.4, 0.6, −0.2, −0.3, −0.5)}, N + (Alvin) ∩ N + (Colin) = {(DOHR, 0.2, 0.4, 0.6, −0.2, −0.3, −0.6)}, N + (Alvin) ∩ N + (Alina) = {(CEO, 0.2, 0.4, 0.5, −0.4, −0.5, −0.5)}, N + (Colin) ∩ N + (Alina)={(DOM, 0.4, 0.4, 0.5, −0.2, −0.3, −0.4)}, N + (Angus) ∩ N + (Alina) = {(BOD, 0.5, 0.4, 0.6, −0.1, −0.4, −0.5), (CEO, 0.5, 0.4, 0.5, −0.1, −0.5, −0.7), (DOM, 0.4, 0.4, 0.4, −0.1, −0.4, −0.2)}. The bipolar neutrosophic competition graph is shown in Fig. 9.20. The bipolar single-valued neutrosophic competition graph is shown in Fig. 9.20. The solid lines indicate the strength of competition between two applicants and the dashed lines indicate the applicant competing for the particular designation. For example, Angus and Alina both are competing for the designation BOD and strength of competition between them is (0.30, 0.20, 0.32, −0.05, −0.24, −0.32). In Table 9.9, T (y, d) represents the strength of competition of applicant y for designation d with respect to hard work in order to fulfill the goals of that organization. The strength to compete with other applicants with respect to particular designations is calculated in Table 9.9. From Table 9.9, Angus and Alina have equal strength to compete with others for designation BOD. Angus competes with others for the designations DOM and CEO, while Colin competes with others for the designation DOHR. The method to calculate the strength of competition among applicants is given in Algorithm 9.4.1. Algorithm 9.4.1 Strength of competition among applicants Step 1. Input the bipolar single-valued neutrosophic A of q applicants y1 , y2 , . . . , yq . Step 2. If for any two distinct vertices yi and y j , A(yi y j ) = (0, 0, 1, 0, 0, −1), then

9.4 Applications of Competition Graphs in Bipolar Neutrosophic Environment

0. 0, .3 (0

381

Angus

) 32 0. ,− 24 0. ,− 05 0. − 2, .3 0 , 20

(0.7, 0.5, 0.2, −0.1, −0.7, −0.8)

BOD

CEO

(0.9, 0.6, 0.8, −0.1, −0.6, −0.5)

(0.7, 0.7, 0.7, −0.5, −0.6, −0.6)

Alma (0.4, 0.5, 0.7, −0.8, −0.7, −0.6)

DOHR

) , −0.40 , −0.20 −0.08

(0.06, 0.1 6, 0.42, −0.14, −0 .28,

−0.40 )

40, 0.24, 0. (0.20,

DOM

(0.6, 0.7, 0.5, −0.6, −0.5, −0.4)

−0.42)

(0.7, 0.8, 0.6, −0.6, −0.7, −0.5) (0 .0 6, 0. 16 ,0 .4 0, − 0. 14 ,− 0. 24 ,− 0. 35 )

(0.06, 0.16, 0.30, −0.02, −0.28, −0.40)

(0.20, 0.20, 0.40, −0.04 , −0.20 ,

Alina (0.6, 0.7, 0.8, −0.7, −0.6, −0.5)

Colin

(0.5, 0.6, 0.8, −0.2, −0.5, −0.8)

Alvin (0.3, 0.4, 0.6, −0.7, −0.8, −0.7) (0.06, 0.16, 0. 48,

−0.04 , −0.20 (0.1 , −0.48 2, 0 ) .20 , 0.4 8, − 0.0 6, − 0.2 0, − 0.4 8)

Fig. 9.20 Bipolar neutrosophic competition graph Table 9.9 Strength of competition of applicants for particular designations (Applicant, designation)

In competition

T (Applicant, designation)

S (Applicant, designation)

(Angus, BOD )

Alina

(0.30, 0.20, 0.32, −0.05, −0.24, −0.32)

1.29

(Alina, BOD)

Angus

(0.30, 0.20, 0.32, −0.05, −0.24, −0.32)

1.29

(Angus, DOM)

Alina, Colin

(0.25, 0.20, 0.36, −0.045, −0.22, −0.36)

1.225

(Alina, DOM)

Angus, Colin

(0.25, 0.22, 0.36, −0.065, −0.22, −0.36)

1.185

(Colin, DOM)

Angus, Alina

(0.20, 0.22, 0.40, −0.060, −0.20, −0.40)

1.120

(Angus, CEO)

Alina, Alvin

(0.180, 0.180, 0.310, −0.035, −0.260, −0.360)

1.275

(Alina, CEO)

Angus, Alvin

(0.180, 0.180, 0.360, −0.095, −0.24, −0.335)

1.120

(Alvin, CEO)

Angus, Alina

(0.06, 0.16, 0.35, −0.080, −0.260, −0.375)

1.105

(Alma, DOHR)

Alvin, Colin

(0.09, 0.180, 0.450, −0.100, −0.240, −0.450)

1.050

(Alvin, DOHR)

Colin, Alma

(0.06, 0.16, 0.450, −0.09, −0.240, −0.450)

0.050

(Colin, DOHR)

Alma, Alvin

(0.09, 0.180, 0.480, −0.05, −0.20, −0.48)

1.060

382

9 Bipolar Neutrosophic Competition Graphs

(y j , T p (yi y j ), I p (yi y j ), F p (yi y j ), T n (yi y j ), I n (yi y j ), F n (yi y j )) ∈ N + (yi ). Step 3. Repeat Step 2 for all vertices yi and y j to calculate bipolar single-valued neutrosophic out-neighborhoods N + (yi ). Step 4. Calculate N + (yi ) ∩ N + (y j ) for each pair of distinct vertices yi and y j . Step 5. Calculate h[N + (yi ) ∩ N + (y j )] for each pair of distinct vertices yi and y j . Step 6. If N + (yi ) ∩ N + (y j ) = ∅, then draw an edge yi y j . Step 7. Repeat Step 6 for all pairs of distinct vertices. Step 8. Assign membership values to each edge yi y j using the conditions T R (yi y j ) = (yi ∧ y j )h 1 [N + (yi ) ∩ N + (y j )],

T Rn (yi y j ) = (yi ∨ y j )h 4 [N + (yi ) ∩ N + (y j )],

I R (yi y j ) = (yi ∧ y j )h 2 [N + (yi ) ∩ N + (y j )],

I Rn (yi y j ) = (yi ∨ y j )h 5 [N + (yi ) ∩ N + (y j )],

p FR (yi y j ) = (yi ∨ y j )h 3 [N + (yi ) ∩ N + (y j )],

FRn (yi y j ) = (yi ∧ y j )h 6 [N + (yi ) ∩ N + (y j )].

p

p

Step 9. If y, r1 , r2 , r3 , . . ., rq are the applicants competing for designation d, then strength of competition T (y, d) = (T p (y, d), I p (y, d), F p (y, d), T n (y, d), I n (y, d), F n (y, d)) of each applicant y for the designation d is 

T (y, d) =

q

i=1

p

T R (yri ),

 q q q q q p p I R (yri ), FR (yri ), T Rn (yri ), I Rn (yri ), FRn (yri )

i=1

i=1

i=1

q

i=1

i=1

.

Step 10. Calculate S(y, d), the strength of competition of each applicant y for designation d. S(y, d) = T p (y, d) − (I p (y, d) + F p (y, d)) + 1 + T n (y, d) − (I n (y, d) + F n (y, d)).

9.4.2 Competition in Textile Market Brands hold great meaning and value for both the buyer and the seller. Brands are centered on what the seller promises to deliver to its customers and the consumers’ expectations for a particular product. A brand can be defined as a name, slogan, or anything that can be used to identify and differentiate a particular product or service. Strong brands such as Stylo shoes and Borjan shoes evoke sound emotional and physiological responses from customers. Branding has several benefits for sellers. By having brands for their products, sellers get product recognition and product differentiation. It means that consumers would be able to accentuate the value they receive from one seller’s product in comparison with other products of a similar nature and they would also be able to easily spot these products among other products. Branding aligns the seller’s advertising and promotional activities. It allows the seller to form emotional relationships with their customers which is important because people base their purchasing decisions mainly on emotions and not logic.

9.4 Applications of Competition Graphs in Bipolar Neutrosophic Environment

383

Within the business, brands serve as guidance for employees since they clearly dictate what the company product is about. Branding is also of value to sellers in terms of profitability. A product with a well-known brand attached to it would bring more revenue to a seller than a product without a brand. For example, a cup of coffee without a brand could cost 70, however, a cup of coffee with the Kaymu brand attached to it could cost 999. For the seller, a good brand increases the perceived value of its product and allows it to have a better price and better sales. For the buyer, a brand represents the satisfaction and value that he or she anticipates and desires from the process of buying and using a product or service. Brands often appeal to buyers emotionally, this in turn helps build relationships between buyers and sellers. Brands provide buyers with value which is why they are oftentimes prepared to pay more for branded products. Among buyers, brands are often a key factor behind purchasing decisions; people often gravitate toward brands that they are familiar with and that are trusted because there is oftentimes the reassurance of quality. In conclusion, brands exist as feelings and experiences that extend beyond the product or service which create relationships between the buyer and seller and have great value for both the parties. We take the following brands in order to discuss the competition between them. 1. At the top of the list, we would bring up the name of the famous Deepak Perwani. He has been linked with the fashion designing career since the year 1994 and working in offering women wear along with men clothing collections. He is considered the best with his designing skills as adorns the dresses with eastern and western images. This designer has even got his name listed in a record book as he made a kurta of 53m for a person. This designer has even got his name listed in a record book as he made a kurta of 53 m for a person. 2. On the second spot of expensive clothing brands in Pakistan, we would mention Maria B. She is one of the topmost wanted fashion designers of Pakistan. She has been infused into offering the collection line of bridal, casual, and western wear. She has done her graduation from the Pakistan School Of Fashion Design and then she organized her company in the year 1999. 3. Khaadi is one of the topmost famous clothing brands of Pakistan. Khaadi was started by Shamoon Sultan in the year 1998. This brand has been staying in the frontline for the purpose of offering high quality of dress designs in magnificent designing concepts. It offers clothing collections for both men and women along with kids wear. 4. On the fourth spot on our list, we would bring up the name of the designer brand Aamir Adnan. This brand is one of the topmost wanted brands of clothing for men. It has made its renowned name all through its valuable fashion for sherwani kurtas. Consider the bipolar single-valued neutrosophic digraph as shown in Fig. 9.21 representing the competition between brands. Let {Deepak Perwani, Khaadi, Maria B, Aamir Adnan} be the set of brands and {Uniqueness, Passion, Consistency} be the qualities for the popularity of a particular brand. The positive degree of membership T p (x) of each brand represents the percentage of productivity of traditional clothes

384

9 Bipolar Neutrosophic Competition Graphs Uniqueness

(0.9, 0.8, 0.1, −0.1, −0.2, −0.7)

(0.8, 0.8, 0.1, −0.1, −0.1, −0.1)

(0.9, 0.8, 0.1, −0.2, −0.3, −0.5)

Deepak Perwani (0.9, 0.8, 0.1, −0.1, −0.2, −0.7)

(0.9, 0.8, 0.1, −0.1, −0.1, −0.5)

(0.8, 0.7, 0.1, −0.1, −0.2, −0.5)

(0.7, 0.8, 0.1, −0.1, −0.1, −0.1)

Passion (0.9, 0.8, 0.1, −0.2, −0.1, −0.1) (0.8, 0.7, 0.1, −0.2, −0.1, −0.6)

Maria B (0.8, 0.7, 0.2, −0.2, −0.3, −0.8)

(0.8, 0.7, 0.2, −0.1, −0.2, −0.5)

Khaadi (0.8, 0.9, 0.3, −0.1, −0.1, −0.1)

Consistency

(0.6, 0.6, 0.1, −0.2, −0.1, −0.1) Aamir Adnan (0.6, 0.7, 0.4, −0.4, −0.5, −0.1)

(0.6, 0.7, 0.2, −0.1, −0.2, −0.2)

(0.8, 0.7, 0.2, −0.1, −0.2, −0.3)

Fig. 9.21 Bipolar neutrosophic market model (0.64, 0.64, 0.03, −0.08, −0.08, −0.07)

Uniqueness (0.9, 0.8, 0.1, −0.2, −0.3, −0.5) Khaadi (0.8, 0.9, 0.3, −0.1, −0.1, −0.1)

Deepak Perwani (0.9, 0.8, 0.1, −0.1, −0.2, −0.7)

(0.36, 0.49, 0.04, −0.06, −0.14, −0.07)

Passion (0.9, 0.8, 0.1, −0.2, −0.1, −0.1)

(0.64, 0.49, 0.02, −0.08, −0.14, −0.08)

8) −0.0 0.07, .07, − 3, −0 9, 0.0 .4 0 , (0.56 (0.42, 0.56, 0.04, −0.07, −0.08, −0.01)

Aamir Adnan (0.6, 0.7, 0.4, −0.4, −0.5, −0.1)

Maria B (0.8, 0.7, 0.2, −0.2, −0.3, −0.8)

Consistency (0.8, 0.7, 0.2, −0.1, −0.2, −0.3)

(0.36, 0.49, 0.04, −0.12, −0.21, −0.08)

Fig. 9.22 Bipolar neutrosophic market competition graph

toward the satisfaction of customers, and I p (x) and F p (x) represent the indeterminacy and falsity in this percentage. The negative degree of membership T n (x) represents the percentage that the brand is not effective to satisfy the customer, and I n (x) and F n (x) represent the indeterminacy and falsity in this percentage. The positive degree of membership T p (x) of each directed edge between brands and the qualities represents the percentage of having that quality, and I p (x) and F p (x) represent the indeterminacy and falsity in this percentage. The negative degree of membership T n (x) of each directed edge between brands and qualities represents the percentage of having no qualities, and I n (x) and F n (x) represent the indeterminacy and falsity in this percentage. Thus, a bipolar single-valued neutrosophic competition graph can be used in order to find competitions between different brands. The bipolar single-valued neutrosophic competition graph is shown in Fig. 9.22. The solid lines indicate the strength of competition between two brands and the dashed lines indicate that the brands are competing for a particular quality. Deepak Perwani and Khaadi are both competing for the quality and uniqueness of their products. The strength of competition between them is (0.64, 0.64, 0.03, −0.08, −0.08,

9.4 Applications of Competition Graphs in Bipolar Neutrosophic Environment

385

Table 9.10 Strength of competition of brands for particular quality (Brand, quality)

In competition

T (Brand, quality)

S (Brand, quality)

(Deepak Perwani, uniqueness)

Khaadi

(0.64, 0.64, 0.03, −0.08, −0.08, −0.07)

1.04

(Khaadi, uniqueness)

Deepak Perwani

(0.64, 0.64, 0.03, −0.08, −0.08, −0.07)

1.04

(Deepak Perwani, passion)

Khaadi, Maria B, Aamir Adnan

(0.54, 0.54, 0.03, −0.07, −0.12, −0.07)

1.09

(Khaadi, passion)

Aamir Adnan, Deepak Perwani, Maria B

(0.54, 0.56, 0.03, −0.07, −0.07, −0.05)

1

(Maria B, passion)

Khaadi, Deepak Perwani, Aamir Adnan

(0.52, 0.51, 0.03, −0.09, −0.14, −0.05)

1.08

(Aamir Adnan, passion)

Khaadi, Deepak Perwani, Maria B

(0.38, 0.51, 0.21, −0.08, −0.14, −0.05)

0.77

(Deepak Perwani, consistency)

Maria B, Aamir Adnan

(0.50, 0.49, 0.03, −0.07, −0.14, −0.07)

1.12

(Maria B, consistency)

Deepak Perwani, Aamir Adnan

(0.50, 0.49, 0.03, −0.10, −0.17, −0.08)

1.13

(Aamir Adnan, consistency)

Maria B, Deepak Perwani

(0.36, 0.49, 0.04, −0.09, −0.17, −0.07)

0.99

−0.07). The strength of competition among brands using Algorithm 9.4.1 is calculated in Table 9.10. In Table 9.10, T (y, d) represents the value of strength of competition of brand y for quality d with respect to level of satisfaction for the customer. From Table 9.10, Deepak Perwani and Khaadi have equal strength to compete with the others for quality and uniqueness. Deepak Perwani competes with the others for quality passion, while Maria B competes for quality consistency.

9.4.3 Competition in Sports Sports are very important; every society has its own special kinds of sports. Sports and games have now come to stay in our civilization as an essential feature of human activity, and their objective is not merely fun, they also instill the spirit of discipline and teamwork. Sports like cricket, hockey, and football are popular because of the spirit of teamwork which they inspire. This is no doubt true. The discipline that is gained in playing sports is invaluable in later life. It makes for a life of cooperation and teamwork which could be used for building up a great society and a nation. The key components of sports are goals, rules, challenges, and interactions. Sports generally involve mental or physical stimulation, and often both. Many sports help develop practical skills, serve as a form of exercise, or otherwise perform an educational or psychological role, etc. Many sports require special equipment and

386

9 Bipolar Neutrosophic Competition Graphs

dedicated playing fields, leading to the involvement of a community much larger than the group of players. A city or town may set aside such resources for the organization of sports leagues, like tabletop games, board games, etc. All these types of sports are called local sports. These sports can be extended to provisional-level sports. After provisional-level sports, there are national-level sports. Every nation has different sports, such as baseball which is known as the national game of the United States, cricket in England, and hockey in Pakistan. After national level, there are international level of sports. International sports are games in which the participants represent different countries. The most well-known international sports event is the Olympic Games, FIFA World Cup, and the Paralympic Games. Consider the set consisting of three countries {C1 , C2 , C3 } and also consider the set of players {(Abigail, 0.9, 0.8, 0.5, −0.6, −0.5, −0.2), (Alex, 0.6, 0.3, 0.4, −0.2, −0.4, −0.4), (Amelia, 0.8, 0.7, 0.2, −0.7, −0.8, −0.5), (Agatha, 0.9, 0.8, 0.5, −0.6, −0.5, −0.2), (Angela, 0.9, 0.8, 0.5, −0.6, −0.5, −0.2), (Belinda, 0.9, 0.8, 0.5, −0.6, −0.5, −0.2), (Ann, 0.5, 0.3, 0.5, −0.5, −0.3, −0.2), (Arlene, 0.8, 0.8, 0.9, −0.8, −0.9, −0.8), (Bella, 0.6, 0.4, 0.9, −0.6, −0.7, −0.5), (Anne, 0.9, 0.7, 0.8, −0.8, −0.8, −0.8), (April, 0.5, 0.3, 0.5, −0.5, −0.3, −0.2), (Abbey, 0.5, 0.3, 0.5, −0.5, −0.3, −0.2)}, who are taking part in their local-, provisional-, national-, and international-level games, as shown in Fig. 9.23. The positive degree of membership T p (x) of each player represents the percentage of hard work to achieve the success in a particular game, and I p (x) and F p (x) represent the indeterminacy and falsity in this percentage. The negative degree of membership T n (x) represents the percentage that the player faces failure in the achievement of success in a particular game, and I n (x) and F n (x) represent the indeterminacy and falsity in this percentage. The positive degree of membership T p (x) of each directed edge between a player and local-, provisional-, national-, and international-level games represents the percentage of having stamina for that level of sports in an international game, and I p (x) and F p (x) represent the indeterminacy and falsity in this percentage. The negative degree of membership T n (x) of each directed edge between a player and local-, provisional-, national-, and international-level games represents the percentage of having no stamina for that level of sports in an international game, and I n (x) and F n (x) represent the indeterminacy and falsity in this percentage. The 4-step bipolar single-valued neutrosophic competition graph can be used to find the best results. The 4-step bipolar single-valued neutrosophic outneighborhoods are calculated in Table 9.11. Therefore, N4+ (Abigail) ∩ N4+ (Alex) = {(I nter national games, 0.2, 0.2, 0.6, −0.1, −0.2, −0.7)}, N4+ (Abigail) ∩ N4+ (Amelia)={(I nter national games, 0.2, 0.2, 0.6, −0.1, −0.2, −0.8)}, and N4+ (Alex) ∩ N4+ (Amelia) = {(I nter national games, 0.4, 0.2, 0.6, −0.1, −0.2, −0.8)}. Further, h(N4+ (Abigail) ∩ N4+ (Alex)) = (0.2, 0.2, 0.6, 0.2, 0.2, 0.6), h(N4+ (Abigail) ∩ N4+ (Amelia)) = (0.2, 0.2, 0.6, 0.2, 0.2, 0.6), and h(N4+ (Amelia) ∩ N4+ (Alex)) = (0.4, 0.2, 0.6, 0.4, 0.2, 0.6). The 4-step bipolar single-valued neutrosophic competition graph is shown in Fig. 9.24. The strength to compete with the others players with respect to hard work in order to achieve success is calculated in Table 9.12. In Table 9.12, T (x, y) represents the value of strength of competition between players x and y with respect to hard work to

9.4 Applications of Competition Graphs in Bipolar Neutrosophic Environment Abigail

0.5 ,

−0 .2,

−0 .5,

−0 .2)

Angela (0.9, 0.8, 0.5, −0.6, −0.5, −0.2)

(0 .5,

0.3 ,

0.5 ,

−0 .5,

−0 .3,

−0 .2)

−0 .5,

−0 .3,

−0 .2)

Arlene

(0.8, 0.8, 0.9, −0.8, −0.9, −0.8)

(0 .5,

0.3 ,

0.5 ,

−0 .4,

−0 .3,

0.3 ,

0.5 ,

−0 .2,

−0 .3,

−0 .2)

0.1 ,

−0 .1,

−0 .1,

−0 .1)

April

(0.5, 0.3, 0.5, −0.5, −0.3, −0.2)

(0 .1,

0.1 ,

National games in C1

0.1 ,

−0 .1,

(0.5, 0.3, 0.7, −0.5, −0.3, −0.2)

Bella

(0.6, 0.4, 0.9, −0.6, −0.7, −0.5)

(0 .5,

−0 .1,

−0 .1)

(0.7, 0.4, 0.3, −0.2, −0.4, −0.3)

0.5 ,

−0 .5,

−0 .3,

(0.7, 0.3, 0.4, −0.4, −0.6, −0.6)

Abbey (0.5, 0.3, 0.5, −0.5, −0.3, −0.2)

(0 .1,

0.1 ,

0.1 ,

−0 .1,

National games in C2

Local games in C3 (0.7, 0.6, 0.5, −0.5, −0.4, −0.9)

−0 .2)

Provisional games in C2

−0 .1,

(0.8, 0.3, 0.2, −0.4, −0.5, −0.8)

(0.8, 0.2, 0.1, −0.2, −0.3, −0.4)

(0 .7 ,0 .3 ,0 .2 ,− 0. 2, − 0. 4, − 0. 6)

0.3 ,

(0.5, 0.6, 0.6, −0.2, −0.2, −0.8)

(0 .5,

(0.6, 0.7, 0.8, −0.7, −0.6, −0.5)

(0.6, 0.2, 0.3, −0.3, −0.4, −0.7)

(0.2, 0.3, 0.4, −0.2, −0.3, −0.4)

(0.9, 0.7, 0.8, −0.8, −0.8, −0.8)

0.1 ,

−0 .2)

Provisional games in C1 Anne

(0 .1,

(0.6, 0.5, 0.6, −0.3, −0.2, −0.7)

0.5 ,

(0.5, 0.6, 0.7, −0.2, −0.5, −0.3)

(0.4, 0.2, 0.5, −0.3, −0.2, −0.5)

0.3 ,

(0.3, 0.4, 0.6, −0.1, −0.3, −0.4)

(0. 5,

Belinda

(0.9, 0.8, 0.5, −0.6, −0.5, −0.2)

Local games in C2

Local games in C1

Ann

(0.5, 0.3, 0.5, −0.5, −0.3, −0.2)

(0.8, 0.7, 0.2, −0.7, −0.8, −0.5)

(0.6, 0.6, 0.3, −0.3, −0.4, −0.5)

0.6 ,

(0.4, 0.2, 0.6, −0.1, −0.2, −0.3)

(0 .5,

(0.4, 0.5, 0.5, −0.1, −0.3, −0.1)

Agatha (0.9, 0.8, 0.5, −0.6, −0.5, −0.2)

Amelia

Alex

(0.6, 0.3, 0.4, −0.2, −0.4, −0.4)

(0.9, 0.8, 0.5, −0.6, −0.5, −0.2)

387

−0 .1)

Provisional games in C3 (0.7, 0.8, 0.9, −0.5, −0.3, −0.2)

National games in C3 (0.6, 0.7, 0.8, −0.3, −0.2, −0.9)

7) 0. − 2, 0. − 3, 0. ,− .5 ,0 .6 ,0 .5 (0

International games (0.9, 0.7, 0.2, −0.3, −0.5, −0.6)

Fig. 9.23 Bipolar neutrosophic digraph of players Table 9.11 4-step bipolar single-valued neutrosophic out-neighborhoods x∈X N4+ (x) Abigail Alex Amelia

{(International games, 0.2, 0.2, 0.6, −0.1, −0.3, −0.4)} {(International games, 0.4, 0.2, 0.6, −0.1, −0.2, −0.7)} {(International games, 0.5, 0.5, 0.6, −0.2, −0.2, −0.8)}

Table 9.12 Strength of competition of players for international games (x, y) T (x y) (Abigail, Alex) (Abigail, Amelia) (Alex, Amelia)

(0.12, 0.06, 0.30, −0.04, −0.08, −0.24) (0.16, 0.14, 0.30, −0.12, −0.10, −0.30) (0.24, 0.06, 0.24, −0.08, −0.08, −0.30)

S(x y) 1.04 1 1.24

388

9 Bipolar Neutrosophic Competition Graphs (0.12, 0.06, 0.30, −0.04, −0.08, −0.24)

Abigail

Agatha (0.9, 0.8, 0.5, −0.6, −0.5, −0.2)

Amelia

Alex

(0.8, 0.7, 0.2, −0.7, −0.8, −0.5)

(0.6, 0.3, 0.4, −0.2, −0.4, −0.4)

(0.9, 0.8, 0.5, −0.6, −0.5, −0.2)

Angela

Belinda

(0.9, 0.8, 0.5, −0.6, −0.5, −0.2)

(0.9, 0.8, 0.5, −0.6, −0.5, −0.2)

(0.24, 0.06, 0.24, −0.08, −0.08, −0.30)

(0.16, 0.14, 0.30, −0.12, −0.10, −0.30)

Local games in C2

Local games in C1

Ann

(0.5, 0.3, 0.5, −0.5, −0.3, −0.2)

(0.5, 0.3, 0.7, −0.5, −0.3, −0.2)

(0.5, 0.6, 0.7, −0.2, −0.5, −0.3)

Bella

Arlene

(0.6, 0.4, 0.9, −0.6, −0.7, −0.5)

(0.8, 0.8, 0.9, −0.8, −0.9, −0.8)

Provisional games in C1

Provisional games in C2

(0.7, 0.3, 0.4, −0.4, −0.6, −0.6)

(0.6, 0.7, 0.8, −0.7, −0.6, −0.5)

Anne

(0.9, 0.7, 0.8, −0.8, −0.8, −0.8)

April

(0.5, 0.3, 0.5, −0.5, −0.3, −0.2)

National games in C1

(0.7, 0.4, 0.3, −0.2, −0.4, −0.3)

Local games in C3

(0.7, 0.6, 0.5, −0.5, −0.4, −0.9)

Provisional games in C3 (0.7, 0.8, 0.9, −0.5, −0.3, −0.2)

Abbey (0.5, 0.3, 0.5, −0.5, −0.3, −0.2)

National games in C2

(0.8, 0.3, 0.2, −0.4, −0.5, −0.8)

National games in C3

(0.6, 0.7, 0.8, −0.3, −0.2, −0.9)

International games (0.9, 0.7, 0.2, −0.3, −0.5, −0.6)

Fig. 9.24 4-step bipolar single-valued neutrosophic competition graph

achieve success in a particular game. From Table 9.12, it is clear that the strength of competition between Alex and Amelia to achieve success in a particular game at the international level is 1.24, while strength of competition between Abigail and Amelia is 1, and strength of competition between Abigail and Alex is 1.04. It is also clear from Table 9.12 that Alex and Amelia are the strongest contestants, as the strength of competition between them has the largest value than that of the other contestants. The method to calculate the net strength of competition among applicants is shown in Algorithm 9.4.2. Algorithm 9.4.2 Strength of competition of players for international games Step 1. Input the bipolar single-valued neutrosophic A of q applicants y1 , y2 , . . . , yq . Step 2. If for any two distinct vertices yi and y j , A(yi y j ) = (0, 0, 1, 0, 0, −1), then (y j , T p (yi y j ), I p (yi y j ), F p (yi y j ), T n (yi y j ), I n (yi y j ), F n (yi y j )) ∈ N + (yi ). Step 3. Repeat Step 2 for all vertices yi and y j to calculate bipolar single-valued neutrosophic-out-neighborhoods N + (yi ).

9.4 Applications of Competition Graphs in Bipolar Neutrosophic Environment

Step 4. Step 5. Step 6. Step 7. Step 8.

389

Calculate N + (yi ) ∩ N + (y j ) for each pair of distinct vertices yi and y j . Calculate h[N + (yi ) ∩ N + (y j )] for each pair of distinct vertices yi and y j . If N + (yi ) ∩ N + (y j ) = ∅, then draw an edge yi y j . Repeat Step 6 for all pairs of distinct vertices. Assign membership values to each edge yi y j using the conditions

T R (yi y j ) = (yi ∧ y j )h 1 [N + (yi ) ∩ N + (y j )] T Rn (yi y j ) = (yi ∨ y j )h 4 [N + (yi ) ∩ N + (y j )] p I R (yi y j ) = (yi ∧ y j )h 2 [N + (yi ) ∩ N + (y j )] I Rn (yi y j ) = (yi ∨ y j )h 5 [N + (yi ) ∩ N + (y j )] p

FR (yi y j ) = (yi ∨ y j )h 3 [N + (yi ) ∩ N + (y j )] FRn (yi y j ) = (yi ∧ y j )h 6 [N + (yi ) ∩ N + (y j )]. p

Step 9.

Calculate S(x, y), the strength of competition between players x and y. p

p

p

S(x y) = T R (x, y) − (I R (x, y) + FR (x, y)) + 1 + T Rn (x, y) − (I Rn (x, y) + FRn (x, y)).

Step 11. Maximum value of S(x, y) gives that x and y are the strongest players than the others.

9.5 Conclusions The fuzzy set theory is very successful in handling uncertainties arising from vagueness or partial belongingness of elements but it cannot model all sorts of uncertainties prevailing in various physical problems. There are many situations containing six components of uncertainty, that is, existence/non-existence/neutral and their counter properties. A bipolar neutrosophic set is a useful notion in this sense which is obtained by combining the theory of neutrosophic sets with bipolar fuzzy information. In this chapter, we have presented the notion of bipolar neutrosophic graphs and bipolar neutrosophic digraphs by applying the technique of bipolar neutrosophic sets to graphs and digraphs. We have studied bipolar neutrosophic competition graphs and presented certain characterizations of bipolar neutrosophic out-neighborhoods, bipolar neutrosophic in-neighborhoods, bipolar neutrosophic open neighborhood graphs, bipolar neutrosophic closed neighborhood graphs, bipolar neutrosophic pcompetition graphs, m-step bipolar neutrosophic competition graphs, and strong preys and strong independent predator–prey relations. We have described various methods for the construction of bipolar neutrosophic competition graphs of certain products of bipolar neutrosophic digraphs. We have elaborated on the importance of bipolar neutrosophic graphs with various real-world applications.

390

9 Bipolar Neutrosophic Competition Graphs

Exercises 9 1. What is the difference between a neutrosophic set and a single-valued neutrosophic set? 2. Derive a method for the construction of a bipolar single-valued neutrosophic − → − → competition graph C( G 1 × G 2 ) of the direct product of bipolar single-valued − → − → − → − → neutrosophic digraphs G 1 and G 2 using C( G 1 ) and C( G 2 ). 3. If all the edges of a bipolar single-valued neutrosophic digraph are strong, then prove using a contradiction example that all the edges of the corresponding bipolar single-valued neutrosophic competition graph may not be strong. 4. What do you mean by (1, 1, 0, −1, −1, 0) and (0, 0, 1, 0, 0, −1) in a bipolar neutrosophic environment? 5. Let A, C and D be three bipolar single-valued neutrosophic sets, then classify whether the following statements are true or false. (a) (b) (c) (d) (e) (f)

(A ∪ B)c = Ac ∩ B c , (A ∩ B)c = Ac ∪ B c , (A ∪ B) ∪ C = A ∪ (B ∪ C), (Ac )c = A, (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C), (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).

References 1. Akram, M.: Certain bipolar neutrosophic competition graphs. J. Indones. Math. Soc. 24(1), 1–25 (2017) 2. Akram, M.: Single-valued neutrosophic planar graphs. Int. J. Algebra Stat. 2, 157–167 (2016) 3. Akram, M.: Single-Valued Neutrosophic Graphs. Infosys Science Foundation Series in Mathematical Sciences. Springer, Singapore (2018) 4. Akram, M., Nasir, M.: Interval-valued neutrosophic competition graphs. Ann. Fuzzy Math. Inform. 14(1), 99–120 (2017) 5. Akram, M., Nasir, M., Shum, K.P.: Novel applications of bipolar single-valued neutrosophic competition graphs. Appl. Math.-A J. Chin. Univ. 33(4), 436–467 (2018) 6. Akram, M., Sarwar, M.: Novel multiple criteria decision making methods based on bipolar neutrosophic sets and bipolar neutrosophic graphs. Ital. J. Pure Appl. Math. 38, 368–389 (2017) 7. Akram, M., Shahzadi, G.: Operations on single-valued neutrosophic graphs. J. Uncertain Syst. 11(1), 1–26 (2017) 8. Akram, M., Shum, K.P.: Bipolar single-valued neutrosophic planar graphs. J. Math. Res. Appl. 36(6), 631–648 (2017) 9. Akram, M., Siddique, S.: Neutrosophic competition graphs with applications. J. Intell. Fuzzy Syst. 33(2), 921–935 (2017) 10. Akram, M., Siddique, S., Shum, K.P.: Certain properties of bipolar neutrosophic graphs. Southeast Asian Bull. Math. 42(4), 463–490 (2018) 11. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 12. Broumi, S., Bakali, A., Talea, M., Smarandache, F., ALi, M.: Shortest path problem under bipolar neutrosophic setting. Appl. Mech. Mater. 859, 59–66 (2017)

References

391

13. Broumi, S., Talea, M., Bakali, A., Smarandache, F.: Single valued neutrosophic graphs. J. New Theory 10, 86–101 (2016) 14. Deli, I., Ali, M., Smarandache, F.: Bipolar neutrosophic sets and their application based on multi-criteria decision making problems. In: IEEE International Conference on Advanced Mechatronic Systems, pp. 249–254 (2015) 15. Smarandache, F.: A unifying field in logics: neutrosophic logic. Philosophy. American Research Press 1–141 (1999) 16. Smarandache, F.: Neutrosophic sets generalization of the intuitionistic fuzzy set. J. Def. Resour. Manag. 1(1), 107–116 (2010) 17. Smarandache, F.: Types of neutrosophic graphs and neutrosophic algebraic structures together with their applications in technology. Seminar, Universitatea Transilvania din Brasov, Facultatea de Design de Produs si Mediu, Brasov, Romania (2015) 18. Wang, H., Smarandache, F., Zhang, Y., Sunderraman, R.: Single valued neutrosophic sets. Infinite Study (2010) 19. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 20. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 21. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998) 22. Zhang, W.-R., Wang, P., Peace, K., Zhan, J., Zhang, Y.: On truth, uncertainty, equilibrium, and harmony–a taxonomy for YinYang scientific computing. New Math. Nat. Comput. 4(2), 207–229 (2008)

Chapter 10

Bipolar Neutrosophic Graph Structures

In this chapter, we apply the powerful technique of bipolar neutrosophic set to graph structures and present a framework of bipolar neutrosophic graph structures with certain operations. We discuss the notions of Bk −edges, strong and complete bipolar neutrosophic graph structures, and bipolar neutrosophic subgraph structures. Using ϕ-complement, we describe certain relations and isomorphism properties of selfcomplementary, totally self-complementary, and totally strong self-complementary bipolar neutrosophic graph structures. We study the importance of bipolar neutrosophic graph structures with a number of real-world applications in international relations, psychology, and global terrorism. This chapter is basically due to [4, 14].

10.1 Introduction A graph structure is a generalization of a graph introduced by Sampathkumar [11] in 2006 and is widely useful in the study of graphs, signed graphs, semi-graphs, edge-colored graphs, and edge-labeled graphs. Graph structures are very useful in various domains of computer science and computational intelligence. Denish and Ramakrishnan [10] introduced the concept of fuzzy graph structure, fuzzy labeling matrices, fuzzy index matrices, incidence algebras, and their fuzzy analogs. The notions of maximal products of fuzzy graph structures and regular fuzzy graph structures are given in [12]. In some applications such as expert system, belief system, and information fusion, we should consider not only the truth membership supported by the evidence but also the falsity membership against the evidence. That is beyond the scope of fuzzy sets and interval-valued fuzzy sets. In 1983, Atanassov [8] introduced the notion of intuitionistic fuzzy set as a generalization of a fuzzy set. In an intuitionistic fuzzy set A, both truth membership value (belongingness) T A (x) and falsity membership value (non-belongingness) FA (x) are attached with each element x with the restriction that the sum of these two membership values © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6_10

393

394

10 Bipolar Neutrosophic Graph Structures

is less or equal to 1. In real-world problems, there are many situations containing tricomponents of uncertainty, e.g., games (winning, defeating, or tie scores), voting (pro, contra, null/black votes), positive/negative/zero numbers, yes/no/NA, decision-making and control theory (making a decision, not making, or hesitating), and accepted/rejected/pending. Smarandache [13], in 1995, proposed the term “neutrosophic" means knowledge of neutral thought and neutral represents the main distinction between a fuzzy/intuitionistic fuzzy set and a neutrosophic set. In contrast to the intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set, the indeterminacy is characterized explicitly in a neutrosophic set which is independent of truth and falsity membership values. Neutrosophic sets are indeed more general than intuitionistic fuzzy sets as there are no constraints between truth, indeterminacy, and falsity membership values. All these degrees can individually vary within [0, 1]. The concept of bipolar neutrosophic sets was introduced in [9] and bipolar singlevalued neutrosophic graph in [3]. Akram and Akmal [1, 2] initiated the concept of bipolar fuzzy graph structures and discussed their operations. The discussion of this chapter is based on bipolar single-valued neutrosophic graph structures which are from Akram and Sitara’s work [4, 14]. For further terminologies and studies, readers are referred to [5–7].

10.2 Operations on Bipolar Neutrosophic Graph Structures A graph structure Gˇ s = (X, X 1 , X 2 , . . . , X m ) consists of a non-empty set X together with relations X 1 , X 2 , . . . , X m on X which are mutually disjoint such that each X k , 1 ≤ k ≤ m is symmetric and irreflexive. A graph structure can be represented in the plane just like a graph, where each edge is labeled as X k , 1 ≤ k ≤ m. Definition 10.1 A bipolar single-valued neutrosophic graph structure of a graph structure Gˇ s = (X, X 1 , X 2 , . . . , X m ) is denoted by Gˇ bn = (B, B1 , B2 , . . . , Bm ), where B =< b, T p (b), I p (b), F p (b), T n (b), I n (b), F n (b) >, p

p

p

Bk =< (bd, Tk (bd), Ik (bd), Fk (bd), Tkn (bd), Ikn (bd), Fkn (bd) > are bipolar single-valued neutrosophic sets on X and X k , respectively, such that p

p

p

Tk (bd) ≤ min{T p (b), T p (d)}, Ik (bd) ≤ min{I p (b), I p (d)}, Fk (bd) ≤ max{F p (b), F p (d)}, Tkn (bd)

≥ max{T (b), T (d)}, n

n

Ikn (bd) p

≥ max{I (b), I (d)}, n

n

p

Fkn (bd)

≥ min{F n (b), F n (d)},

p

for all b, d ∈ X. Note that 0 ≤ Tk (bd) + Ik (bd) + Fk (bd) ≤ 3, −3 ≤ Tkn (bd) + Ikn (bd) + Fkn (bd) ≤ 0 for all bd ∈ X k . Example 10.1 Consider a graph structure Gˇ s = (X, X 1 , X 2 , X 3 ) such that X = {b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 } and X 1 = {b1 b2 , b2 b7 , b4 b8 , b6 b8 , b5 b6 , b3 b4 }, X 2 =

10.2 Operations on Bipolar Neutrosophic Graph Structures Table 10.1 Bipolar neutrosophic set B B b1 b2 b3 p

T Ip Fp Tn In Fn

0.4 0.3 0.4 −0.4 −0.3 −0.4

0.3 0.2 0.3 −0.3 −0.2 −0.3

0.3 0.3 0.2 −0.3 −0.3 −0.2

b4

b5

b6

b7

b8

0.4 0.3 0.4 −0.4 −0.3 −0.4

0.2 0.1 0.2 −0.2 −0.1 −0.2

0.3 0.3 0.5 −0.3 −0.3 −0.5

0.4 0.4 0.2 −0.4 −0.4 −0.2

0.2 0.3 0.3 −0.2 −0.3 −0.3

Table 10.2 Bipolar neutrosophic sets B1 , B2 , and B3 B1 b1 b2 b2 b7 b4 b8 p

T Ip Fp Tn In Fn

0.3 0.2 0.4 −0.3 −0.2 −0.4

0.3 0.2 0.3 −0.3 −0.2 −0.3

395

0.2 0.3 0.4 −0.2 −0.3 −0.4

b6 b8

b5 b6

b3 b4

0.2 0.3 0.5 −0.2 −0.3 −0.5

0.2 0.1 0.5 −0.2 −0.1 −0.5

0.3 0.3 0.4 −0.3 −0.3 −0.4

B2

b1 b5

b5 b7

b3 b6

b7 b8

Tp Ip Fp Tn In Fn

0.2 0.1 0.4 −0.2 −0.1 −0.4

0.2 0.1 0.2 −0.2 −0.1 −0.2

0.3 0.3 0.5 −0.3 −0.3 −0.5

0.2 0.3 0.3 −0.2 −0.3 −0.3

B3

b1 b3

b2 b4

Tp Ip Fp Tn In Fn

0.3 0.3 0.4 −0.3 −0.3 −0.4

0.3 0.2 0.4 −0.3 −0.2 −0.4

{b1 b5 , b5 b7 , b3 b6 , b7 b8 }, X 3 = {b1 b3 , b2 b4 }. Let B be a bipolar single-valued neutrosophic set on X given in Table 10.1 and B1 , B2 , B3 be bipolar single-valued neutrosophic sets on X 1 , X 2 , and X 3 , respectively, given in Table 10.2. Routine calculations show that Gˇ bn = (B, B1 , B2 , B3 ) is a bipolar single-valued neutrosophic graph structure, as shown in Fig. 10.1. Example 10.2 Consider a graph structure Gˇ s = (X, X 1 , X 2 ) such that X = {b1 , b2 , b3 , b4 }, X 1 = {b1 b3 , b1 b2 , b3 b4 }, X 2 = {b1 b4 , b2 b3 }. By defining bipolar single-

396

10 Bipolar Neutrosophic Graph Structures

) .5) .3 0.3, −0 −0 2) 0.5) −0.2, − −0.3, − 3, −0. , −0.3, .2, 0.3, 0.5, 0. 0 ( 0.3, 0.5 − .4, ) B1 b6(0.3, 0 2, 5 0. , − 0. − , − 0.4 1, .3 0. , 0 .2, − − .3 2, , 0 4, 0 0. . .2 (0 , 0 ,− .5 B 2 (0.4 ,0 .1 b7 ,0 .2) .2 .1, −0 .2, −0 (0 .2, −0 B1 , 0.1, 0 B 2 (0.2

) , −0.4 4) , −0.3 0. , −0.3 − 0.3, 0.4 2, 4) (0.3, 0. 0. B 1 − 2) − 3, 3, −0. 0. 0. , − .3 ,− 0 3, .4 0. ,− ,0 .2 , − 0.3 .4 ,0 − .3 , 0 .2, .3 (0 0 3 , 0 3, B 0. .3 .4) (0 .3, .2, −0 2, −0.3) .3, −0 B 3 3(0 .4, −0 .3, 0.2, 0.3, −0.3, −0. b , 0.2, 0 b2 (0 B 1(0.3

.4) −0 , 3 . −0 .4, 0 , − B2 (0.2, 0 .1, 0. 0.4 , 4, −0 b5 (0.2, 3 0 .2, − 0. , 0.1, − .1, 0.2, − 4 0.2, 0.4) 0. ( −0.1 b1 , −0 .2)

B2 (0 .3, 0.3 , 0.5,

B1 (0 .3, 0.2 , 0.3,

b4 (0 .4, 0.

3, 0.4 ,

−0.3 , −0.3 , −0.5 )

−0.3 , −0.2 , −0.3 )

3) 0. − , 3 0. − , −0.4 2 ) 0. ,− 3 0. 3, 0. , .2 (0 b8

B −0.4 1 (0.2, 0 .3 , −0. 3, −0 , 0.4, −0. 2, −0 .4) .3,

Fig. 10.1 Bipolar neutrosophic graph structure no. 1

valued neutrosophic sets B, B1 and B2 on X , X 1 , and X 2 , respectively, a bipolar single-valued neutrosophic graph structure can be drawn as shown in Fig. 10.2. Definition 10.2 Let Gˇ bn = (B, B1 , B2 , . . . , Bm ) be a bipolar single-valued neutrosophic graph structure of graph structure Gˇ s . If Hˇ bn = (B  , B1 , B2 , . . . , Bm ) is a bipolar single-valued neutrosophic graph structure of Gˇ s such that T  p (b) ≤ T p (k), I  p (b) ≤ I p (b), F  p (b) ≥ F p (b), T n (b) ≥ T p (k), I n (b) ≥ I p (b), F n (b) ≤ F n (b), p

p

p

p

p

p

Tk (bd) ≤ Tk (bd), Ik (bd) ≤ Ik (bd), Fk (bd) ≥ Fk (bd), Tkn (bd) ≥ Tkn (bd), Ikn (bd) ≥ Ikn (bd), Fkn (bd) ≤ Fkn (bd), b ∈ X and bd ∈ X k , k = 1, 2, . . . , m.

) , −0.3 , −0.2 , −0.3 0.2, −0.3) .3 0 , , 0.2 ,− b4 (0.3 −0.3 ) , 0.3, 2 . 0 0.3 , .3 ,− B 1(0 2 . 0 ,− 0.2 − , 0.3 .2, 0 , 2 (0. B2

b2 (0.2, 0.2, 0.3,

0.2,

0.2,

0.4, 0.3,

0.4, −0. 2, − 0.3, −0. 4)

b3 (0.3, 0.4, 0.3,

B1 (0.2, 0.2, 0.4, −0.2, −0.2, −0.4)

4) −0. 0.2, 2, − −0. 0.4, 0.2, 0.2, B 2(

b1 (0.2, 0.3, 0.4, −0.2, −0.3, −0.4) B1 ( 0.2, 0.3,

397

0.3)

10.2 Operations on Bipolar Neutrosophic Graph Structures

0.3)

b2 (0.1, 0.1, 0.4,

0.4) 0.1, − 0.2, − , −0.4) − , .4 0.1 , 0.1, 0 0.2, − b4 (0.2 0.4, − , 1 .4) . 0 .2, −0 , B 1(0 1 . −0 .1, 0 ,− 0.4 1, . 0 1, (0. B2

0.1,

0.1,

0.3, 0.2,

0.5, −0. 1, − 0.2, −0. 5)

b3 (0.2, 0.3, 0.4,

B1 (0.1, 0.1, 0.5, −0.1, −0.1, −0.5)

5) −0. 0.1, 1, − −0. 0.5, 0.1, 0.1, B 2(

b1 (0.1, 0.2, 0.5, −0.1, −0.2, −0.5) B ( 1 0. 1, 0 .2,

0.4)

Fig. 10.2 Bipolar neutrosophic graph structure no. 2

0.4)

Fig. 10.3 Bipolar neutrosophic subgraph structure

Then Hˇ bn is called a bipolar single-valued neutrosophic subgraph structure of the bipolar single-valued neutrosophic graph structure Gˇ bn denoted by Hˇ bn ⊆ Gˇ bn . Example 10.3 Consider a bipolar single-valued neutrosophic graph structure Hˇ bn = (B  , B1 , B2 ) of graph structure Gˇ s = (X, X 1 , X 2 ) as shown in Fig. 10.3. Routine calculations indicate that Hˇ bn is a bipolar single-valued neutrosophic subgraph structure of Fig. 10.2.

398

10 Bipolar Neutrosophic Graph Structures

b2 (0.2, 0.2, 0.3, −0.2, −0.2, −0.3)

4) 0. − 3, 0. − 2, 0. ,− .4 ,0 .3 ,0 .2 (0 b1

Fig. 10.4 Bipolar neutrosophic induced subgraph structure

.4) , −0 −0.2 , 2 . , −0 , 0.4 , 0.2 2 . 0 (

B1

B

(0 B1 (0.2, 0.3, 0.4, −0.2, −0.3, −0.4) .2 ,0 .2 ,0 .3 ,− 0. 2, − 0. 2, − 0. 3)

b3

(0 .3 ,0 .4 ,0 .3 ,− 0. 3, − 0. 4, − 0. 3)

2

Definition 10.3 A bipolar single-valued neutrosophic subgraph structure Hˇ bn = (B  , B1 , B2 , . . . , Bm ) of Gˇ bn = (B, B1 , B2 , . . . , Bm ) is called a bipolar singlevalued neutrosophic induced subgraph structure of a bipolar single-valued neutrosophic graph structure Gˇ bn by Q ⊆ X if T  p (b) = T p (b), I  p (b) = I p (b), F  p (b) = F p (b), T n (b) = T n (b), I n (b) = I n (b), F n (b) = F n (b), p

p

p

Tk (bd) = Tk (bd), Ik (bd) = Ik (bd), Fk (bd) = Fk (bd), Tkn (bd) = Tkn (bd), p

Ikn (bd)

=

Ikn (bd),

p

Fkn (bd)

=

p

Fkn (bd),

for each k = 1, 2, . . . , m and for all b, d ∈ Q. Example 10.4 A bipolar single-valued neutrosophic graph structure shown in Fig. 10.4 is a bipolar single-valued neutrosophic induced subgraph structure of a bipolar single-valued neutrosophic graph structure given in Fig. 10.2. Definition 10.4 A bipolar single-valued neutrosophic subgraph structure Hˇ bn = (B  , B1 , B2 , . . . , Bm ) of Gˇ bn = (B, B1 , B2 , . . . , Bm ) is called bipolar single-valued neutrosophic spanning subgraph structure of Gˇ bn if B  = B and p

p

p

p

p

p

Tk (bd) ≤ Tk (bd), Ik (bd) ≤ Ik (bd), Fk (bd) ≥ Fk (bd),

Tkn (bd) ≥ Tkn (bd), Ikn (bd) ≥ Ikn (bd), Fkn (bd) ≤ Fkn (bd), for each k = 1, 2, . . . , m.

Example 10.5 A bipolar single-valued neutrosophic graph structure given in Fig. 10.5 is a bipolar single-valued neutrosophic spanning subgraph structure of Fig. 10.2.

10.2 Operations on Bipolar Neutrosophic Graph Structures

399

−0.3, − 0.4) ) .4 0 ,− 4) 0.1 −0. − , , 1 0.1 0. ,− − 4 , . 2 0 0. .1, ,0 ,− 1 4 . . 0 (0 1, B1 0B. , b4 .2 1 (0. 2, (0.3 (0 0. B2 1, , 0.2 0. B 3, , 0. 2 (0 − 3, .1, 0. 0.1 2, −0. , 0. − 3, 3, − 0. 0.1 1, −0 ) ,− − .2, 0.1 0.3 0. − ,− − 3 0.3 ) 0.3 .4, ) ) −0 , 3 . −0 .3, 0 , 0.4 .3, 0 ( b3 .4) .2, −0 .1, −0 .4, −0 , 0.2, 0 B 1(0.1

b2 (0.2 , 0.2, 0 .3, −0 .2, −0 .2, −0 .3)

b1 (0.2 , 0.3, 0 .4, −0.2 ,

Fig. 10.5 Bipolar neutrosophic spanning subgraph structure

Definition 10.5 Let Gˇ bn = (B, B1 , B2 , . . . , Bm ) be a bipolar single-valued neutrosophic graph structure, then bd ∈ X k is called a bipolar single-valued neutrosophic Bk -edge or shortly Bk -edge if Bk (bd) = (0, 0, 1, 0, 0, −1). Consequently, the support of Bk is defined as supp(Bk ) = {bd ∈ X k |Bk (bd) = (0, 0, 1, 0, 0, −1)}, for each k = 1, 2, . . . , m. Definition 10.6 A bipolar single-valued neutrosophic Bk -path in a bipolar singlevalued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , . . . , Bm ) is a sequence of distinct vertices b1 , b2 , . . . , bm (except bm = b1 ) in X such that bk−1 bk is a bipolar single-valued neutrosophic Bk -edge for all k = 2, 3, . . . , m. Definition 10.7 A bipolar single-valued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , . . . , Bm ) is Bk -strong, for any k ∈ {1, 2, . . . , m}, if p

p

p

Tk (bd) = min{T p (b), T p (d)}, Ik (bd) = min{I p (b), I p (d)}, Fk (bd) = max{F p (b), F p (d)}, Tkn (bd)

= max{T (b), T (d)}, n

n

Ikn (bd)

= max{I (b), I (d)}, n

n

Fkn (bd)

= min{F n (b), F n (d)},

for all bd ∈ X k . If Gˇ bn is Bk -strong for all k ∈ {1, 2, . . . , m}, then Gˇ bn is called a strong bipolar single-valued neutrosophic graph structure.

400

10 Bipolar Neutrosophic Graph Structures

) 0.6 0. , − .6, − 3 . 0 0 , .3 − .3, .2, 0 −0 B 1(0 , .6 ,0 0.3

b5 (0.2, 0.1, 0.3, −0.2, −0.1, −0.3) B −0.1, −0.5) 1 (0 B 2 (0.2, 0.1, 0.5, −0.2, .2, B 3( 0.1 0 ,0 b3 .3, 0 .6, (0 . 3 .3, ,0 −0 0.3 .5, .2, −0 ,0 −0 .3, .3 .1, −0 , −0 −0 .3, .3, .6) −0 −0 .5) .3, −0 0.6, −0.3, −0.3, −0.6) .3) B2 (0.3, 0.3, 3, (0. b6

B 2( −0.4)b7 ( 0.2, 2, −0. 3, −0. 0.4, 0.2, , B B1 (0.3 0.4 0.3 3( , ,0 0.3 .4, 0.4, ,0 0.3 −0 .2, , − .2, 0.5 0.4 −0 ,− , − .3, 0.3 0.4 −0 ,− , − .4) 0.2 0.3 ,− ) ) −0.5 0.5 B1 (0.2, 0.3, 0.5, −0.2, −0.3, ) b4 (0.4, 0.3, 0.5, −0.4, −0.3, −0.5)

b8 (0.2, 0.3, 0.4, −0.2, −0.3, −0.4)

2, (0. B2

, 0.2 3, − . 0 , 0.1

, 0.3 2, −

.6) −0

) 0.3 ,− 1 . −0

3, (0. B1

5, , 0. 0.3

. −0

.5) −0 , 3 0. 3, −

) 3, 0.5 0. , − .4, − 2 . 0 0 , − .2, 0.3 0.3, 0 − , ( 5 b2 , 0. 0.2

) 0.4 ,− 2 . −0

3, (0. B1 b1 (0.4, 0.3, 0.5, −0.4, −0.3, −0.5)

Fig. 10.6 Strong bipolar single-valued neutrosophic graph structure

Example 10.6 Consider a bipolar single-valued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , B3 ) as shown in Fig. 10.6. Then Gˇ bn is a strong bipolar single-valued neutrosophic graph structure since it is Bk -strong for all k ∈ 1, 2, 3. Definition 10.8 A bipolar single-valued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , . . . , Bm ) is called a complete bipolar single-valued neutrosophic graph structure if 1. Gˇ bn is a strong bipolar single-valued neutrosophic graph structure, 2. supp(Bk ) = ∅, for all k = 1, 2, . . . , m, 3. For all b, d ∈ X , bd is a Bk − edge, for some k. Example 10.7 Let Gˇ bn = (B, B1 , B2 ) be a bipolar single-valued neutrosophic graph structure of graph structure Gˇ = (X, X 1 , X 2 ) such that X = {b1 , b2 , b3 , b4 },

10.2 Operations on Bipolar Neutrosophic Graph Structures

401

b1 (0.3, 0.4, 0.5, −0.3, −0.4, −0.5)

0.5 ,

0.2 ,

0.2 ,

2(

−0 .5) −0 .4, −0 .1, 0.5 , 0.3 , 0.1 ,

2(

.5) −0 b3 (0.1, 0.4, 0.5,

B

B

B2 (0.2, 0.2, 0.4, −0.2, −0.2, −0.4)

B2 (0.1, 0.4, 0.5, −0.1, −0.4, −0.5)

.3, −0

b2 (0.2, 0.3, 0.3, −0.2, −0.3, −0.3) .5) −0

.2, −0

, 0.2 ,− 0.1 ,− 0.5 .2, 1, 0 (0. B1 b4 (0.2, 0.2, 0.4, −0.2, −0.2, −0.4)

, 0.5

−0 .2,

, 0.3

−0 .2,

2, (0. B1

−0 .5)

Fig. 10.7 Complete bipolar neutrosophic graph structure

0.1,

0.4,

0.5)

X 1 = {b1 b2 , b3 b4 }, X 2 = {b1 b3 , b2 b3 , b1 b4 , b2 b4 } as shown in Fig. 10.7. Through direct calculations, it is easy to check that Gˇ bn is a strong bipolar single-valued neutrosophic graph structure. Moreover, supp(B1 ) = ∅, supp(B2 ) = ∅, and each pair bk bl of vertices in X is either a B1 -edge or B2 -edge. Hence, Gˇ bn is a complete bipolar single-valued neutrosophic graph structure, that is, B1 − B2 −complete bipolar single-valued neutrosophic graph structure. Definition 10.9 Let Gˇ b1 = (B1 , B11 , B12 , . . . , B1m ) and Gˇ b2 = (B2 , B21 , B22 , . . . , B2m ) be two bipolar single-valued neutrosophic graph structures. The lexicographic product of Gˇ b1 and Gˇ b2 is denoted by Gˇ b1 • Gˇ b2 = (B1 • B2 , B11 • B21 , B12 • B22 , . . . , B1m • B2m ) and defined as ⎧ p p p p p ⎨ T(B1 •B2 ) (bd) = (TB1 • TB2 )(bd) = TB1 (b) ∧ TB2 (d) p p p p p I (bd) = (I B1 • I B2 )(bd) = I B1 (b) ∧ I B2 (d) 1. ⎩ (Bp1 •B2 ) p p p p F(B1 •B2 ) (bd) = (FB1 • FB2 )(bd) = FB1 (b) ∨ FB2 (d) ⎧ n n n n n ⎨ T(B1 •B2 ) (bd) = (TB1 • TB2 )(bd) = TB1 (b) ∨ TB2 (d) n n n n n I (bd) = (I B1 • I B2 )(bd) = I B1 (b) ∨ I B2 (d) 2. ⎩ (Bn1 •B2 ) F(B1 •B2 ) (bd) = (FBn1 • FBn2 )(bd) = FBn1 (b) ∧ FBn2 (d) for all bd ∈ X 1 × X 2 , ⎧ p p p p p ⎨ T(B1k •B2k ) (bd1 )(bd2 ) = (TB1k • TB2k )(bd1 )(bd2 ) = TB1 (b) ∧ TB2k (d1 d2 ) p p p p p I (bd1 )(bd2 ) = (I B1k • I B2k )(bd1 )(bd2 ) = I B1 (b) ∧ I B2k (d1 d2 ) 3. ⎩ (Bp1k •B2k ) p p p p F(B1k •B2k ) (bd1 )(bd2 ) = (FB1k • FB2k )(bd1 )(bd2 ) = FB1 (b) ∨ FB2k (d1 d2 )

−0 .2, −0 .3, 0.5 , 0.2 , .3,

2 (0

B2

−0 .1, −0 .5, 0.8 , 0.1 , .5,

1 (0

B1

d3 (0.5, 0.3, 0.5, −0.5, −0.3, −0.5)

4) −0. 0.1, .2, − , −0 , 0.4 , 0.1 (0.2 B 21

d1 (0.2, 0.1, 0.3, −0.2, −0.1, −0.3)

6) −0. 0.2, .4, − , −0 , 0.6 , 0.2 (0.4 B 12

b3 (0.4, 0.2, 0.4, −0.4, −0.2, −0.4) b1 (0.5, 0.1, 0.6, −0.5, −0.1, −0.6)

−0 .5)

10 Bipolar Neutrosophic Graph Structures

−0 .8)

402

Fig. 10.8 Bipolar neutrosophic graph structures

⎧ n n n n n ⎪ ⎨ T(B1k •B2k ) (bd1 )(bd2 ) = (TB1k • TB2k )(bd1 )(bd2 ) = TB1 (b) ∨ TB2k (d1 d2 ) n n n n n I(B •B ) (bd1 )(bd2 ) = (I B • I B )(bd1 )(bd2 ) = I B (b) ∨ I B (d1 d2 ) 4. for all 1 1k 2k 2k 1k 2k ⎪ n n n n ⎩ Fn (B •B ) (bd1 )(bd2 ) = (FB • FB )(bd1 )(bd2 ) = FB (b) ∧ FB (d1 d2 ) 1k

2k

1k

2k

1k

1

2k

2k

b ∈ X 1 , d1 d2 ∈ X 2k , ⎧ p p p p p ⎪ ⎨ T(B1k •B2k ) (b1 d1 )(b2 d2 ) = (TB1k • TB2k )(b1 d1 )(b2 d2 ) = TB1k (b1 b2 ) ∧ TB2k (d1 d2 ) p p p p p I(B •B ) (b1 d1 )(b2 d2 ) = (I B • I B )(b1 d1 )(b2 d2 ) = I B (b1 b2 ) ∧ I B (d1 d2 ) 5. 2k 1k 2k ⎪ p p p p ⎩ F p1k 2k (b d )(b d ) = (F 1k 1 1 2 2 B1k • FB2k )(b1 d1 )(b2 d2 ) = FB1k (b1 b2 ) ∨ FB2k (d1 d2 ) 1k •B2k ) ⎧ (B n n n n n ⎪ ⎨ T(B1k •B2k ) (b1 d1 )(b2 d2 ) = (TB1k • TB2k )(b1 d1 )(b2 d2 ) = TB1k (b1 b2 ) ∨ TB2k (d1 d2 ) n I(B (b1 d1 )(b2 d2 ) = (I Bn • I Bn )(b1 d1 )(b2 d2 ) = I Bn (b1 b2 ) ∨ I Bn (d1 d2 ) 6. 1k 2k 1k 2k 1k •B2k ) ⎪ n n n n ⎩ Fn (B •B ) (b1 d1 )(b2 d2 ) = (FB • FB )(b1 d1 )(b2 d2 ) = FB (b1 b2 ) ∧ FB (d1 d2 ) 1k

for all b1 b2 ∈ X 1k , d1 d2 ∈ X 2k .

2k

1k

2k

Example 10.8 Consider two bipolar single-valued neutrosophic graph structures Gˇ b1 = (B1 , B11 , B12 ) and Gˇ b2 = (B2 , B21 , B22 ) of graph structures Gˇ s1 = (X 1 , X 11 , X 12 ) and Gˇ s2 = (X 2 , X 21 , X 22 ), respectively, as shown in Fig. 10.8, where X 11 = {b1 b2 }, X 12 = {b3 b4 }, X 21 = {d1 d2 }, X 22 = {d2 d3 }. The lexicographic product of bipolar single-valued neutrosophic graph structures Gˇ b1 and Gˇ b2 shown in Fig. 10.8 is defined as Gˇ b1 • Gˇ b2 = {B1 • B2 , B11 • B21 , B12 • B22 } and is given in Fig. 10.9. Theorem 10.1 The lexicographic product Gˇ b1 • Gˇ b2 = (B1 • B2 , B11 • B21 , B12 • B22 , . . . , B1m • B2m ) of two bipolar single-valued neutrosophic graph structures of graph structures Gˇs1 and Gˇs2 is a bipolar single-valued neutrosophic graph structure of Gˇs1 • Gˇs2 . Proof It is given that Gˇ b1 • Gˇ b2 = (B1 • B2 , B11 • B21 , B12 • B22 , . . . , B1m • B2m ) are two bipolar single-valued neutrosophic graph structures of graph structures Gˇs1 and Gˇs2 . Here there arise two cases.

10.2 Operations on Bipolar Neutrosophic Graph Structures

403

B11 • B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8)

b2 d3 (0 .5, .6) 0 0.2 − , .1 , 0 8) 0.8, − ) , 0. .8 3 0 −0 . − − , 0 B • B (0.2, 0.1, 0.8, −0.2, −0.1, −0.8) 1 . , 0 .5, − 2 11 − 21 , , 0. −0 −0.2 0.6 , − , 8 . , .2, ,0 .1 1 3 . 0 0 . , −0 0 .2, 3 0 . ( − .8) (0 b 2d 1 8, 2 . d 0 b1 , 2 . 0 3, B11 • B21 (0.2, 0.1, 0.6, −0.2, −0.1, −0.6) (0. 22 B B1 • 2 • B2 B 12 2 (0 b1 .3, 0.1 d1 , 0. b2 d (0 6, − 2 (0 .2, .3, 0 0.3 .6) 0.1 .2, 0 ,− −0 0.1 ,0 .8, − 1, . , .6, − 0 0.3, 0.6 ,− −0 B11 • B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8) −0. ) 0.5 2, − .2, ,− 0.8) 6 −0 . ,0 .1, 0.1 −0 5, .6) (0. 3 d b1

B12 • B22 (0.3, 0.2, 0.5, −0.3, −0.2, −0.5)

b4 d3 (0 ) 6 .5, . 0 0.2 − b 3 d2 , 1 ,0 ( . 0.3, 0 .6, 0.2, − , 0 −0 .4, − B12 • B22 (0.3, 0.2, 0.6, −0.3, −0.2, −0.6) 2 . 0 .5, 0.3, − −0. , −0 6 2 . ) , −0 .2, .4 ,0 .4) 0 1 −0 . 0 − , , .6) 1 .2 . 0 0 ( − 1 d , B • B (0.3, 2 4 0.2, 12 0.6, 22 −0.3, −0.2, −0.6) b 0. B1 − 1 • 4, B2 0. 1 (0 , .2, .1 0.1 ,0 , 0. .2 6, − (0 0.2 1 2 ,− 0.1 .6) ) •B ,− 0.5 b3 d 1 − −0 , 0.6 .2 0 ) B1 − .2, 1 (0 , 4 0 . 0 .2, − B • B (0.3, 0.2, 0.6, −0.3, −0.2, −0.6) − 0.1 0.5, 12 22 .3, 0.2, , 0. −0 (0.4, 4, − 6, . 3d 3 b 0 0.2 , ,− 0.2 0.1 .3, ,− (0 0.4 2 d ) b4

Fig. 10.9 Lexicographic product Gˇ b1 • Gˇ b2

Case 1:

For b ∈ X 1 , d1 d2 ∈ X 2k , p

p

p

T(B1k •B2k ) ((bd1 )(bd2 )) = TB1 (b) ∧ TB2k (d1 d2 ) p

p

p

≤ TB1 (b) ∧ [TB2 (d1 ) ∧ TB2 (d2 )] p

p

p

p

= [TB1 (b) ∧ TB2 (d1 )] ∧ [TB1 (b) ∧ TB2 (d2 )] p

p

= T(B1 •B2 ) (bd1 ) ∧ T(B1 •B2 ) (bd2 ), n T(B ((bd1 )(bd2 )) = TBn1 (b) ∨ TBn2k (d1 d2 ) 1k •B2k )

≥ TBn1 (b) ∨ [TBn2 (d1 ) ∨ TBn2 (d2 )] = [TBn1 (b) ∨ TBn2 (d1 )] ∨ [TBn1 (b) ∨ TBn2 (d2 )] n n = T(B (bd1 ) ∨ T(B (bd2 ), 1 •B2 ) 1 •B2 )

404

10 Bipolar Neutrosophic Graph Structures

p

p

p

I(B1k •B2k ) ((bd1 )(bd2 )) = I B1 (b) ∧ I B2k (d1 d2 ) p

p

p

≤ I B1 (b) ∧ [I B2 (d1 ) ∧ I B2 (d2 )] p

p

p

p

= [I B1 (b) ∧ I B2 (d1 )] ∧ [I B1 (b) ∧ I B2 (d2 )] p

p

= I(B1 •B2 ) (bd1 ) ∧ I(B1 •B2 ) (bd2 ), n ((bd1 )(bd2 )) = I Bn1 (b) ∨ I Bn2k (d1 d2 ) I(B 1k •B2k )

≥ I Bn1 (b) ∨ [I Bn2 (d1 ) ∨ I Bn2 (d2 )] = [I Bn1 (b) ∨ I Bn2 (d1 )] ∨ [I Bn1 (b) ∨ I Bn2 (d2 )] n n = I(B (bd1 ) ∨ I(B (bd2 ), 1 •B2 ) 1 •B2 ) p

p

p

F(B1k •B2k ) ((bd1 )(bd2 )) = FB1 (b) ∨ FB2k (d1 d2 ) p

p

p

≤ FB1 (b) ∨ [FB2 (d1 ) ∨ FB2 (d2 )] p

p

p

p

= [FB1 (b) ∨ FB2 (d1 )] ∨ [FB1 (b) ∨ FB2 (d2 )] p

p

= F(B1 •B2 ) (bd1 ) ∨ F(B1 •B2 ) (bd2 ), n F(B ((bd1 )(bd2 )) = FBn1 (b) ∧ FBn2k (d1 d2 ) 1k •B2k )

≥ FBn1 (b) ∧ [FBn2 (d1 ) ∧ FBn2 (d2 )] = [FBn1 (b) ∧ FBn2 (d1 )] ∧ [FBn1 (b) ∧ FBn2 (d2 )] n n = F(B (bd1 ) ∧ F(B (bd2 ), for bd1 , bd2 ∈ X 1 • X 2 . 1 •B2 ) 1 •B2 )

Case 2:

For b1 b2 ∈ X 1k , d1 d2 ∈ X 2k p

p

p

T(B1k •B2k ) ((b1 d1 )(b2 d2 )) = TB1k (b1 b2 ) ∧ TB2k (d1 d2 ) p

p

p

p

p

p

≤ [TB1 (b1 ) ∧ TB1 (b2 ] ∧ [TB2 (d1 ) ∧ TB2 (d2 )] p

p

= [TB1 (b1 ) ∧ TB2 (d1 )] ∧ [TB1 (b2 ) ∧ TB2 (d2 )] p

p

= T(B1 •B2 ) (b1 d1 ) ∧ T(B1 •B2 ) (b2 d2 ), n T(B ((b1 d1 )(b2 d2 )) = TBn1k (b1 b2 ) ∨ TBn2k (d1 d2 ) 1k •B2k )

≥ [TBn1 (b1 ) ∨ TBn1 (b2 ] ∨ [TBn2 (d1 ) ∨ TBn2 (d2 )] = [TBn1 (b1 ) ∨ TBn2 (d1 )] ∨ [TBn1 (b2 ) ∨ TBn2 (d2 )] n n = T(B (b1 d1 ) ∨ T(B (b2 d2 ), 1 •B2 ) 1 •B2 )

10.2 Operations on Bipolar Neutrosophic Graph Structures p

p

405

p

I(B1k •B2k ) ((b1 d1 )(b2 d2 )) = I B1k (b1 b2 ) ∧ I B2k (d1 d2 ) p

p

p

p

p

p

≤ [I B1 (b1 ) ∧ I B1 (b2 ] ∧ [I B2 (d1 ) ∧ I B2 (d2 )] p

p

= [I B1 (b1 ) ∧ I B2 (d1 )] ∧ [I B1 (b2 ) ∧ I B2 (d2 )] p

p

= I(B1 •B2 ) (b1 d1 ) ∧ I(B1 •B2 ) (b2 d2 ), n I(B ((b1 d1 )(b2 d2 )) = I Bn1k (b1 b2 ) ∨ I Bn2k (d1 d2 ) 1k •B2k )

≥ [I Bn1 (b1 ) ∨ I Bn1 (b2 ] ∨ [I Bn2 (d1 ) ∨ I Bn2 (d2 )] = [I Bn1 (b1 ) ∨ I Bn2 (d1 )] ∨ [I Bn1 (b2 ) ∨ I Bn2 (d2 )] n n = I(B (b1 d1 ) ∨ I(B (b2 d2 ), 1 •B2 ) 1 •B2 )

p

p

p

F(B1k •B2k ) ((b1 d1 )(b2 d2 )) = FB1k (b1 b2 ) ∨ FB2k (d1 d2 ) p

p

p

p

p

p

≤ [FB1 (b1 ) ∨ FB1 (b2 ] ∨ [FB2 (d1 ) ∨ FB2 (d2 )] p

p

= [FB1 (b1 ) ∨ FB2 (d1 )] ∨ [FB1 (b2 ) ∨ FB2 (d2 )] p

p

= F(B1 •B2 ) (b1 d1 ) ∨ F(B1 •B2 ) (b2 d2 ), n F(B ((b1 d1 )(b2 d2 )) = FBn1k (b1 b2 ) ∧ FBn2k (d1 d2 ) 1k •B2k )

≥ [FBn1 (b1 ) ∧ FBn1 (b2 ] ∧ [FBn2 (d1 ) ∧ FBn2 (d2 )] = [FBn1 (b1 ) ∧ FBn2 (d1 )] ∧ [FBn1 (b2 ) ∧ FBn2 (d2 )] n n = F(B (b1 d1 ) ∧ F(B (b2 d2 ), 1 •B2 ) 1 •B2 )

for b1 d1 , b2 d2 ∈ X 1 • X 2 and k ∈ {1, 2, . . . , m}. It completes the proof.



Definition 10.10 Let Gˇ b1 = (B1 , B11 , B12 , . . . , B1m ) and Gˇ b2 = (B2 , B21 , B22 , . . . , B2m ) be two bipolar single-valued neutrosophic graph structures. The strong product of Gˇ b1 and Gˇ b2 is denoted by Gˇ b1  Gˇ b2 = (B1  B2 , B11  B21 , B12  B22 , . . . , B1m  B2m ) and defined as ⎧ p p p p p ⎨ T(B1 B2 ) (bd) = (TB1  TB2 )(bd) = TB1 (b) ∧ TB2 (d) p p p p p I (bd) = (I B1  I B2 )(bd) = I B1 (b) ∧ I B2 (d) 1. ⎩ (Bp1 B2 ) p p p p F 1 B2 ) (bd) = (FB1  FB2 )(bd) = FB1 (b) ∨ FB2 (d) ⎧ (B n n n n n ⎨ T(B1 B2 ) (bd) = (TB1  TB2 )(bd) = TB1 (b) ∨ TB2 (d) n n n n n (bd) = (I B1  I B2 )(bd) = I B1 (b) ∨ I B2 (d) I 2. ⎩ (Bn1 B2 ) p F(B1 B2 ) (bd) = (FBn1  FBn2 )(bd) = FBn1 (b) ∧ FB2 (d) for all (bd) ∈ X 1 × X 2 .

406

3.

4.

5.

6.

10 Bipolar Neutrosophic Graph Structures

⎧ p p p p p ⎨ T(B1k B2k ) (bd1 )(bd2 ) = (TB1k  TB2k )(bd1 )(bd2 ) = TB1 (b) ∧ TB2k (d1 d2 ) p p p p p (bd1 )(bd2 ) = (I B1k  I B2k )(bd1 )(bd2 ) = I B1 (b) ∧ I B2k (d1 d2 ) I ⎩ (Bp1k B2k ) p p p p F(B1k B2k ) (bd1 )(bd2 ) = (FB1k  FB2k )(bd1 )(bd2 ) = FB1 (b) ∨ FB2k (d1 d2 ) ⎧ n n n n n ⎨ T(B1k B2k ) (bd1 )(bd2 ) = (TB1k  TB2k )(bd1 )(bd2 ) = TB1 (b) ∨ TB2k (d1 d2 ) n n n n n (bd1 )(bd2 ) = (I B1k  I B2k )(bd1 )(bd2 ) = I B1 (b) ∨ I B2k (d1 d2 ) I ⎩ (Bn1k B2k ) F(B1k B2k ) (bd1 )(bd2 ) = (FBn1k  FBn2k )(bd1 )(bd2 ) = FBn1 (b) ∧ FBn2k (d1 d2 ) for all b ∈ X 1 , (d1 d2 ) ∈ X 2k . ⎧ p p p p p ⎨ T(B1k B2k ) (b1 d)(b2 d) = (TB1k  TB2k )(b1 d)(b2 d) = TB2 (d) ∧ TB1k (b1 b2 ) p p p p p (b1 d)(b2 d) = (I B1k  I B2k )(b1 d)(b2 d) = I B2 (d) ∧ I B2k (b1 b2 ) I ⎩ (Bp1k B2k ) p p p p F 1k B2k ) (b1 d)(b2 d) = (FB1k  FB2k )(b1 d)(b2 d) = FB2 (d) ∨ FB2k (b1 b2 ) ⎧ (B n n n n n ⎨ T(B1k B2k ) (b1 d)(b2 d) = (TB1k  TB2k )(b1 d)(b2 d) = TB2 (d) ∨ TB1k (b1 b2 ) n n n n n (b1 d)(b2 d) = (I B1k  I B2k )(b1 d)(b2 d) = I B2 (d) ∨ I B2k (b1 b2 ) I ⎩ (Bn1k B2k ) F(B1k B2k ) (b1 d)(b2 d) = (FBn1k  FBn2k )(b1 d)(b2 d) = FBn2 (d) ∧ FBn2k (b1 b2 ) for all d ∈ X 2 , (b1 b2 ) ∈ X 1k . ⎧ p

p

p

p

p

⎪ ⎨ T(B1k B2k ) (b1 d1 )(b2 d2 ) = (TB1k  TB2k )(b1 d1 )(b2 d2 ) = TB1k (b1 b2 ) ∧ TB2k (d1 d2 ) p p p p p I(B B ) (b1 d1 )(b2 d2 ) = (I B  I B )(b1 d1 )(b2 d2 ) = I B (b1 b2 ) ∧ I B (d1 d2 ) 7. 1k 2k 1k 2k ⎪ ⎩ F p1k 2k (b d )(b d ) = (F p  F p )(b d )(b d ) = F p (b b ) ∨ F p (d d ) 1 1 2 2 1 1 2 2 1 2 B B B B2k 1 2 B ) (B 1k 2k 1k 1k 2k ⎧ n n  T n )(b d )(b d ) = T n (b b ) ∨ T n (d d ) T (b d )(b d ) = (T ⎪ 1 1 2 2 1 1 2 2 1 2 B1k B2k B1k B2k 1 2 ⎨ (B1k B2k ) n n  I n )(b d )(b d ) = I n (b b ) ∨ I n (d d ) (b d )(b d ) = (I I(B 8. 1 1 2 2 1 1 2 2 1 2 B1k B2k B1k B2k 1 2 B ) ⎪ ⎩ F n 1k 2k (b d )(b d ) = (F n  F n )(b d )(b d ) = F n (b b ) ∧ F n (d d ) 2 2 1 1 2 2 1 2 1 2 B B B B (B B ) 1 1 1k

2k

1k

for all (b1 b2 ) ∈ X 1k , (d1 d2 ) ∈ X 2k .

2k

1k

2k

Example 10.9 The strong product of bipolar single-valued neutrosophic graph structures Gˇ b1 and Gˇ b2 shown in Fig. 10.8 is defined as Gˇ b1  Gˇ b2 = {B1  B2 , B11  B21 , B12  B22 } and is computed in Figs. 10.10 and 10.11. Theorem 10.2 The strong product Gˇ b1  Gˇ b2 = (B1  B2 , B11  B21 , B12  B22 , . . . , B1m  B2m ) of two bipolar single-valued neutrosophic graph structures of structures Gˇs1 and Gˇs2 is a bipolar single-valued neutrosophic graph structure of Gˇs1  Gˇs2 . Proof There are three cases regarding vertices and edges in the strong product. Case 1:

For b ∈ X 1 , d1 d2 ∈ X 2k , p

p

p

T(B1k B2k ) ((bd1 )(bd2 )) = TB1 (b) ∧ TB2k (d1 d2 ) p

p

p

≤ TB1 (b) ∧ [TB2 (d1 ) ∧ TB2 (d2 )] p

p

p

p

= [TB1 (b) ∧ TB2 (d1 )] ∧ [TB1 (b) ∧ TB2 (d2 )] p

p

= T(B1 B2 ) (bd1 ) ∧ T(B1 B2 ) (bd2 ),

10.2 Operations on Bipolar Neutrosophic Graph Structures

d1

(0 .2 ,0 .1 ,0 B 11 .8 ,− 0. 2, − 0. 1, − 0. 8)

B2

B21 (0.2, 0.1, 0.6, −0.2, −0.1, −0.6)

b2 d3 (0.5, 0.2, 0.8, −0.5, −0.2, −0.8) 1 (0 .5,

0.1 ,

6) 0. ,− 1 0. −0 − −0 .8) .1, 3, −0 0. .8) − , .8) .6 −0 ,0 .1, .1 −0 0 , , 2 . .3 −0 (0 .8, 2 1, 0 B2 , 0. 2 . (0 2 B 21 B1

1, 0 .

8, − 0.2 ,

0.8 ,

−0 .5,

−0 .1,

b1 d3 (0.5, 0.1, 0.6, −0.5, −0.1, −0.6)

B 11

b2

B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8)

b

d1 1

B11

6) 0. − 1, 0. − 3, 0. ,− .6 , 0B12 B22 (0.5, 0.1, 0.8, −0.5, −0.1, −0.8) .1 ,0 .3 (0 B11 B21 (0.3, 0.1, 0.8, −0.3, −0.1, −0.8) d2 b1

6) 0. − 1, . 0 − 2, 0. B1 − 1 6, B1 0. 1 1, . B2 0 , 1 (0 .2 .2, 0. (0

407

B11

B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8)

d b2

8) 0. − 2, . 0 − 3, 0. − 8, 0. 2, 0. , .3 (0 2

Fig. 10.10 Strong product Gˇ b1  Gˇ b2 (0 .5 ,0 .2 ,0 .6 ,− 0. b3 d1 (0.2, 0.1, 0.4, −0.2, −0.1, −0.4) ) 5, .6 0 − ,− 0. 0.2 2, − ) , 4 .3 − . 0 0 0. ,− 6 − . 6) 0 , , 2 1 . 0 , 0. 3 . 0 − ( , 2 22 . B 0 − B 12 4, 0. B1 1, . 2 ,0 B2 .2 2 (0 (0 B1 .3, 1 2 0.2 B2 B2 , 0. 2 (0 6, − .2, 0 0.3 .1, 0 ,− .6, − 0.2 0.2, ,− 0.6 −0. ) 1 , −0. b4 d1 (0.2, 0.1, 0.6, −0.2, −0.1, −0.6) 6)

B12

− 0. 2, − 0. 5)

1

(0 .4 ,0 .2 , 0 B 12 .5 ,− 0. 4,

B1

B22 (0.4, 0.2, 0.6, −0.4, −0.2, −0.6)

B12 × B22 (0.3, 0.2, 0.6, −0.3, −0.2, −0.6)

B22 (0.3, 0.2, 0.5, −0.3, −0.2, −0.5)

b3

(0 .3 ,0 .2 ,0 .4 ,− 0. 3, − 0. 2, − 0. 4)

d3

d3

B22 (0.3, 0.2, 0.6, −0.3, −0.2, −0.6)

B12

d2

B21 (0.2, 0.1, 0.6, −0.2, −0.1, −0.6)

b3

, 0.2

B11

d b4

.3, (0 2

b4

.6) −0 .2, 0 ,− 0.3 ,− 6 . 0

Fig. 10.11 Strong product Gˇ b1  Gˇ b2

n T(B ((bd1 )(bd2 )) = TBn1 (b) ∨ TBn2k (d1 d2 ) 1k B2k )

≥ TBn1 (b) ∨ [TBn2 (d1 ) ∨ TBn2 (d2 )] = [TBn1 (b) ∨ TBn2 (d1 )] ∨ [TBn1 (b) ∨ TBn2 (d2 )] n n = T(B (bd1 ) ∨ T(B (bd2 ), 1 B2 ) 1 B2 ) p

p

p

I(B1k B2k ) ((bd1 )(bd2 )) = I B1 (b) ∧ I B2k (d1 d2 ) p

p

p

≤ I B1 (b) ∧ [I B2 (d1 ) ∧ I B2 (d2 )]

408

10 Bipolar Neutrosophic Graph Structures p

p

p

p

= [I B1 (b) ∧ I B2 (d1 )] ∧ [I B1 (b) ∧ I B2 (d2 )] p

p

= I(B1 B2 ) (bd1 ) ∧ I(B1 B2 ) (bd2 ), n I(B ((bd1 )(bd2 )) = I Bn1 (b) ∨ I Bn2k (d1 d2 ) 1k B2k )

≥ I Bn1 (b) ∨ [I Bn2 (d1 ) ∨ I Bn2 (d2 )] = [I Bn1 (b) ∨ I Bn2 (d1 )] ∨ [I Bn1 (b) ∨ I Bn2 (d2 )] n n = I(B (bd1 ) ∨ I(B (bd2 ), 1 B2 ) 1 B2 ) p

p

p

F(B1k B2k ) ((bd1 )(bd2 )) = FB1 (b) ∨ FB2k (d1 d2 ) p

p

p

≤ FB1 (b) ∨ [FB2 (d1 ) ∨ FB2 (d2 )] p

p

p

p

= [FB1 (b) ∨ FB2 (d1 )] ∨ [FB1 (b) ∨ FB2 (d2 )] p

p

= F(B1 B2 ) (bd1 ) ∨ F(B1 B2 ) (bd2 ), n ((bd1 )(bd2 )) = FBn1 (b) ∧ FBn2k (d1 d2 ) F(B 1k B2k )

≥ FBn1 (b) ∧ [FBn2 (d1 ) ∧ FBn2 (d2 )] = [FBn1 (b) ∧ FBn2 (d1 )] ∧ [FBn1 (b) ∧ FBn2 (d2 )] n n = F(B (bd1 ) ∧ F(B (bd2 ), 1 B2 ) 1 B2 )

for bd1 , bd2 ∈ X 1 × X 2 . Case 2: For b ∈ X 2 , d1 d2 ∈ X 1k , p

p

p

T(B1k B2k ) ((d1 b)(d2 b)) = TB2 (b) ∧ TB1k (d1 d2 ) p

p

p

≤ TB2 (b) ∧ [TB1 (d1 ) ∧ TB1 (d2 )] p

p

p

p

= [TB2 (b) ∧ TB1 (d1 )] ∧ [TB2 (b) ∧ TB1 (d2 )] p

p

= T(B1 B2 ) (d1 b) ∧ T(B1 B2 ) (d2 b),

n T(B ((d1 b)(d2 b)) = TBn2 (b) ∨ TBn1k (d1 d2 ) 1k B2k ) p

≥ TBn2 (b) ∨ [TBn1 (d1 ) ∨ TB1 (d2 )] = [TBn2 (b) ∨ TBn1 (d1 )] ∨ [TBn2 (b) ∨ TBn1 (d2 )] n n = T(B (d1 b) ∨ T(B (d2 b), 1 B2 ) 1 B2 )

10.2 Operations on Bipolar Neutrosophic Graph Structures

p

p

409

p

I(B1k B2k ) ((d1 b)(d2 b)) = I B2 (b) ∧ I B1k (d1 d2 ) p

p

p

≤ I B2 (b) ∧ [I B1 (d1 ) ∧ I B1 (d2 )] p

p

p

p

= [I B2 (b) ∧ I B1 (d1 )] ∧ [I B2 (b) ∧ I B1 (d2 )] p

p

= I(B1 B2 ) (d1 b) ∧ I(B1 B2 ) (d2 b), n ((d1 b)(d2 b)) = I Bn2 (b) ∨ I Bn1k (d1 d2 ) I(B 1k B2k )

≥ I Bn2 (b) ∨ [I Bn1 (d1 ) ∨ I Bn1 (d2 )] = [I Bn2 (b) ∨ I Bn1 (d1 )] ∨ [I Bn2 (b) ∨ I Bn1 (d2 )] n n = I(B (d1 b) ∨ I(B (d2 b), 1 B2 ) 1 B2 ) p

p

p

F(B1k B2k ) ((d1 b)(d2 b)) = FB2 (b) ∨ FB1k (d1 d2 ) p

p

p

≤ FB2 (b) ∨ [FB1 (d1 ) ∨ FB1 (d2 )] p

p

p

p

= [FB2 (b) ∨ FB1 (d1 )] ∨ [FB2 (b) ∨ FB1 (d2 )] p

p

= F(B1 B2 ) (d1 b) ∨ F(B1 B2 ) (d2 b), n ((d1 b)(d2 b)) = FBn2 (b) ∧ FBn1k (d1 d2 ) F(B 1k B2k )

≥ FBn2 (b) ∧ [FBn1 (d1 ) ∧ FBn1 (d2 )] = [FBn2 (b) ∧ FBn1 (d1 )] ∧ [FBn2 (b) ∧ FBn1 (d2 )] n n = F(B (d1 b) ∧ F(B (d2 b), 1 B2 ) 1 B2 )

for d1 b, d2 b ∈ X 1  X 2 . Case 3: For b1 b2 ∈ X 1k , d1 d2 ∈ X 2k , p

p

p

T(B1k B2k ) ((b1 d1 )(b2 d2 )) = TB1k (b1 b2 ) ∧ TB2k (d1 d2 ) p

p

p

p

p

p

≤ [TB1 (b1 ) ∧ TB1 (b2 ] ∧ [TB2 (d1 ) ∧ TB2 (d2 )] p

p

= [TB1 (b1 ) ∧ TB2 (d1 )] ∧ [TB1 (b2 ) ∧ TB2 (d2 )] p

p

= T(B1 B2 ) (b1 d1 ) ∧ T(B1 B2 ) (b2 d2 ), n ((b1 d1 )(b2 d2 )) = TBn1k (b1 b2 ) ∨ TBn2k (d1 d2 ) T(B 1k B2k )

≥ [TBn1 (b1 ) ∨ TBn1 (b2 ] ∨ [TBn2 (d1 ) ∨ TBn2 (d2 )] = [TBn1 (b1 ) ∨ TBn2 (d1 )] ∨ [TBn1 (b2 ) ∨ TBn2 (d2 )] n n = T(B (b1 d1 ) ∨ T(B (b2 d2 ), 1 B2 ) 1 B2 )

410

10 Bipolar Neutrosophic Graph Structures p

p

p

I(B1k B2k ) ((b1 d1 )(b2 d2 )) = I B1k (b1 b2 ) ∧ I B2k (d1 d2 ) p

p

p

p

p

p

≤ [I B1 (b1 ) ∧ I B1 (b2 ] ∧ [I B2 (d1 ) ∧ I B2 (d2 )] p

p

= [I B1 (b1 ) ∧ I B2 (d1 )] ∧ [I B1 (b2 ) ∧ I B2 (d2 )] p

p

= I(B1 B2 ) (b1 d1 ) ∧ I(B1 B2 ) (b2 d2 ), n ((b1 d1 )(b2 d2 )) = I Bn1k (b1 b2 ) ∨ I Bn2k (d1 d2 ) I(B 1k B2k )

≥ [I Bn1 (b1 ) ∨ I Bn1 (b2 ] ∨ [I Bn2 (d1 ) ∨ I Bn2 (d2 )] = [I Bn1 (b1 ) ∨ I Bn2 (d1 )] ∨ [I Bn1 (b2 ) ∨ I Bn2 (d2 )] n n = I(B (b1 d1 ) ∨ I(B (b2 d2 ), 1 B2 ) 1 B2 )

p

p

p

F(B1k B2k ) ((b1 d1 )(b2 d2 )) = FB1k (b1 b2 ) ∨ FB2k (d1 d2 ) p

p

p

p

p

p

≤ [FB1 (b1 ) ∨ FB1 (b2 ] ∨ [FB2 (d1 ) ∨ FB2 (d2 )] p

p

= [FB1 (b1 ) ∨ FB2 (d1 )] ∨ [FB1 (b2 ) ∨ FB2 (d2 )] p

p

= F(B1 B2 ) (b1 d1 ) ∨ F(B1 B2 ) (b2 d2 ), n ((b1 d1 )(b2 d2 )) = FBn1k (b1 b2 ) ∧ FBn2k (d1 d2 ) F(B 1k B2k )

≥ [FBn1 (b1 ) ∧ FBn1 (b2 ] ∧ [FBn2 (d1 ) ∧ FBn2 (d2 )] = [FBn1 (b1 ) ∧ FBn2 (d1 )] ∧ [FBn1 (b2 ) ∧ FBn2 (d2 )] n n = F(B (b1 d1 ) ∧ F(B (b2 d2 ), 1 B2 ) 1 B2 )

for b1 d1 , b2 d2 ∈ X 1 × X 2 . All cases hold for all k ∈ {1, 2, . . . , m}.



Definition 10.11 Let Gˇ b1 = (B1 , B11 , B12 , . . . , B1m ) and Gˇ b2 = (B2 , B21 , B22 , . . . , B2m ) be bipolar single-valued neutrosophic graph structures. The union of Gˇ b1 and Gˇ b2 is denoted by Gˇ b1 ∪ Gˇ b2 = (B1 ∪ B2 , B11 ∪ B21 , B12 ∪ B22 , . . . , B1m ∪ B2m ) and is defined as ⎧ p p p p p ⎨ T(B1 ∪B2 ) (b) = (TB1 ∪ TB2 )(b) = TB1 (b) ∨ TB2 (b), p p p p p I (b) = (I B1 ∪ I B2 )(b) = (I B1 (b) + I B2 (b))/2, 1. ⎩ (Bp1 ∪B2 ) p p p p F(B1 ∪B2 ) (b) = (FB1 ∪ FB2 )(b) = FB1 (b) ∧ FB2 (b), ⎧ n n n n n ⎨ T(B1 ∪B2 ) (b) = (TB1 ∪ TB2 )(b) = TB1 (b) ∧ TB2 (b), In (b) = (I Bn1 ∪ I Bn2 )(b) = (I Bn1 (b) + I Bn2 (b))/2, 2. ⎩ (Bn1 ∪B2 ) F(B1 ∪B2 ) (b) = (FBn1 ∪ FBn2 )(b) = FBn1 (b) ∨ FBn2 (b) for all b ∈ X 1 ∪ X 2 ,

10.2 Operations on Bipolar Neutrosophic Graph Structures b2 (0.5, 0.1, 0.8, −0.5, −0.1, −0.8)

B12 ∪ B22 (0.3, 0.1, 0.5, −0.3, −0.1, −0.5) B11 ∪ B21 (0.5, 0.05, 0.8, −0.5, −0.05, −0.8)

b1 (0.5, 0.05, 0.6,

0.5,

0.05,

0.6)

d2 (0.3, 0.1, 0.4,

b4 (0.5, 0.1, 0.6, −0.5, −0.1, −0.6)

d3 (0.5, 0.15, 0.5, −0.5, −0.15, −0.5)

0.3,

411

d1 (0.2, 0.05, 0.3, −0.2, −0.05, −0.3)

B11 ∪ B21 (0.2, 0.05, 0.4, −0.2, −0.05, −0.4) B12 ∪ B22 (0.4, 0.1, 0.6, −0.4, −0.1, −0.6)

0.1,

0.4)

b3 (0.4, 0.1, 0.4,

0.4,

0.1,

0.4)

Fig. 10.12 Union Gˇ b1 ∪ Gˇ b2

⎧ p p p p p ⎨ T(B1k ∪B2k ) (bd) = (TB1k ∪ TB2k )(bd) = TB1k (bd) ∨ TB2k (bd), p p p p p I (bd) = (I B1k ∪ I B2k )(bd) = (I B1k (bd) + I B2k (bd))/2, 3. ⎩ (Bp1k ∪B2k ) p p p p F(B1k ∪B2k ) (bd) = (FB1k ∪ FB2k )(bd) = FB1k (bd) ∧ FB2k (bd), ⎧ n n n n n ⎨ T(B1k ∪B2k ) (bd) = (TB1k ∪ TB2k )(bd) = TB1k (bd) ∧ TB2k (bd), In (bd) = (I Bn1k ∪ I Bn2k )(bd) = (I Bn1k (bd) + I Bn2k (bd))/2, 4. ⎩ (Bn1k ∪B2k ) F(B1k ∪B2k ) (bd) = (FBn1k ∪ FBn2k )(bd) = FBn1k (bd) ∨ FBn2k (bd) for all bd ∈ X 1k ∪ X 2k . Example 10.10 The union of two bipolar single-valued neutrosophic graph structures Gˇ b1 and Gˇ b2 shown in Fig. 10.8 is defined as Gˇ b1 ∪ Gˇ b2 = {B1 ∪ B2 , B11 ∪ B21 , B12 ∪ B22 } and is given in Fig. 10.12. Theorem 10.3 The union Gˇ b1 ∪ Gˇ b2 = (B1 ∪ B2 , B11 ∪ B21 , B12 ∪ B22 , . . . , B1m ∪ B2m ) of two bipolar single-valued neutrosophic graph structures of the graph structures Gˇ 1 and Gˇ 2 is a bipolar single-valued neutrosophic graph structure of Gˇ 1 ∪ Gˇ 2 . Proof Let b1 b2 ∈ X 1k ∪ X 2k , then there are the following two cases. For b1 , b2 ∈ X 1 , by Definition 10.11 p p p p p p p TB2 (b1 ) = TB2 (b2 ) = TB2k (b1 b2 ) = 0, I B2 (b1 ) = I B2 (b2 ) = I B2k (b1 b2 ) = 0, FB2 (b1 ) = p p FB2 (b2 ) = FB2k (b1 b2 ) = 1, TBn2 (b1 ) = TBn2 (b2 ) = TBn2k (b1 b2 ) = 0, I Bn2 (b1 ) = I Bn2 (b2 ) = I Bn2k (b1 b2 ) = 0, FBn2 (b1 ) = FBn2 (b2 ) = FBn2k (b1 b2 ) = −1, so

Case 1:

p

p

p

T(B1k ∪B2k ) (b1 b2 ) = TB1k (b1 b2 ) ∨ TB2k (b1 b2 ) p

= TB1k (b1 b2 ) ∨ 0 p

p

≤ [TB1 (b1 ) ∧ TB1 (b2 )] ∨ 0 p

p

= [TB1 (b1 ) ∨ 0] ∧ [TB1 (b2 ) ∨ 0] p

p

p

p

= [TB1 (b1 ) ∨ TB2 (b1 )] ∧ [TB1 (b2 ) ∨ TB2 (b2 )] p

p

= T(B1 ∪B2 ) (b1 ) ∧ T(B1 ∪B2 ) (b2 ),

412

10 Bipolar Neutrosophic Graph Structures n T(B (b1 b2 ) = TBn1k (b1 b2 ) ∧ TBn2k (b1 b2 ) 1k ∪B2k )

= TBn1k (b1 b2 ) ∧ 0 ≥ [TBn1 (b1 ) ∨ TBn1 (b2 )] ∧ 0 = [TBn1 (b1 ) ∧ 0] ∨ [TBn1 (b2 ) ∧ 0] = [TBn1 (b1 ) ∧ TBn2 (b1 )] ∨ [TBn1 (b2 ) ∧ TBn2 (b2 )] n n = T(B (b1 ) ∨ T(B (b2 ), 1 ∪B2 ) 1 ∪B2 )

p

p

p

F(B1k ∪B2k ) (b1 b2 ) = FB1k (b1 b2 ) ∧ FB2k (b1 b2 ) p

= FB1k (b1 b2 ) ∧ 1 p

p

≤ [FB1 (b1 ) ∨ FB1 (b2 )] ∧ 1 p

p

= [FB1 (b1 ) ∧ 1] ∨ [FB1 (b2 ) ∧ 1] p

p

p

p

= [FB1 (b1 ) ∧ FB2 (b1 )] ∨ [FB1 (b2 ) ∧ FB2 (b2 )] p

p

= F(B1 ∪B2 ) (b1 ) ∨ F(B1 ∪B2 ) (b2 ),

n F(B (b1 b2 ) = FBn1k (b1 b2 ) ∨ FBn2k (b1 b2 ) 1k ∪B2k )

= FBn1k (b1 b2 ) ∨ −1 ≥ [FBn1 (b1 ) ∧ FBn1 (b2 )] ∨ −1 = [FBn1 (b1 ) ∨ −1] ∧ [FBn1 (b2 ) ∨ −1] = [FBn1 (b1 ) ∨ FBn2 (b1 )] ∧ [FBn1 (b2 ) ∨ FBn2 (b2 )] n n = F(B (b1 ) ∧ F(B (b2 ), 1 ∪B2 ) 1 ∪B2 )

p

p

I(B1k ∪B2k ) (b1 b2 ) = = ≤

p

I B1k (b1 b2 ) + I B2k (b1 b2 ) 2 p I B1k (b1 b2 ) + 0 2 p p [I B1 (b1 ) ∧ I B1 (b2 )] + 0 2 p

=[ = =

I B1 (b1 ) 2

p [I B1 (b1 )

p

+ 0] ∧ [ +

I B1 (b2 )

p I B2 (b1 )]

2 p I(B1 ∪B2 ) (b1 )

∧

2 ∧

+ 0]

p [I B1 (b2 )

p I(B1 ∪B2 ) (b2 ),

p

+ I B2 (b2 )] 2

10.2 Operations on Bipolar Neutrosophic Graph Structures n I(B (b1 b2 ) = 1k ∪B2k )

= ≥

413

I Bn1k (b1 b2 ) + I Bn2k (b1 b2 ) 2 I Bn1k (b1 b2 ) + 0 2 [I Bn1 (b1 ) ∨ I Bn1 (b2 )] + 0 I Bn1 (b1 )

2 + 0] ∨ [

I Bn1 (b2 )

+ 0] 2 2 [I Bn1 (b1 ) + I Bn2 (b1 )] [I Bn1 (b2 ) + I Bn2 (b2 )] = ∨ 2 2 n n = I(B (b ) ∨ I (b ), 1 (B1 ∪B2 ) 2 1 ∪B2 )

=[

for b1 , b2 ∈ X 1 ∪ X 2 . Case 2: For b1 , b2 ∈ X 2 , by Definition 10.11, p

p

p

p

p

p

p

TB1 (b1 ) = TB1 (b2 ) = TB1k (b1 b2 ) = 0, I B1 (b1 ) = I B1 (b2 ) = I B1k (b1 b2 ) = 0, FB1 (b1 ) = p p FB2 (b2 ) = FB1k (b1 b2 ) = 1, n n TB1 (b1 ) = TB1 (b2 ) = TBn1k (b1 b2 ) = 0, I Bn1 (b1 ) = I Bn1 (b2 ) = I Bn1k (b1 b2 ) = 0, FBn1 (b1 ) = FBn2 (b2 ) = FBn1k (b1 b2 ) = −1, so p

p

p

T(B1k ∪B2k ) (b1 b2 ) = TB1k (b1 b2 ) ∨ TB2k (b1 b2 ) p

= TB2k (b1 b2 ) ∨ 0 p

p

≤ [TB2 (b1 ) ∧ TB2 (b2 )] ∨ 0 p

p

= [TB2 (b1 ) ∨ 0] ∧ [TB2 (b2 ) ∨ 0] p

p

p

p

= [TB2 (b1 ) ∨ TB1 (b1 )] ∧ [TB2 (b2 ) ∨ TB1 (b2 )] p

p

= T(B1 ∪B2 ) (b1 ) ∧ T(B1 ∪B2 ) (b2 ), n T(B (b1 b2 ) = TBn1k (b1 b2 ) ∧ TBn2k (b1 b2 ) 1k ∪B2k )

= TBn2k (b1 b2 ) ∧ 0 ≥ [TBn2 (b1 ) ∨ TBn2 (b2 )] ∧ 0 = [TBn2 (b1 ) ∧ 0] ∨ [TBn2 (b2 ) ∧ 0] = [TBn2 (b1 ) ∧ TBn1 (b1 )] ∨ [TBn2 (b2 ) ∧ TBn1 (b2 )] n n = T(B (b1 ) ∨ T(B (b2 ), 1 ∪B2 ) 1 ∪B2 )

p

p

p

F(B1k ∪B2k ) (b1 b2 ) = FB1k (b1 b2 ) ∧ FB2k (b1 b2 ) p

= FB2k (b1 b2 ) ∧ (1) p

p

≤ [FB2 (b1 ) ∨ FB2 (b2 )] ∧ (1) p

p

= [FB2 (b1 ) ∧ (1)] ∨ [FB2 (b2 ) ∧ (1)]

414

10 Bipolar Neutrosophic Graph Structures p

p

p

p

= [FB2 (b1 ) ∧ FB1 (b1 )] ∨ [FB2 (b2 ) ∧ FB1 (b2 )] p

p

= F(B1 ∪B2 ) (b1 ) ∨ F(B1 ∪B2 ) (b2 ), n F(B (b1 b2 ) = FBn1k (b1 b2 ) ∨ FBn2k (b1 b2 ) 1k ∪B2k )

= FBn2k (b1 b2 ) ∨ (−1) ≥ [FBn2 (b1 ) ∧ FBn2 (b2 )] ∨ (−1) = [FBn2 (b1 ) ∨ (−1)] ∧ [FBn2 (b2 ) ∨ (−1)] = [FBn2 (b1 ) ∨ FBn1 (b1 )] ∧ [FBn2 (b2 ) ∨ FBn1 (b2 )] n n = F(B (b1 ) ∧ F(B (b2 ), 1 ∪B2 ) 1 ∪B2 )

p

p

I(B1k ∪B2k ) (b1 b2 ) = = ≤

2 p I B2k (b1 b2 ) + 0 2 p p [I B2 (b1 ) ∧ I B2 (b2 )] + 0 2

=[ =

p

I B1k (b1 b2 ) + I B2k (b1 b2 )

p I B2 (b1 )

2

p [I B2 (b1 )

p

+ 0] ∧ [ +

I B2 (b2 ) 2

p I B1 (b1 )]

∧

+ 0]

p [I B2 (b2 )

2 p p = I(B1 ∪B2 ) (b1 ) ∧ I(B1 ∪B2 ) (b2 ),

n I(B (b1 b2 ) = 1k ∪B2k )

= ≥

2

I Bn1k (b1 b2 ) + I Bn2k (b1 b2 ) 2 I Bn2k (b1 b2 ) + 0 2 [I Bn2 (b1 ) ∨ I Bn2 (b2 )] + 0

=[ =

p

+ I B1 (b2 )]

I Bn2 (b1 ) 2

[I Bn2 (b1 )

2 + 0] ∨ [ +

I Bn2 (b2 )

I Bn1 (b1 )]

2 ∨

+ 0]

[I Bn2 (b2 )

2 n n = I(B (b 1 ) ∨ I(B1 ∪B2 ) (b2 ), 1 ∪B2 )

+ I Bn1 (b2 )] 2

for b1 , b2 ∈ X 1 ∪ X 2 . Both cases hold for all k ∈ {1, 2, . . . , m}. It completes the proof.

10.2 Operations on Bipolar Neutrosophic Graph Structures

415

Theorem 10.4 Let Gˇ s = (X 1 ∪ X 2 , X 11 ∪ X 21 , X 12 ∪ X 22 , . . . , X 1m ∪ X 2m ) be the union of graph structues Gˇs1 = (X 1 , X 11 , X 12 , . . . , X 1m ) and Gˇs2 = (X 2 , X 21 , X 22 , . . . , X 2m ). Then every bipolar single-valued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , . . . , Bm ) of Gˇ s is the union of two bipolar single-valued neutrosophic graph structures Gˇ b1 and Gˇ b2 of graph structures Gˇs1 and Gˇs2 , respectively. Proof Firstly, define B1 , B2 , B1k , and B2k for k ∈ {1, 2, . . . , m} as p

p

p

p

p

p

p

p

TB1 (b) = TB (b), I B1 (b) = I B (b), FB1 (b) = FB (b), I Bn (b), FBn1 (b) = FBn (b), if b ∈ X 1 . p

p

p

TBn1 (b) = TBn (b), I Bn1 (b) =

p

TB2 (b) = TB (b), I B2 (b) = I B (b), FB2 (b) = FB (b), TBn2 (b) = TBn (b), I Bn2 (b) = I Bn (b), FBn2 (b) = FBn (b), if b ∈ X 2 . p

p

p

p

p

p

TBn1k

p

TBn2k

TB1k (b1 b2 ) = TBk (b1 b2 ), I B1k (b1 b2 ) = I Bk (b1 b2 ), FB1k (b1 b2 ) = FBk (b1 b2 ), (b1 b2 ) = TBnk (b1 b2 ), I Bn1k (b1 b2 ) = I Bnk (b1 b2 ), FBn1k (b1 b2 ) = FBnk (b1 b2 ), if b1 b2 ∈ X 1k . p

p

p

p

p

TB2k (b1 b2 ) = TBk (b1 b2 ), I B2k (b1 b2 ) = I Bk (b1 b2 ), FB2k (b1 b2 ) = FBk (b1 b2 ), (b1 b2 ) = TBnk (b1 b2 ), I Bn2k (b1 b2 ) = I Bnk (b1 b2 ), FBn2k (b1 b2 ) = FBnk (b1 b2 ), if b1 b2 ∈ X 2k .

Then B = B1 ∪ B2 and Bk = B1k ∪ B2k , k ∈ {1, 2, . . . , m}. Now for b1 b2 ∈ X tk , t = 1, 2, k ∈ {1, 2, . . . , m}: p

p

p

p

p

p

TBtk (b1 b2 ) = TBk (b1 b2 ) ≤ TB (b1 ) ∧ TB (b2 ) = TBt (b1 ) ∧ TBt (b2 ), p

p

p

p

p

p

I Btk (b1 b2 ) = I Bk (b1 b2 ) ≤ I B (b1 ) ∧ I B (b2 ) = I Bt (b1 ) ∧ I Bt (b2 ), p

p

p

p

p

p

FBtk (b1 b2 ) = FBk (b1 b2 ) ≤ FB (b1 ) ∨ FB (b2 ) = FBt (b1 ) ∨ FBt (b2 ), TBntk (b1 b2 ) = TBnk (b1 b2 ) ≥ TBn (b1 ) ∨ TBn (b2 ) = TBnt (b1 ) ∨ TBnt (b2 ), I Bntk (b1 b2 ) = I Bnk (b1 b2 ) ≥ I Bn (b1 ) ∨ I Bn (b2 ) = I Bnt (b1 ) ∨ I Bnt (b2 ), FBntk (b1 b2 ) = FBnk (b1 b2 ) ≥ FBn (b1 ) ∧ FBn (b2 ) = FBnt (b1 ) ∧ FBnt (b2 ). That is, Gˇ bl = (Bl , Bl1 , Bl2 , . . . , Blm ) is a bipolar single-valued neutrosophic graph structure of Gˇ t , t = 1, 2. Thus Gˇ bn = (B, B1 , B2 , . . . , Bm ), a bipolar single-valued neutrosophic graph structure of Gˇ s = Gˇ s1 ∪ Gˇ s2 , is the union of two bipolar singlevalued neutrosophic graph structures Gˇ b1 and Gˇ b2 .

416

10 Bipolar Neutrosophic Graph Structures

Definition 10.12 Let Gˇ b1 = (B1 , B11 , B12 , . . . , B1m ) and Gˇ b2 = (B2 , B21 , B22 , . . . , B2m ) be bipolar single-valued neutrosophic graph structures and let X 1 ∩ X 2 = ∅. The join of Gˇ b1 and Gˇ b2 is denoted by Gˇ b1 + Gˇ b2 = (B1 + B2 , B11 + B21 , B12 + B22 , . . . , B1m + B2m ) and defined as ⎧ p p ⎨ T(B1 +B2 ) (b) = T(B1 ∪B2 ) (b), p p I (b) = I(B1 ∪B2 ) (b), 1. ⎩ (Bp1 +B2 ) p F(B1 +B2 ) (b) = F(B1 ∪B2 ) (b), ⎧ n n ⎨ T(B1 +B2 ) (b) = T(B1 ∪B2 ) (b), n n I (b) = I(B1 ∪B2 ) (b), 2. ⎩ (Bn1 +B2 ) n F(B1 +B2 ) (b) = F(B (b), 1 ∪B2 ) for all b ∈ X ∪ X , 1 2 ⎧ p p ⎨ T(B1k +B2k ) (bd) = T(B1k ∪B2k ) (bd), p p I (bd) = I(B1k ∪B2k ) (bd), 3. ⎩ (Bp1k +B2k ) p F(B1k +B2k ) (bd) = F(B1k ∪B2k ) (bd), ⎧ n n ⎨ T(B1k +B2k ) (bd) = T(B1k ∪B2k ) (bd), n n I (bd) = I(B1k ∪B2k ) (bd), 4. ⎩ (Bn1k +B2k ) n F(B1k +B2k ) (bd) = F(B (bd), 1k ∪B2k ) for all (bd) ∈ X ∪ X , 1k 2k ⎧ p p p p p ⎨ T(B1k +B2k ) (bd) = (TB1k + TB2k )(bd) = TB1 (b) ∧ TB2 (d), p p p p p I (bd) = (I B1k + I B2k )(bd) = I B1 (b) ∧ I B2 (d), 5. ⎩ (Bp1k +B2k ) p p p p F(B1k +B2k ) (bd) = (FB1k + FB2k )(bd) = FB1 (b) ∨ FB2 (d), ⎧ n n n n n ⎨ T(B1k +B2k ) (bd) = (TB1k + TB2k )(bd) = TB1 (b) ∨ TB2 (d), n n n n n I (bd) = (I B1k + I B2k )(bd) = I B1 (b) ∨ I B2 (d), 6. ⎩ (Bn1k +B2k ) F(B1k +B2k ) (bd) = (FBn1k + FBn2k )(bd) = FBn1 (b) ∧ FBn2 (d), for all b ∈ X 1 , d ∈ X 2 . Example 10.11 The join of two bipolar single-valued neutrosophic graph structures Gˇ b1 and Gˇ b2 shown in Fig. 10.8 is defined as Gˇ b1 + Gˇ b2 = {B1 + B2 , B11 + B21 , B12 + B22 } and is given in Fig. 10.13. Theorem 10.5 The join Gˇ b1 + Gˇ b2 = (B1 + B2 , B11 + B21 , B12 + B22 , . . . , B1m + B2m ) of two bipolar single-valued neutrosophic graph structures of the graph structures Gˇ 1 and Gˇ 2 is a bipolar single-valued neutrosophic graph structure of Gˇ 1 + Gˇ 2 . The proof of Theorem 10.5 is a direct consequence of Theorem 10.3 and Definition 10.12.

10.3 Isomorphism in Bipolar Neutrosophic Graph Structures In this section, isomorphic properties of bipolar single-valued neutrosophic graph structures are discussed.

−0 .3,

.3, −0

.4) −0

0.1 ,

0.8 ,

.1, −0

(0. 3,

0.1, .4,

d1 (0.2, 0.05, 0.3, −0.2, −0.05, −0.3)

−0 .8)

, 0.4

−0 .1,

d3 (0.5, 0.15, 0.5, −0.5, −0.15, −0.5)

, 0.1

B12 + B22 (0.4, 0.1, 0.6,

B1 1 ) 0.5 d2 (0.3, 0.1, 0.4, −0.3, −0.1, −0.4) + B2 − , 1 (0 ()0 .1 .2, 0 6 − 0.0 0. .3, 0 .6) .3, b4 5, 0 − . 0 0 1 , (0 − ,0 − 5 .3, , , 0 5 . . . .6, 5, 5 −0 0 0 0 , . 0.1 1 −0 .2, − . 0 , 0 ,0 , −0 − 3 . . 3 , 3 . 0 .6, .05 5 , 0 . ( − − 0 2 ,− −0 , 0 − 6 . B2 0.3 . 1, , . 0 6 5 + . , ) −0 ,− 2 0 5 1 , 0 B .6) (0 . 0.1 5 ) 0 0 6 .2, . , . , 0 3 0 − . , 0.0 0 5 − 0 . ( , .6) 5, 5 (0 0.6 b1 0.0 ,− ,− 5 0 . . 2, −0 −0 .6, .05 (0.2, 0 ) ,0 .05, 0 , −0.6 ,− 5 .1 0 .6 0 − , −0.2 , . 0.6 , −0.0 , −0.5 ,0 .6 ) 0 5 , 5 . , −0.6 0.1 , .5 (0(0 ) ) 0 ( .5, 0.4 − , 0.1 5 ) (0 0 .8 .4, 0.1, . ,0 , −0 0.5, −0 .8, −0 −0.05 .4, −0.1 −0.2, −0 .2, , −0.5) 5, 0.8, .5, −0 .2, 0.0 , (0 .4 −0 ,0 .1, .05 −0 ,0 2 . .8) (0 b3 (0.4, 0.1, 0.4, 0.4, 0.1, 0.4) b2 (0.5, 0.1, 0.8, 0.5, 0.1, 0.8)

.6)

417

3, (0.

B11 + B21 (0.5, 0.05, 0.8, −0.5, −0.05, −0.8)

10.3 Isomorphism in Bipolar Neutrosophic Graph Structures

Fig. 10.13 Join Gˇ b1 + Gˇ b2

Definition 10.13 A bipolar single-valued neutrosophic graph structure Gˇ b1 = (B1 , B11 , B12 , . . . , B1m ) of a graph structure Gˇs1 = (X 1 , X 11 , X 12 , . . . , X 1m ) is isomorphic to a bipolar single-valued neutrosophic graph structure Gˇ b2 = (B2 , B21 , B22 , . . . , B2m ) of a graph structure Gˇs2 = (X 2 , X 21 , X 22 , ..., X 2m ) if there exists a pair (h, ϕ), where h : X 1 → X 2 is a bijection and ϕ is a permutation on {1, 2, . . . , m} such that 1. 2. 3. 4.

p

p

p

p

p

p

p

p

TB1 (b) = TB2 (h(b)), I B1 (b) = I B2 (h(b)), FB1 (b) = FB2 (h(b)), TBn1 (b) = TBn2 (h(b)), I Bn1 (b) = I Bn2 (h(b)), FBn1 (b) = FBn2 (h(b)), p

p

p

p

TB1k (bd) = TB2ϕ(k) (h(b)h(d)), I B1k (bd) = I B2ϕ(k) (h(b)h(d)), FB1k (bd) = FB2ϕ(k) (h(b)h(d)), TBn1k (bd) = TBn2ϕ(k) (h(b)h(d)), I Bn1k (bd) = I Bn2ϕ(k) (h(b)h(d)), FBn1k (bd) = FBn2ϕ(k) (h(b)h(d)),

for all b ∈ X 1 , bd ∈ B1k , k ∈ {1, 2, . . . , m}. Example 10.12 Let Gˇ b1 = (B, B1 , B2 ) and Gˇ b2 = (B  , B1 , B2 ) be two bipolar singlevalued neutrosophic graph structures of two graph structures Gˇ s1 = (X, X 1 , X 2 ) and Gˇ s2 = (X  , X 1 , X 2 ) as shown in Figs. 10.14 and 10.15, respectively. Gˇ b1 and Gˇ b2 are isomorphic under (h, ϕ), where h : X → X  is a bijection and ϕ is a permutation on the set {1, 2} defined as ϕ(1) = 2, ϕ(2) = 1 and satisfy the following conditions: p

p

p

p

p

p

1. TB (bi ) = TB  (h(bi )), I B (bi ) = I B  (h(bi )), FB (bi ) = FB  (h(bi )), 2. TBn (bi ) = TBn (h(bi )), I Bn (bi ) = I Bn  (h(bi )), FBn (bi ) = FBn  (h(bi )), p p p p 3. TBk (bi b j ) = TB  (h(bi )h(b j )), I Bk (bi b j ) = I B  (h(bi )h(b j )), p

ϕ(k)

ϕ(k)

FB  (h(bi )h(b j )), ϕ(k) 4. TBnk (bi b j ) = TBn (h(bi )h(b j )), I Bnk (bi b j ) = I Bn  (h(bi )h(b j )), ϕ(k) ϕ(k) FBn  (h(bi )h(b j )), ϕ(k)

p

FBk (bi b j ) = FBnk (bi b j ) =

10 Bipolar Neutrosophic Graph Structures

, 0.5

B1

0. 3, 0. 3, 0. 0. 4, 2, − 0. 0. 3, 3, 0. − 4, 0. − 3, 0. − 2, 0. − 4) 0. 3, − 0. 4) 1(

B

B

1(

.5) −0

.5) −0

.5, −0

.7, −0

.6, −0

.3, −0

, 0.5

B2 (0.2, 0.2, 0.5, −0.2, −0.2, −0.5)

0.

, 0.7

, 0.5

B2 (0.3, 0.2, 0.5, −0.3, −0.2, −0.5)

.2, 3, 0 (0.

, 0.3 b 3(

, 0.6 b 2(

b4 (0 .3, 0 .2, 0 .5, − 0.3, −0.2 , −0 .5) ) 0.4 − , 2 . 0 ,− 0.3 − , 4

418

b1 (0 .4, 0 .3, 0 .4, − 0.4, −0. 3, − 0.4)

Fig. 10.14 Bipolar single-valued neutrosophic graph structure Gˇ b1

for all bi ∈ X , bi b j ∈ X k , k = 1, 2. Definition 10.14 A bipolar single-valued neutrosophic graph structure Gˇ b1 = (B1 , B11 , B12 , . . . , B1m ) of a graph structure Gˇs1 = (X 1 , X 11 , X 12 , . . . , X 1m ) is said to be identical to the bipolar single-valued neutrosophic graph structure Gˇ b2 = (B2 , B21 , B22 , ..., B2m ) of a graph structure Gˇs2 = (X 2 , X 21 , X 22 , . . . , X 2m ) if h : X 1 → X 2 is a bijection such that 1. 2. 3. 4.

p

p

p

p

TB (h(b)h(d)), 2k TBn (h(b)h(d)),

p

p

p

TB1 (b) = TB2 (h(b)), I B1 (b) = I B2 (h(b)), FB1 (b) = FB2 (h(b)), TBn1 (b) = TBn2 (h(b)), I Bn1 (b) = I Bn2 (h(b)), FBn1 (b) = FBn2 (h(b)),

TB (bd) = 1k TBn (bd) = 1k

p

2k

p

p

p

p

I B (bd) = I B (h(b)h(d)), FB (bd) = FB (h(b)h(d)), 1k 2k 1k 2k I Bn (bd) = I Bn (h(b)h(d)), FBn (bd) = FBn (h(b)h(d)), 1k

2k

1k

2k

for all b ∈ X 1 , bd ∈ X 1k , k ∈ {1, 2, . . . , m}. Example 10.13 Let Gˇ b1 = (B, B1 , B2 ) and Gˇ b2 = (B  , B1 , B2 ) be two bipolar singlevalued neutrosophic graph structures of the graph structures Gˇs1 = (X, X 1 , X 2 ) and Gˇs2 = (X  , X 1 , X 2 ), respectively, as shown in Figs. 10.16 and 10.17. The bipolar single-valued neutrosophic graph structure Gˇ b1 is identical to Gˇ b2 under h : X → X  defined as h(b1 ) = d2 , h(b2 ) = d1 , h(b3 ) = d4 , h(b4 ) = d3 , h(b5 ) = d5 , h(b6 ) = d8 , h(b7 ) = d7 , h(b8 ) = d6 . Moreover,

−0 .5)

−0 .7,

−0 .3,

0.7 ,

.4) −0

.4) −0

0.2 ,

.3, −0

.3, −0

0.4 ,

.2, −0

−0 .3,

, 0.4

.3, −0

.3,

d3

−0 .4) −0 .2,

, 0.3

, 0.4

B

2 (0

−0 .6,

0.5 ,

0.5 , 0.5 , (0. 3,

(0. 6,

2, −0.5)

.2, (0 B2

, 0.3

0.5) 0.2, − 0.3, − 0.5, −

B1 (0.3, 0.2, 0.5, −0.3, −0.

, 3, 0.2

B1 (0.2, 0.2, 0.5, −0 .2, −0.2, −0.5)

.3, (0 B2

d2

419

d 4(0.

−0 .5,

−0 .5)

10.3 Isomorphism in Bipolar Neutrosophic Graph Structures

d1

) −0.4 0.3, − , 4 0. .4, − 0.3, 0 , 4 . 0 (

Fig. 10.15 Bipolar neutrosophic graph structure Gˇ b2

1. 2. 3. 4.

p

p

p

p

p

p

TB (bi ) = TB  (h(bi )), I B (bi ) = I B  (h(bi )), FB (bi ) = FB  (h(bi )), TBn (bi ) = TBn (h(bi )), I Bn (bi ) = I Bn  (h(bi )), FBn (bi ) = FBn  (h(bi )), p

p

p

p

p

p

TBk (bi b j ) = TB  (h(bi )h(b j )), I Bk (bi b j ) = I B  (h(bi )h(b j )), FBk (bi b j ) = FB  (h(bi )h(b j )), k

k

k

k

k

k

TBnk (bi b j ) = TBn (h(bi )h(b j )), I Bnk (bi b j ) = I Bn  (h(bi )h(b j )), FBnk (bi b j ) = FBn  (h(bi )h(b j )),

for all bi ∈ X , bi b j ∈ X k , k = 1, 2. Definition 10.15 Let Gˇ bn = (B, B1 , B2 , . . . , Bm ) be a bipolar single-valued neutrosophic graph structure and ϕ be a permutation on {B1 , B2 , . . . , Bm } and also on set {1, 2, . . . , m}, that is, ϕ(Bk ) = Bl if and only if ϕ(k) = l for all k. If bd ∈ X k and  n TBnϕ (bd) = TBn (b) ∨ TBn (d) − Tϕ(Bl ) (bd), k k l =k l =k   p n 2. I Bpϕ (bd) = I Bp (b) ∧ I Bp (d) − Iϕ(B (bd), I Bn ϕ (bd) = I Bn (b) ∨ I Bn (d) − Iϕ(B (bd), l) l) k k l =k l =k  p  p p p n 3. FB ϕ (bd) = FB (b) ∨ FB (d) − Tϕ(Bl ) (bd), FBn ϕ (bd) = FBn (b) ∧ FBn (d) − Tϕ(B (bd), l)

1. TBpϕ (bd) = TBp (b) ∧ TBp (d) −



p

Tϕ(Bl ) (bd),

l =k

k

for each k ∈ {1, 2, . . . , m}, then bd ∈ 1. 2.

p TBeϕ (bd) n TBeϕ (bd)

p TB ϕ (bd), k n TB ϕ (bd), k

≥ ≤ {1, 2, . . . , m}.

p I Beϕ (bd) n I Beϕ (bd)

l =k

k

≥ ≤

ϕ Be ,

where e is selected such that

p I B ϕ (bd), k n I B ϕ (bd), k

p

p

FBeϕ (bd) ≥ FB ϕ (bd), k FBneϕ (bd) ≤ FBn ϕ (bd), k

for all k ∈

420

10 Bipolar Neutrosophic Graph Structures .6, (0 b5 , 0.6

) 0.3 ,− 4 . −0

B2 ( 0.7, 0.5, 0.5, −0. 7, − 0.5, −0. 5)

.5) −0

B2 ( 0.5, 0.3, 0.6, −0. 5, − 0.3, −0. 6)

.6, −0

B1 ( 0.3, 0.2, 0.4, −0. 3, − 0.2, −0. 4)

.6, −0

, 0.4

4) 0. − 3, 0. − 3, 0. ,− .4 ,0 .3 ,0 .3 (0 b1

.4, (0 b2

B2 ( 0.2, 0.3, 0.5, −0. 2, − 0.3, −0. 5)

, 0.5

.4) .6, ) −0 , −0 , −0.4, −0.5) 3 0.6 3 . . 6, 0.4, 0.5, −0.6, ,− −0 .4, 0 B2 (0. 3 , . 0 .3 ,0 ,− −0 (0.6 0.6 .4, b3 − 0 , .6 .3, 0.2) ,0 ,0 − 0.3 0.3 .5, , ( .6 B 2 , −0 (0 b7 5 0. − −0.5) 2, 5, −0.4, −0.2, 0. B1 (0.4, 0.2, 0. 5, . ,0 .5 (0 4) 0. b6 − 5, . 0 −0.3) − 3, −0.2, −0.2, ) 0.7, 5 B2 (0.2, 0.2, 0. . 0 − , − .4, .4) 0.3 5, 0 −0 − , . , 0 .3 0.4 .7, −0 ,− (0 .3, 5 0 . 4 b ,0 ,− 0.3 0.3) 0.4 .4, .3, 0 0 ,− ( −0.2) B 2 .3, 0.6 (0 2, −0.4, −0.4, 0. − 4, 0. , 1 .4 , (0 B1 B .5 −0 .3, 0 , 0.6 .5, 0 ( b8

, 0.5

.4, −0

.4, −0

.5) −0

Fig. 10.16 Bipolar neutrosophic graph structure Gˇ b1

ϕ

ϕ

ϕ

The bipolar single-valued neutrosophic graph structure (B, B1 , B2 , . . . , Bm ) is called ϕ-complement of the bipolar single-valued neutrosophic graph structure Gˇ bn ϕc and denoted by Gˇ bn . Example 10.14 Let B = {(b1 , 0.4, 0.4, 0.7, −0.4, −0.4, −0.7), (b2 , 0.6, 0.6, 0.4, −0.6, −0.6, −0.4), (b3 , 0.8, 0.5, 0.3, −0.8, −0.5, −0.3)}, B1 = {(b1 b3 , 0.4, 0.4, 0.3, −0.4, −0.4, −0.3)}, B2 = {(b2 b3 , 0.6, 0.4, 0.3, −0.6, −0.4, −0.3)}, and B3 = {(b1 b2 , 0.4, 0.3, 0.4, −0.4, −0.3, −0.4)} be bipolar single-valued neutrosophic sets on X = {b1 , b2 , b3 }, X 1 = {b1 b3 }, X 2 = {b2 b3 }, and X 3 = {b1 b2 }, respectively. Consider a bipolar single-valued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , B3 ) of graph structure Gˇ = (X, X 1 , X 2 , X 3 ). Let ϕ(B1 ) = B2 , ϕ(B2 ) = B3 , ϕ(B3 ) = B1 , where ϕ is a permutation on {1, 2, 3}. Through simple calculations for edges b1 b3 , b2 b3 , b1 b2 ∈ B1 , B2 , B3 , respecϕ ϕ ϕ ϕc ϕ ϕ tively, it is easy to see that b1 b3 ∈ B3 , b2 b3 ∈ B1 , b1 b2 ∈ B2 . So, Gˇ bn =(B, B1 , B2 , ϕ B3 ) is a ϕ-complement of the bipolar single-valued neutrosophic graph structure Gˇ bn as shown in Fig.10.18. Proposition 10.1 The ϕ-complement of a bipolar single-valued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , . . . , Bm ) is a strong bipolar single-valued neutrosophic

10.3 Isomorphism in Bipolar Neutrosophic Graph Structures

421

4) 0. − 3, 0. 3) − 0. 3, ,− 0. .4 ,− 0.5) , −0 .4 0.4, − 6 ,0 0.6, − 0. .3 , 0.5, − ,− ,0 .6, 0.4 .3 .3 B 2(0 ,0 (0 .4 B1 ,0 .6 (0 d4

. ,0 .3 ,0 .3 (0 B1 4) 0. − 5, 0. − 7, 0. ,− .4 ,0 .5 ,0 .7 (0 d3

5) 0. − 6, 0. ,− 6 0. ,− .5 0 , B2 (0 .6 .7, 0.5 , 0.5, ,0 −0.7, .6 0 ( −0.5, −0.5) d5

0.6) 0.3, − 0.5, − − , .6 , 0.3, 0 B 2(0.5

B1 (0 .3, 0.2 , 0.4,

) , −0.3 , −0.2 , −0.2 .3 0 , , 0.2 B 2(0.2

) , −0.5 , −0.3 , −0.2 .5 0 , , 0.3 B 2(0.2

d1

3) 0. 4) − 0. 6, 5) − 0. 0. 3, − − 0. 5, 3, − 0. 0. − − 3, 3, 0. 4, 0. − 0. 6, 4, . ,− .5 ,0 .5 ,0 (0 .3 d6 ,0 .4 (0 B2

5) 0. − , 4 0. − 4, . 0 − B1 (0. 4, 0.4 5, 0. , 0.2, , −0.4 4 0. , −0. , 4, −0 4 . .2) (0

2) 0. − 5, 0. − 5, 0. ,− .2 ,0 .5 ,0 .5 (0 d8

4) 0. − , 3 −0.3 0. , −0.2 − , −0.4 3, . ) 0 − 4, 0. , 3 0. 3, 0. ( d2

6) 0. − , 3 0. − 6, . 0 ,− .6 0 3, 0. 6, . (0

B1 (0.4 , 0.2, 0 .5, −0 .4, −0 .2, −0 .5)

d7

Fig. 10.17 Bipolar neutrosophic graph structure Gˇ b2 , 0.6 b 2(

ϕ

2

B

−0 .3)

−0 .4,

−0 .4,

0.3 , 0.4 , 0.4 ,

1(

B

(0 .6 ,0 .5 ,0 .4 ,− 0. 6, − 0. 5, B3ϕ( − 0.4, 0. 0.4, 4) 0.7, −0.4 , −0 .4, − 0.7) b3 (0.8, 0.5, 0.3 , 0.8, 0.5, 0 .3) .4) −0

.4) −0

ψc Fig. 10.18 Gˇ bn and ϕ-complement Gˇ bn

.6, −0

.3, −0

0.7)

ϕ

.6, −0

.4, −0

0.4,

1

, 0.4

, 0.4

0.4,

(0 .4 ,0 .4 ,0 .7 ,− 0. 4, − 0. 4, − 0. 7)

, 0.6

, 0.3

b1 (0.4, 0.4, 0.7,

B

0.7) 4, − −0. 0.4, .7, − .4, 0 .4, 0 b 1(0

4, (0. B3

b3 (0.8, 0.5, 0.3, −0.8, −0.5, −0.3) .3) b2 (0.6, 0.6, 0.4, −0.6, −0.6, −0.4) −0 .4, 0 ,− 0.6 ,− 0.3 , 0.4 6, (0. B2

422

10 Bipolar Neutrosophic Graph Structures

graph structure. Moreover, if ϕ(k) = e, where k, e ∈ {1, 2, . . . , m}, then all Be edges in the bipolar single-valued neutrosophic graph structure (B, B1 , B2 , . . . , Bm ) ϕ ϕ ϕ ϕ become Bk -edges in (B, B1 , B2 , . . . , Bm ). Proof By Definition 10.15, p

p



p

TB ϕ (bd) = TB (b) ∧ TB (d) − k

p

p

p

l=k

I B ϕ (bd) = I B (b) ∧ I B (d) − k

p

p

p

FB ϕ (bd) = FB (b) ∨ FB (d) − k

 l=k

l=k



p

Tϕ(Bl ) (bd), TBnϕ (bd) = TBn (b) ∨ TBn (d) − k

p

Iϕ(Bl ) (bd), p

Fϕ(Bl ) (bd),

I Bn ϕ (bd) = I Bn (b) ∨ I Bn (d) − k

l=k

l=k

FBn ϕ (bd) = FBn (b) ∧ FBn (d) − k

n Tϕ(B (bd), l)

n Iϕ(B (bd), l)

 l=k

n Fϕ(B (bd), l)

p

where k ∈ {1, 2, . . . , m}. For expression of TB ϕ : k  p p p p p p As TB (b) ∧ TB (d) ≥ 0, Tϕ(Bl ) (bd) ≥ 0 and TBk (bd) ≤ TB (b) ∧ TB (d) for all l=k  p p p p p Tϕ(Bl ) (bd) ≤ TB (b) ∧ TB (d). This shows that TB (b) ∧ TB (b) − Bk . This implies l=k  p p p Tϕ(Bl ) (bd) ≥ 0. Hence, TB ϕ (bd) ≥ 0 for all k. Furthermore, TB ϕ (bd) obtains maxk k l=k  p Tϕ(Bl ) (bd) is zero. Undoubtedly, when ϕ(Bk ) = Be and bd is imum value when l=kp Tϕ(Bl ) (bd) acquires zero value. Hence, a Be -edge, then l=k

p

p

p

TB ϕ (bd) = TB (b) ∧ TB (d), f or bd ∈ Be , ϕ(Bk ) = Be . k

For expression of TBnϕ : k  n Since TBn (b) ∨ TBn (d) ≤ 0, Tϕ(Bl ) (bd) ≤ 0 and TBnk (bd) ≥ TBn (b) ∨ TBn (d) for l=k  n Tϕ(Bl ) (bd) ≥ TBn (b) ∨ TBn (d). It indicates TBn (b) ∨ TBn (d) − all Bk . This implies l=k  Tϕ(Bl ) (bd) ≤ 0. Hence, TBnϕ (bd) ≤ 0 for all k. Furthermore, TBnϕ (bd) is minimum k k l=k  n Tϕ(Bl ) (bd) is zero. Certainly when ϕ(Bk ) = Be and bd is a Be -edge, then when  n l=k Tϕ(Bl ) (bd) is zero. Hence, l=k

TBnϕ (bd) = TBn (b) ∨ TBn (d), f or bd ∈ Be , ϕ(Bk ) = Be . k

p

Similarly, for expression of I B ϕ and I Bn ϕ results are as follows: k k  p p p p p p Since I B (b) ∧ I B (d) ≥ 0, Iϕ(Bl ) (bd) ≥ 0 and I Bk (bd) ≤ I B (b) ∧ I B (d) ∀Bk . l=k  p p p p p Iϕ(Bl ) (bd) ≤ I B (b) ∧ I B (d), which shows that I B (b) ∧ I B (d) − This implies l=k

10.3 Isomorphism in Bipolar Neutrosophic Graph Structures



p

p

423 p

Iϕ(Bl ) (bd) ≥ 0. Hence, I B ϕ (bd) ≥ 0 for all k. Furthermore, I B ϕ (bd) obtains maxk k  p Iϕ(Bl ) (bd) is zero. Undoubtedly, when ϕ(Bk ) = Be and bd is a imum value when  l=p k Iϕ(Bl ) (bd) acquires zero value. Hence, Be -edge, then

l=k

l=k p

p

p

I B ϕ (bd) = I B (b) ∧ I B (d), f or bd ∈ Be , ϕ(Bk ) = Be . k

 n Moreover, as I Bn (b) ∨ I Bn (d) ≤ 0, Iϕ(Bl ) (bd) ≤ 0 and I Bnk (bd) ≥ I Bn (b) ∨ I Bn (d)  n l=k Iϕ(Bl ) (bd) ≥ I Bn (b) ∨ I Bn (d). This indicates I Bn (b) ∨ for all Bk . This implies l=k  Iϕ(Bl ) (bd) ≤ 0. Hence, I Bn ϕ (bd) ≤ 0 for all k. Furthermore, I Bn ϕ (bd) is I Bn (d) − k k l=k  n Iϕ(Bl ) (bd) is zero. Certainly, when ϕ(Bk ) = Be and bd is a Be minimum when  n l=k Iϕ(Bl ) (bd) is zero. Hence, edge, then l=k

I Bn ϕ (bd) = I Bn (b) ∨ I Bn (d) f or bd ∈ Be , ϕ(Bk ) = Be . k

p

On similar argument, derive expressions for FB ϕ and FBn ϕ as follows: k k  p p p p p p Since FB (b) ∨ FB (d) ≥ 0, Fϕ(Bl ) (bd) ≥ 0 and FBk (bd) ≤ FB (b) ∨ FB (d) for l=k  p p p p p Fϕ(Bl ) (bd) ≤ FB (b) ∨ FB (d). It indicates FB (b) ∨ FB (d) − all Bk . This implies l=k  p p F Pϕ(Bl ) (bd) ≥ 0. Hence, FB ϕ (bd) ≥ 0 for all k. Furthermore, FB ϕ (bd) is maxk k l=k  p Fϕ(Bl ) (bd) is zero. When ϕ(Bk ) = Be and bd is a Be -edge, then imum when l=k  p Fϕ(Bl ) (bd) is zero. Hence, l=k

p

p

p

FB ϕ (bd) = FB (b) ∨ FB (d) f or bd ∈ Be , ϕ(Bk ) = Be . k

 n Moreover, as FBn (b) ∧ FBn (d) ≤ 0, Fϕ(Bl ) (bd) ≤ 0 and FBnk (bd) ≥ FBn (b) ∧ FBn (d) l=k  n Fϕ(Bl ) (bd) ≥ FBn (b) ∧ FBn (d). This shows that FBn (b) ∧ FBn (b) − ∀Bk . This implies l=k  n Fϕ(Bl ) (bd) ≤ 0. Hence, FBn ϕ (bd) ≤ 0 ∀ k. Furthermore, FBn ϕ (bd) obtains minik k l=k  n Fϕ(Bl ) (bd) is zero. Undoubtedly, when ϕ(Bk ) = Be and bd is a mum value when  l=nk Fϕ(Bl ) (bd) acquires zero value. Hence, Be -edge, then l=k

FBn ϕ (bd) = FBn (b) ∧ FBn (d), f or bd ∈ Be , ϕ(Bk ) = Be . k

10 Bipolar Neutrosophic Graph Structures 0.5) 0.5) 5, − 4, − .6) −0. −0. 0 0.5, 0.5, .5, − .5, − 0.6) .4, − .5, 0 5, 0.4, 0 0 − .5, 0 0. .5, .5, − 0 b 4(0 B 2( , − −0 0.5 .6, ,− ,0 0.6 0.4 .5, (0 B1

5, (0. b2

, 0.5

b3

(0 .3 ,0 .3 ,0 .4 ,− 0. 3, B − 1( 0.3 0.3, − ,0 0. .3, 0.5 4) ,− 0.3 ,− 0.3 ,

−0 .5) b1 (0.8, 0.4, 0.5,

b5 (0 B2 ( .3, 0. 3, 0.3 B 0.3, 0.3, 3( , −0. 0.5 b6 ( 0.5, −0. 3, −0 3, − 0 ,0 .3, − 0.3, .4, .5, 0 0.3) −0.5 0.6 .5, ) , − 0.6 0.5 , − , − 0.5 0.4 , − , − 0.5 0.6 , − 0.6 ) )

424

4) 0. − 3, 0. − , 3 0. ,− ) 4 . .5 ,0 .3 , −0 0 , .3 0.3 (0 − b 7 .3, 0 − 5, . 0 3, 0. , .3 (0

B3 0.8, 0.4,

0.5)

Fig. 10.19 Bipolar neutrosophic graph structure

It completes the proof.



Definition 10.16 Let Gˇ bn = (B, B1 , B2 , . . . , Bm ) be a bipolar single-valued neutrosophic graph structure and ϕ be a permutation on {1, 2, . . . , m}, then 1. Gˇ bn is a self-complementary bipolar single-valued neutrosophic graph structure ϕc if Gˇ bn is isomorphic to Gˇ bn . 2. Gˇ bn is a strong self-complementary bipolar single-valued neutrosophic graph structure if Gˇ bn is identical to Gˇ ϕc bn . Definition 10.17 Let Gˇ bn = (B, B1 , B2 , . . . , Bm ) be a bipolar single-valued neutrosophic graph structure, then 1. Gˇ bn is a totally self-complementary bipolar single-valued neutrosophic graph ϕc structure if Gˇ bn is isomorphic to Gˇ bn for all permutations ϕ on {1, 2, . . . , m}. 2. Gˇ bn is a totally strong self-complementary bipolar single-valued neutrosophic graph structure if Gˇ bn is identical to Gˇ ϕc bn for all permutations ϕ on {1, 2, . . . , m}. Example 10.15 A bipolar single-valued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , B3 ) shown in Fig.10.19 is totally strong self-complementary, since it is identical to its ϕ−complement for all permutations ϕ on {1, 2, 3}. Theorem 10.6 A bipolar single-valued neutrosophic graph structure is totally selfcomplementary if and only if it is a strong bipolar single-valued neutrosophic graph structure. Proof Consider a strong bipolar single-valued neutrosophic graph structure Gˇ bn and a permutation ϕ on {1,2, …, m}. By proposition 10.1, ϕ-complement of a bipolar

10.3 Isomorphism in Bipolar Neutrosophic Graph Structures

425

single-valued neutrosophic graph structure Gˇ bn = (B, B1 , B2 , . . . , Bm ) is a strong bipolar single-valued neutrosophic graph structure. Moreover, if ϕ −1 (e) = k, where k, e ∈ {1, 2, . . . , m}, then all Be -edges in the bipolar single-valued neutrosophic ϕ ϕ ϕ ϕ graph structure (B, B1 , B2 , . . . , Bm ) become Bk -edges in (B, B1 , B2 , . . . , Bm ). This leads to the following results. p

p

p

p

p

p

p

p

p

p

p

p

TB (bd) = TB (b) ∧ TB (d) = T ϕ (bd), I B (bd) = I B (b) ∧ I B (d) = I ϕ (bd), FB (bd) = FB (b) ∨ FB (d) = F ϕ (bd), e e e Bk Bk Bk p TBn (bd) = TBn (b) ∨ TBn (d) = T n ϕ (bd), I Bn (bd) = I Bn (b) ∨ I Bn (d) = I n ϕ (bd), FBn (bd) = FBn (b) ∧ FBn (d) = F ϕ (bd). e e e Bk Bk Bk

ϕc Therefore, under h : X → X (identity mapping), Gˇ bn and Gˇ bn are isomorphic such that p p p p p p TB (b) = TB (h(b)), I B (b) = I B (h(b)), FB (b) = FB (h(b)), TBn (b) = TBn (h(b)), I Bn (b) = I Bn (h(b)), FBn (b) = FBn (h(b)). Further, p

p

p

k

k

p

p

k

k

TBe (bd) = TB ϕ (h(b)h(d)) = TB ϕ (bd), TBne (bd) = TBnϕ (h(b)h(d)) = TBnϕ (bd), p

k

k

I Be (bd) = I B ϕ (h(b)h(d)) = I B ϕ (bd), I Bne (bd) = I Bn ϕ (h(b)h(d)) = I Bn ϕ (bd), p

p

p

k

k

k

k

FBe (bd) = FB ϕ (h(b)h(d)) = FB ϕ (bd), FBne (bd) = FBn ϕ (h(b)h(d)) = FBn ϕ (bd), k

k

for all bd ∈ X e , for ϕ −1 (e) = k; e, k = 1, 2, . . . , m. These relations hold for each permutation ϕ on {1, 2, . . . , m}. Hence, Gˇ bn is a totally self-complementary bipolar single-valued neutrosophic graph structure. Conversely, ϕc let Gˇ bn be isomorphic to Gˇ bn for each permutation ϕ on {1, 2, . . . , m}. Moreover, according to the definitions of ϕ-complement and isomorphism of the bipolar singlevalued neutrosophic graph structure, p

p

p

p

p

p

TBe (bd) = TB ϕ (h(b)h(d)) = TB (h(b)) ∧ TB (h(d)) = TB (b) ∧ TB (d), k

TBne (bd) = TBnϕ (h(b)h(d)) = TBn (h(b)) ∨ TBn (h(d)) = TBn (b) ∨ TBn (d), k

p

p

p

p

p

p

I Be (bd) = I B ϕ (h(b)h(d)) = I B (h(b)) ∧ I B (h(d)) = I B (b) ∧ I B (d), k

I Bne (bd) = I Bn ϕ (h(b)h(d)) = I Bn (h(b)) ∨ I Bn (h(d)) = I Bn (b) ∨ I Bn (d), k

p

p

p

p

p

p

FBe (bd) = FB ϕ (h(b)h(d)) = FB (h(b)) ∨ FB (h(d)) = FB (b) ∨ FB (d), k

FBne (bd) = FBn ϕ (h(b)h(d)) = FBn (h(b)) ∧ FBn (h(d)) = FBn (b) ∧ FBn (d), k

426

10 Bipolar Neutrosophic Graph Structures

for all bd ∈ X e , where e = 1, 2, . . . , m. Hence, Gˇ bn is a strong bipolar single-valued neutrosophic graph structure.  Remark 10.1 Every self-complementary bipolar single-valued neutrosophic graph structure is always a totally self-complementary bipolar single-valued neutrosophic graph structure. Theorem 10.7 If Gˇ s = (X, X 1 , X 2 , . . . , X m ) is a totally strong-self-complementary p p p graph structure and B = (TB , I B , FB , TBn , I Bn , FBn ) is a bipolar single-valued neutrop p p sophic subset of X , where TB , I B , FB , TBn , I Bn , FBn are constant functions, then every strong bipolar single-valued neutrosophic graph structure of Gˇ s with bipolar singlevalued neutrosophic vertex set B is necessarily a totally strong self-complementary bipolar single-valued neutrosophic graph structure. Proof Let f , f  ∈ [0, 1], g, g  ∈ [0, 1], and i, i  ∈ [0, 1] be six constants and TB (b) = p p f , I B (b) = g, FB (b) = i, TBn (b) = f  , I Bn (b) = g  , FBn (b) = i  , for all b ∈ X . Since Gˇ s is a totally strong self-complementary graph structure, so for every permutation ϕ −1 on {1, 2, . . . , m} there exists a bijection h : X → X such that for ϕ −1 c every Be -edge bd, h(b)h(d) [a Bk -edge in Gˇ s ] is a Be -edge in Gˇ s . Thus, for every p

−1

ϕ c ϕ Be -edge (bd), (h(b)h(d)) [a Bk -edge in Gˇ bn ] is a Be -edge in Gˇbn . Moreover, Gˇ bn is a strong bipolar single-valued neutrosophic graph structure. So p p p p p p TB (b) = f = TB (h(b)), I B (b) = g = I B (h(b)), FB (b) = i = FB (h(b)), TBn (b) = n n n n n    f = TB (h(b)), I B (b) = g = I B (h(b)), FB (b) = i = FB (h(b)), for all b ∈ X . Also, p

p

p

p

p

p

TBe (bd) = TB (b) ∧ TB (d) = TB (h(b)) ∧ TB (h(d)) = TB ϕ (h(b)h(d)), k

p

p

p

p

p

p

I Be (bd) = I B (b) ∧ I B (d) = I B (h(b)) ∧ I B (h(d)) = I B ϕ (h(b)h(d)), k

p

p

p

p

p

p

FBe (bd) = FB (b) ∨ I B (d) = FB (h(b)) ∨ FB (h(d)) = FB ϕ (h(b)h(d)), k

TBne (bd) = TBn (b) ∨ TBn (d) = TBn (h(b)) ∨ TBn (h(d)) = TBnϕ (h(b)h(d)), k

I Bne (bd) = I Bn (b) ∨ I Bn (d) = I Bn (h(b)) ∨ I Bn (h(d)) = I Bn ϕ (h(b)h(d)), k

FBne (bd) = FBn (b) ∧ I Bn (d) = FBn (h(b)) ∧ FBn (h(d)) = FBn ϕ (h(b)h(d)), k

for all bd ∈ X k , k = 1, 2, . . . , m. This shows that Gˇ bn is a strong self-complementary bipolar single-valued neutrosophic graph structure. It satisfies for every permutation ϕ and ϕ −1 on the set {1, 2, . . . , m}. Thus, Gˇ bn is a totally strong self-complementary bipolar single-valued neutrosophic graph structure. Remark 10.2 The converse of Theorem 10.7 may not be true, e.g., a bipolar singlevalued neutrosophic graph structure depicted in Fig.10.19 is totally strong self-

10.3 Isomorphism in Bipolar Neutrosophic Graph Structures

427

complementary. It is also a strong bipolar single-valued neutrosophic graph strucp p ture with a totally strong self-complementary underlying graph structure but TB , I B , p n n n FB ,TB , I B , FB are not constant functions.

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures In this section, the applications of bipolar single-valued neutrosophic graph structures are discussed in international relations, psychology, and global terrorism.

10.4.1 International Relations Naturally, any living organism cannot be 100% self-reliant, especially a human being who is the noblest of all creatures, but he relies on many things for his survival including, plants, animals, and other human beings. This dependence causes relationships among human beings. On Earth, human population is divided into 195 countries. That’s why the interdependence of human beings results in many relationships among countries. In a relationship, both countries play their role so that these relations can be represented using a bipolar single-valued neutrosophic graph structure, which will show that between two particular countries which type of relationship is the strongest and what are the degrees of its strength from both countries. There are some relationships which are not good in some aspects. A bipolar single-valued neutrosophic graph structure will highlight as to which one country of those two countries is the strongest participant. The relationships and strong partici- pants can be checked. It can be seen which country is the most favorite for a particular relationship. Let X be the set of eight countries: X = {China, Turkey, Bangladesh, America, India, Russia, Pakistan, Afghanistan}. Consider a bipolar single-valued neutrosophic set B on X as shown in Table 10.3. In Table 10.3, T p , F p of a country indicate its positive and negative approaches about the World and I p shows indeterminacy/ambiguity of its approach, whereas T n and F n denote its positive and negative reputation and I n stands to represent the percentage of its uncertain reputation. Use the following abbreviations of country names: Ch = China, TU = Turkey, Ba = Bangladesh, Am = America, In = India, Ru = Russia, Pak = Pakistan, Afg = Afghanistan. Each pair of countries of set X has T p , I p , F p , T n , I n , and F n values of different relationships in Tables 10.4, 10.5, 10.6, 10.7, 10.8, 10.9, 10.10, 10.11, 10.12. Each pair of countries T p , T n of any particular relationship indicates strength of that relationship from the first country and the second country, respectively. Similarly F p , F n represent percentage of weakness and I p , I n demonstrate percentage

428

10 Bipolar Neutrosophic Graph Structures

Table 10.3 Bipolar neutrosophic information of countries Country Tp Ip Fp Tn China Turkey Bangladesh America India Russia Pakistan Afghanistan

0.9 0.8 0.7 0.8 0.6 0.8 0.8 0.5

0.4 0.6 0.6 0.4 0.5 0.4 0.5 0.7

0.2 0.5 0.5 0.4 0.5 0.3 0.4 0.6

−0.9 −0.6 −0.4 −0.8 −0.5 −0.9 −0.5 −0.4

In

Fn

−0.2 −0.5 −0.6 −0.2 −0.6 −0.3 −0.6 −0.5

−0.2 −0.5 −0.5 −0.4 −0.6 −0.4 −0.6 −0.8

Table 10.4 Types of relations between countries Types of relations

(China, Turkey)

(China, Bangladesh)

(China, America)

Project investment

(0.1,0.4,0.3,−0.1,−0.2,−0.3)

(0.2,0.2,0.2,−0.1,−0.2,−0.3)

(0.2,0.1,0.4,−0.1,−0.2,−0.4)

Political support

(0.3,0.3,0.4,−0.3,−0.1,−0.3)

(0.4,0.2,0.1,−0.4,−0.2,−0.3)

(0.4,0.4,0.4,−0.3,−0.2,−0.4)

Warfare activities

(0.1,0.4,0.5,−0.1,−0.2,−0.2)

(0.1,0.3,0.5,−0.1,−0.2,−0.3)

(0.6,0.1,0.1,−0.7,−0.2,−0.1)

Nuclear weapons

(0.4,0.2,0.3,−0.1,−0.2,−0.3)

(0.1,0.2,0.5,−0.1,−0.2,−0.3)

(0.2,0.1,0.4,−0.1,−0.2,−0.4)

Friendship

(0.5,0.3,0.3,−0.5,−0.1,−0.2)

(0.4,0.2,0.1,−0.4,−0.1,−0.3)

(0.2,0.1,0.4,−0.1,−0.2,−0.3)

Respecting religious beliefs

(0.1,0.4,0.4,−0.4,−0.2,−0.3)

(0.4,0.3,0.1,−0.4,−0.2,−0.2)

(0.7,0.1,0.4,−0.6,−0.2,−0.3)

Table 10.5 Types of relations between countries Types of relations

(Turkey, Bangladesh)

(Turkey, America)

(Turkey, India)

Project investment

(0.4,0.3,0.3,−0.1,−0.2,−0.3)

(0.2,0.2,0.5,−0.4,−0.2,−0.5)

(0.4,0.1,0.4,−0.1,−0.2,−0.5)

Political support

(0.5,0.4,0.3,−0.4,−0.3,−0.3)

(0.6,0.4,0.4,−0.5,−0.2,−0.4)

(0.2,0.4,0.4,−0.1,−0.4,−0.5)

Warfare activities

(0.1,0.4,0.4,−0.1,−0.3,−0.5)

(0.1,0.2,0.4,−0.3,−0.2,−0.4)

(0.2,0.4,0.5,−0.3,−0.2,−0.4)

Nuclear weapons

(0.1,0.4,0.3,−0.1,−0.2,−0.3)

(0.1,0.1,0.5,−0.4,−0.1,−0.4)

(0.2,0.1,0.4,−0.1,−0.4,−0.5)

Friendship

(0.6,0.4,0.4,−0.3,−0.2,−0.3)

(0.4,0.4,0.4,−0.5,−0.1,−0.5)

(0.2,0.3,0.5,−0.3,−0.2,−0.3)

Respecting religious beliefs

(0.7,0.3,0.2,−0.4,−0.2,−0.1)

(0.8,0.2,0.1,−0.3,−0.2,−0.4)

(0.2,0.1,0.4,−0.2,−0.2,−0.5)

Table 10.6 Types of relations between countries Types of relations

(Bangladesh, America)

(Bangladesh, India)

(Bangladesh, Russia)

Project investment

(0.1,0.4,0.4,−0.1,−0.2,−0.3)

(0.1,0.5,0.5,−0.1,−0.3,−0.4)

(0.2,0.1,0.5,−0.3,−0.2,−0.3)

Political support

(0.3,0.3,0.4,−0.1,−0.1,−0.2)

(0.4,0.2,0.4,−0.2,−0.4,−0.5)

(0.3,0.2,0.4,−0.2,−0.3,−0.3)

Warfare activities

(0.1,0.3,0.4,−0.1,−0.2,−0.3)

(0.3,0.3,0.4,−0.1,−0.3,−0.4)

(0.4,0.3,0.3,−0.4,−0.5,−0.5)

Nuclear weapons

(0.1,0.4,0.5,−0.1,−0.2,−0.5)

(0.2,0.2,0.4,−0.1,−0.5,−0.6)

(0.5,0.4,0.4,−0.3,−0.2,−0.4)

Friendship

(0.2,0.3,0.5,−0.1,−0.2,−0.5)

(0.3,0.2,0.4,−0.1,−0.3,−0.5)

(0.2,0.1,0.4,−0.3,−0.3,−0.3)

Respecting religious beliefs

(0.4,0.4,0.4,−0.1,−0.2,−0.4)

(0.4,0.3,0.4,−0.1,−0.4,−0.3)

(0.5,0.2,0.4,−0.2,−0.3,−0.3)

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures

429

Table 10.7 Types of relations between countries Types of relations

(America, India)

(America, Russia)

(America, Pakistan)

Project investment

(0.5,0.2,0.3,−0.1,−0.2,−0.5)

(0.1,0.2,0.4,−0.1,−0.1,−0.4)

(0.4,0.3,0.4,−0.1,−0.2,−0.6)

Political support

(0.6,0.2,0.3,−0.5,−0.2,−0.3)

(0.3,0.1,0.3,−0.2,−0.2,−0.4)

(0.6,0.1,0.1,−0.5,−0.1,−0.0)

Warfare activities

(0.1,0.4,0.1,−0.1,−0.2,−0.5)

(0.4,0.2,0.4,−0.5,−0.2,−0.3)

(0.5,0.1,0.1,−0.1,−0.2,−0.6)

Nuclear weapons

(0.6,0.1,0.0,−0.1,−0.2,−0.5)

(0.2,0.2,0.4,−0.1,−0.1,−0.4)

(0.4,0.1,0.4,−0.1,−0.2,−0.5)

Friendship

(0.4,0.2,0.3,−0.5,−0.2,−0.3)

(0.3,0.2,0.4,−0.1,−0.2,−0.4)

(0.4,0.1,0.3,−0.5,−0.1,−0.1)

Respecting religious beliefs

(0.5,0.2,0.2,−0.5,−0.2,−0.3)

(0.8,0.1,0.1,−0.8,−0.1,−0.1)

(0.2,0.1,0.4,−0.5,−0.2,−0.1)

Table 10.8 Types of relations between countries Types of relations

(India, Russia)

(India, Pakistan)

(India, Afghanistan)

Project investment

(0.4,0.1,0.3,−0.5,−0.1,−0.1)

(0.1,0.2,0.5,−0.1,−0.2,−0.3)

(0.3,0.3,0.5,−0.1,−0.2,−0.3)

Political support

(0.3,0.2,0.4,−0.4,−0.3,−0.3)

(0.0,0.2,0.4,−0.2,−0.2,−0.3)

(0.5,0.1,0.1,−0.4,−0.1,−0.1)

Warfare activities

(0.1,0.3,0.5,−0.1,−0.2,−0.5)

(0.6,0.1,0.0,−0.4,−0.2,−0.3)

(0.4,0.4,0.1,−0.1,−0.2,−0.2)

Nuclear weapons

(0.1,0.2,0.5,−0.5,−0.3,−0.2)

(0.0,0.1,0.5,−0.0,−0.3,−0.3)

(0.4,0.1,0.4,−0.1,−0.2,−0.1)

Friendship

(0.4,0.3,0.2,−0.3,−0.2,−0.4)

(0.1,0.2,0.5,−0.3,−0.2,−0.3)

(0.4,0.4,0.2,−0.4,−0.2,−0.2)

Respecting religious beliefs

(0.4,0.2,0.2,−0.4,−0.2,−0.2)

(0.3,0.1,0.4,−0.5,−0.3,−0.3)

(0.2,0.1,0.4,−0.4,−0.2,−0.2)

Table 10.9 Types of relations between countries Types of relations

(Russia, Pakistan)

(Russia, Afghanistan)

(Russia, China)

Project investment

(0.5,0.2,0.4,−0.1,−0.2,−0.6)

(0.4,0.2,0.1,−0.0,−0.3,−0.8)

(0.2,0.1,0.3,−0.1,−0.2,−0.4)

Political support

(0.4,0.2,0.4,−0.4,−0.2,−0.4)

(0.1,0.2,0.4,−0.3,−0.2,−0.3)

(0.3,0.3,0.3,−0.1,−0.2,−0.4)

Warfare activities

(0.1,0.2,0.4,−0.1,−0.1,−0.6)

(0.5,0.0,0.0,−0.4,−0.2,−0.3)

(0.2,0.1,0.3,−0.1,−0.2,−0.3)

Nuclear weapons

(0.7,0.2,0.1,−0.4,−0.2,−0.4)

(0.4,0.2,0.1,−0.1,−0.2,−0.3)

(0.1,0.1,0.3,−0.1,−0.1,−0.4)

Friendship

(0.4,0.2,0.3,−0.1,−0.2,−0.3)

(0.4,0.2,0.1,−0.1,−0.2,−0.3)

(0.2,0.4,0.3,−0.1,−0.2,−0.3)

Respecting religious beliefs

(0.4,0.2,0.4,−0.4,−0.2,−0.4)

(0.4,0.2,0.1,−0.1,−0.2,−0.3)

(0.6,0.1,0.2,−0.7,−0.2,−0.1)

Table 10.10 Types of relations between countries Types of relations

(Pakistan, China)

(Turkey, Pakistan)

(Pakistan, Bangladesh)

Project investment

(0.4,0.2,0.4,−0.5,−0.1,−0.1)

(0.7,0.1,0.2,−0.5,−0.1,−0.1)

(0.1,0.1,0.4,−0.1,−0.2,−0.3)

Political support

(0.8,0.2,0.1,−0.5,−0.0,−0.0)

(0.4,0.2,0.4,−0.5,−0.2,−0.3)

(0.6,0.1,0.4,−0.4,−0.2,−0.3)

Warfare activities

(0.0,0.0,0.4,−0.−0.0,−0.0,−0.6)(0.1,0.2,0.5,−0.0,−0.1,−0.5)

(0.2,0.1,0.4,−0.1,−0.2,−0.3)

Nuclear weapons

(0.4,0.2,0.3,−0.1,−0.2,−0.2)

(0.1,0.1,0.5,−0.1,−0.2,−0.5)

(0.1,0.1,0.5,−0.1,−0.1,−0.6)

Friendship

(0.8,0.1,0.1,−0.5,−0.1,−0.0)

(0.6,0.2,0.1,−0.5,−0.2,−0.3)

(0.5,0.1,0.3,−0.3,−0.2,−0.3)

Respecting religious beliefs

(0.8,0.2,0.2,−0.4,−0.2,−0.1)

(0.5,0.1,0.1,−0.1,−0.1,−0.1)

(0.7,0.1,0.1,−0.4,−0.1,−0.0)

430

10 Bipolar Neutrosophic Graph Structures

Table 10.11 Types of relations between countries Types of relations

(Afghanistan, China)

(Afghanistan, America)

(Afghanistan, Turkey)

Project investment

(0.1,0.2,0.6,−0.1,−0.2,−0.7)

(0.0,0.1,0.6,−0.0,−0.0,−0.6)

(0.0,0.1,0.6,−0.1,−0.2,−0.6)

Political support

(0.3,0.3,0.5,−0.1,−0.2,−0.8)

(0.0,0.1,0.6,−0.0,−0.0,−0.8)

(0.2,0.1,0.4,−0.1,−0.1,−0.5)

Warfare activities

(0.1,0.2,0.6,−0.1,−0.2,−0.5)

(0.2,0.2,0.3,−0.4,−0.1,−0.0)

(0.1,0.1,0.4,−0.1,−0.2,−0.7)

Nuclear weapons

(0.1,0.4,0.6,−0.1,−0.2,−0.8)

(0.0,0.0,0.6,−0.0,−0.0,−0.7)

(0.0,0.1,0.5,−0.1,−0.1,−0.8)

Friendship

(0.4,0.2,0.6,−0.1,−0.2,−0.4)

(0.0,0.0,0.6,−0.0,−0.2,−0.6)

(0.2,0.1,0.6,−0.1,−0.2,−0.7)

Respecting religious beliefs

(0.4,0.2,0.5,−0.1,−0.2,−0.3)

(0.0,0.2,0.4,−0.1,−0.2,−0.5)

(0.5,0.1,0.2,−0.4,−0.1,−0.1)

Table 10.12 Types of relations between countries Types of relations

(China, India)

(Turkey, Russia)

(Bangladesh, Afghanistan)

Project investment

(0.2,0.1,0.5,−0.1,−0.2,−0.6)

(0.4,0.2,0.5,−0.1,−0.2,−0.6)

(0.2,0.1,0.4,−0.1,−0.2,−0.8)

Political support

(0.1,0.1,0.4,−0.1,−0.2,−0.6)

(0.4,0.3,0.4,−0.3,−0.2,−0.5)

(0.2,0.1,0.4,−0.1,−0.2,−0.3)

Warfare activities

(0.5,0.1,0.2,−0.5,−0.1,−0.1)

(0.5,0.2,0.5,−0.1,−0.2,−0.3)

(0.2,0.1,0.4,−0.1,−0.2,−0.6)

Nuclear weapons

(0.1,0.2,0.5,−0.1,−0.1,−0.6)

(0.4,0.1,0.5,−0.1,−0.2,−0.5)

(0.2,0.1,0.4,−0.1,−0.2,−0.8)

Friendship

(0.1,0.1,0.5,−0.1,−0.1,−0.5)

(0.6,0.2,0.1,−0.4,−0.2,−0.4)

(0.3,0.1,0.4,−0.3,−0.2,−0.3)

Respecting religious beliefs

(0.4,0.3,0.1,−0.5,−0.1,−0.1)

(0.4,0.2,0.1,−0.6,−0.2,−0.1)

(0.5,0.1,0.1,−0.5,−0.2,−0.1)

of ambiguity of that relationship from first and second countries, respectively. Since many relations exist on set X , define six relations on set X as X 1 = Project investment, X 2 = Political support, X 3 = Warfare activities, X 4 = Nuclear weapons, X 5 = Friendship, X 6 = Respecting religious beliefs. Define the relations with those elements such that (X, X 1 , X 2 , X 3 , X 4 , X 5 , X 6 ) is a graph structure. Every element of a relation denotes a highly worthwhile relationship in that pair of countries. Since (X, X 1 , X 2 , X 3 , X 4 , X 5 , X 6 ) is a graph structure, every relation has unique elements, that is, they are present in that relation only. Hence, any pair of countries will be an element of that relation for which its value of T p , I n , and F n are high, and T n , I p , and F p values are comparatively low, according to the data of Tables 10.4, 10.5, 10.6, 10.7, 10.8, 10.9, 10.10, 10.11, 10.12. Write down the T p , I p , F p , T n , I n , and F n values of all elements in the relations, using the data of Tables 10.24, 10.25, 10.26, 10.27, 10.28, 10.29, 10.30, 10.31, 10.32, such that B1 , B2 , B3 , B4 , B5 , B6 are bipolar single-valued neutrosophic sets on X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , respectively. Let X 1 = {(Turkey, Pakistan), (Russia, India)}, X 2 = {(India, Afghanistan), (America, Pakistan), (America, Bangladesh)}, X 3 = {(India, Pakistan), (China, India), (Afghanistan, Pakistan), (America, Afghanistan), (Russia, Afg)}, X 4 = {(Russia, Pakistan), (America, India)}, X 5 = {(Pakistan, China), (Turkey, China)}, X 6 = {(Pakistan, Bangladesh), (America, Russia), (Bangladesh, Turkey), (Turkey, Afghanistan)}. Corresponding bipolar single-valued neutrosophic sets B1 , B2 , B3 , B4 , B5 , B6 , respectively, are B1 = {((T u, Pak), 0.5, 0.1, 0.2, −0.7, −0.1, −0.1), ((Ru, I nd), 0.4, 0.1, 0.3, −0.5, −0.1, −0.1)}, B2 = {((I nd, A f g), 0.4, 0.1, 0.1, −0.5, −0.1, −0.1), ((Am, Pak), 0.6, 0.1, 0.1, −0.5, −0.1, 0.0),

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures

431

((Am, Bang), 0.1, 0.1, 0.2, −0.3, −0.3, −0.4)}, B3 = {((I nd, Pak), 0.4, 0.2, 0.3, −0.6, −0.1, 0.0), ((Ch, I nd), 0.5, 0.1, 0.2, −0.5, −0.1, −0.1), ((A f g, Pak), 0.3, 0.4, 0.6, −0.4, −0.3, −0.3), ((Am, A f g), 0.2, 0.2, 0.3, −0.4, −0.1, 0.0), ((Ru, A f g), 0.4, 0.2, 0.3, −0.5, 0.0, 0.0)}, B4 = {((Ru, Pak), 0.4, 0.2, 0.4, −0.7, −0.2, −0.1), ((Am, I nd), 0.6, 0.1, 0.0, −0.3, −0.2, −0.3)}, B5 = {((Pak, Ch), 0.8, 0.1, 0.1, −0.5, −0.1, 0.0), ((T u, Ch), 0.5, 0.1, 0.2, −0.5, −0.3, −0.3)}, B6 = {((Pak, Bang), 0.4, 0.1, 0.0, −0.7, −0.1, −0.1), ((Am, Ru), 0.8, 0.1, 0.1, −0.8, −0.1, −0.1), ((Bang, T u), 0.4, 0.2, 0.1, −0.7, −0.3, −0.2), ((T u, A f g), 0.4, 0.1, 0.1, −0.5, −0.1, −0.2)}. It is easy to check that (B, B1 , B2 , B3 , B4 , B5 , B6 ) is a bipolar single-valued neutrosophic graph structure as shown in Fig.10.20. In a bipolar single-valued neutrosophic graph structure, represented in Fig. 10.20, an edge depicts the most dominating and attention-getting relationship of contiguous countries. For instance, between Turkey and Pakistan most dominating and attention-getting relationship is project investment; participation of Turkey in this relationship’s strength, weakness, and indeterminacy is 70%, 10%, and 10%, respectively, and participation of Pakistan in this relationship’s strength, weakness, and indeterminacy is 50%, 10%, and 20%, respectively. This bipolar single-valued neutrosophic graph structure represents complete status of a relationship from both of its participants. It is very utilitarian for some particular relationships, like warfare activities. It tells a country that in what percentage its warfare activities are answered from a corresponding country and warns it that in case of war which kind of reaction can appear. It guides the United Nations and other peace loving organizations as to which country is most actively involved in peace destroying relationship, for instance, warfare activities and nuclear weapons, and what is its percentage. For instance, India and Pakistan, America, and Afghanistan are involved in warfare activities and the percentage of these edges shows that India and America are more actively involved in it. So it guides the United Nations Security Council that in what percentage it should pressurize India, America and other countries involved in it. It also highlights those countries whose attention-getting relationship is nuclear weapons and spotlights those countries who are its strong participants than their contiguous countries. For example, America and India have nuclear weapons as their dominating relationship and America is a strong participant, as its participation for its strength is high. So, this bipolar single-valued neutrosophic graph structure indicates that in order to reduce nuclear proliferation, the United Nations should pressurize America and Russia. The general procedure for this application is explained in Algorithm 10.1.

Algorithm 10.1 Computing international relations of countries 1. Begin 2. Input membership values B(u i ) of n number of countries u 1 , u 2 , . . . , u n . 3. Input the adjacency matrix of countries with respect to X 1 , X 2 , . . . , X m mutually disjoint, irreflexive, and symmetric relations. 4. do i from 1 → m 5. do j from 1 → m 6. do k from 1 → n

432

10 Bipolar Neutrosophic Graph Structures

gi ing reli Respect

−0.8, − 0.

1, −0.1)

(0.6, 0.1, 0.0, −0.3, −0.2, −0.3)

rf a

re a

Nu clea r

W

ctiv itie s

ar

fa

re

3) −0. 0.3, .4, − , −0 , 0.6 , 0.4 (0.3

ac tiv i

(0.8 , 0.1 , 0.1 , −0 .5, − 0.1, 0.0)

Re (0.4 specti , 0.2 ng r el , 0.1 , −0 ig iou .7, − s bel 0.3, ief s −0.2 )

Turkey

w ea

(0.4, 0.2, 0 .4,

tie s (0. 4,

0.2 ,

0.3 ,

−0.7 , −0.2 , −0 .1)

−0 .6,

−0 .1, 0.0 ) rf a re a (0.5 ctiv , 0.1 , 0.2 i ti e , −0 .5, − s 0.1, −0.1 )

China

ief s

−0.3)

(0.4, 0.

India

1) 0. ,− t .1 or −0 p , up .5 l s −0 ca 1, iti 0. ol 0.1, P , .4 (0

F riendship

religious bel

.1) , −0 −0.1 ent 0.5, m .3, − I nv est 0 , ct , 0.1 (0.4 P roj e

Wa

, −0.5, −0.3, (0.5, 0.1, 0.2

Respecting

Russia

pon s

Afghanistan

−0.2) 0.5, −0.1, 1, 0.1, −

Wa

rf a

re a

i ctiv

(0 .4 ,0 .2 ,0 .3 ,− 0. 5, 0. 0, 0. 0)

−0 .0)

rt

(0.2, 0.2, 0.3 , −0.4, −0 .1, −0.0)

upp o

−0 .1,

i tic al s

Wa Pakistan

p hi

Respecting r elig ious belief s

W arf are activities

N uclear weapons

P ol

0.1 ,

0.1 ,

4) (0 .6,

−0.3, − 0.

America

t ) en 0.1 m ,− st v e 0.1 In , − .7 ct j e −0 , ro P , 0.2 .1 ,0 .5 (0

Bangladesh

(0.8, 0.1, 0.1,

s

t

ds i en

(0.4, 0.1, 0.0, −0.7, −0.1, −0.1)

2, −0.3,

or

Fr

(0.1, 0. 1, 0.

pp su

−0 .5,

P

it ol

al ic

ef ous beli

ti es

Fig. 10.20 International relations of countries

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18.

do l from k → n p p p p p if (i = j, TX i (u k u l ) > TX j (u k u l ), I X i (u k u l ) < I B j (u k u l ), FBi (u k u l ) p < FB j (u k u l ))then if TBni (u k u l ) < TBnj (u k u l ), I Bni (u k u l ) > I Bn j (u k u l ), FBni (u k u l ) > n FB j (u k u l ))then Label u k u l as Bi end if end if end do end do end do end do p p p TBi , FBi , I Bi values of an edge between two different vertices(countries) a and b show participation of vertex(country)a in strength, weakness, and indeterminacy of most conspicuous relationship X i between them, whereas TBni , FBni , I Bni are corresponding values of vertex(country) b. End

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures

433

10.4.2 Psychological Improvement of Patients A society is an enduring and cooperating social group whose members have developed organized patterns of relationships through interactions with one another. In a society, interaction of its members is very important; this compactness causes all types of relationships among society members. As homo sapiens have complex nature, for grouping of human beings we have to focus many psychological aspects and behaviors. Bipolar single-valued neutrosophic graph structures can be used to judge behavior and other social aspects of any group of society. Bipolar single-valued neutrosophic graph structures can be used to compute conspicuous behavioral property of any two group members with each other. The most frequent occurrence of a particular behavioral property reveals whether this group is good or bad in that perspective. This bipolar single-valued neutrosophic graph structure is very helpful for sub-grouping and assigning of tasks. In hospitals, it is difficult to estimate the psychological behavior of patients and psychiatrist can decide which patient has high violence-based relationship with others. He must reduce his interaction with others, and can decide whether the mental stage of his patients is getting better or worse. It helps a psychiatrist in taking precautionary measures and providing a suitable environment for his patients. He can decide by looking at the bipolar single-valued neutrosophic graph structure as to which patient can properly survive in the society with better behavior and which one is still not capable to show moderate behavior. A psychiatrist can use many types of psychological behaviors to estimate improvement in mental state of his patients including, violent, friendly, jealous, helping, forgiving, communicative, and problem sharing. Consider a set X of eight patients in a mental hospital: X = {Albert, Charles, Burton, Calvert, Christopher, David, Chapman, Joseph}. Consider a bipolar single-valued neutrosophic set B on X as given in Table 10.13. In Table 10.13, T p , F p of a patient indicate his positive and negative thinking for the society, and I p shows indeterminacy/ambiguity of his thinking for society, whereas T n , F n denote his positive and negative reputation in the society and I n represents the percentage of his uncertain reputation. Each pair of patients of set X

Table 10.13 Bipolar neutrosophic information of patients Patients Tp Ip Fp Tn Albert Charles Burton Calvert Christopher David Chapman Joseph

0.4 0.5 0.6 0.7 0.5 0.5 0.4 0.7

0.9 0.8 0.7 0.8 0.7 0.8 0.7 0.6

0.9 0.8 0.7 0.7 0.6 0.7 0.7 0.6

−0.4 −0.6 −0.4 −0.5 −0.4 −0.6 −0.4 −0.5

In

Fn

−0.6 −0.7 −0.6 −0.6 −0.5 −0.5 −0.6 −0.5

−0.9 −0.8 −0.7 −0.6 −0.7 −0.8 −0.8 −0.6

434

10 Bipolar Neutrosophic Graph Structures

Table 10.14 Psychological behavior of Albert with other patients Psychological behaviors

(Albert, charles)

(Albert, Burton)

(Albert, Calvert)

Violent

(0.4,0.4,0.7,−0.2,−0.6,−0.8)

(0.2,0.7,0.8,−0.1,−0.6,−0.9)

(0.2,0.3,0.4,−0.1,−0.2,−0.3)

Friendly

(0.3,0.5,0.6,−0.1,−0.5,−0.5)

(0.3,0.6,0.7,−0.2,−0.5,−0.8)

(0.3,0.4,0.5,−0.2,−0.3,−0.4)

Jealousy

(0.3,0.6,0.7,−0.3,−0.4,−0.6)

(0.4,0.5,0.5,−0.3,−0.4,−0.6)

(0.4,0.5,0.6,−0.3,−0.4,−0.5)

Helping

(0.1,0.5,0.8,−0.2,−0.6,−0.7)

(0.3,0.2,0.6,−0.1,−0.6,−0.7)

(0.2,0.6,0.7,−0.4,−0.5,−0.6)

Forgiving

(0.4,0.3,0.4,−0.4,−0.4,−0.5)

(0.4,0.6,0.8,−0.2,−0.5,−0.6)

(0.3,0.7,0.8,−0.1,−0.6,−0.7)

Problem sharing

(0.3,0.6,0.7,−0.3,−0.5,−0.7)

(0.3,0.7,0.7,−0.3,−0.4,−0.9)

(0.4,0.8,0.9,−0.2,−0.2,−0.8)

Communicative

(0.2,0.7,0.6,−0.4,−0.4,−0.8)

(0.2,0.6,0.6,−0.1,−0.6,−0.8)

(0.1,0.1,0.4,−0.3,−0.3,−0.8)

Table 10.15 Psychological behavior of Charles with other patients Psychological behaviors

(Charles, Burton)

(Charles, Calvert)

(Charles, Christopher)

Violent

(0.3,0.7,0.8,−0.1,−0.6,−0.8)

(0.4,0.8,0.7,−0.1,−0.6,−0.8)

(0.2,0.7,0.8,−0.4,−0.5,−0.3)

Friendly

(0.2,0.6,0.7,−0.2,−0.5,−0.7)

(0.3,0.7,0.8,−0.2,−0.5,−0.7)

(0.3,0.6,0.4,−0.3,−0.4,−0.4)

Jealousy

(0.1,0.5,0.6,−0.3,−0.4,−0.6)

(0.2,0.2,0.6,−0.3,−0.4,−0.6)

(0.4,0.5,0.5,−0.2,−0.3,−0.5)

Helping

(0.5,0.4,0.5,−0.4,−0.4,−0.6)

(0.1,0.2,0.5,−0.4,−0.3,−0.5)

(0.5,0.4,0.6,−0.1,−0.2,−0.6)

Forgiving

(0.4,0.4,0.5,−0.1,−0.6,−0.8)

(0.4,0.2,0.4,−0.1,−0.2,−0.4)

(0.2,0.3,0.7,−0.4,−0.4,−0.7)

Problem sharing

(0.5,0.6,0.6,−0.3,−0.5,−0.7)

(0.5,0.3,0.4,−0.5,−0.2,−0.4)

(0.3,0.5,0.4,−0.1,−0.3,−0.4)

Communicative

(0.2,0.6,0.7,−0.2,−0.4,−0.6)

(0.4,0.4,0.5,−0.3,−0.3,−0.5)

(0.4,0.4,0.4,−0.1,−0.2,−0.3)

Table 10.16 Psychological behavior of Burton with other patients Psychological behaviors

(Burton, Calvert)

(Burton, Christopher)

(Burton, David)

Violent

(0.1,0.6,0.7,−0.1,−0.4,−0.7)

(0.4,0.7,0.2,−0.1,−0.5,−0.4)

(0.3,0.7,0.3,−0.1,−0.4,−0.8)

Friendly

(0.4,0.4,0.3,−0.4,−0.2,−0.3)

(0.2,0.6,0.3,−0.2,−0.4,−0.6)

(0.4,0.6,0.4,−0.2,−0.5,−0.7)

Jealousy

(0.2,0.2,0.6,−0.2,−0.5,−0.6)

(0.3,0.5,0.4,−0.1,−0.3,−0.7)

(0.3,0.5,0.5,−0.3,−0.3,−0.6)

Helping

(0.3,0.2,0.5,−0.3,−0.2,−0.5)

(0.5,0.2,0.2,−0.4,−0.3,−0.3)

(0.4,0.4,0.6,−0.1,−0.4,−0.8)

Forgiving

(0.4,0.2,0.6,−0.4,−0.3,−0.4)

(0.4,0.4,0.5,−0.3,−0.5,−0.4)

(0.4,0.3,0.7,−0.3,−0.5,−0.7)

Problem sharing

(0.1,0.5,0.4,−0.1,−0.4,−0.3)

(0.2,0.3,0.6,−0.1,−0.4,−0.6)

(0.5,0.1,0.3,−0.4,−0.3,−0.6)

Communicative

(0.2,0.6,0.7,−0.2,−0.5,−0.4)

(0.3,0.3,0.7,−0.3,−0.3,−0.7)

(0.4,0.2,0.4,−0.3,−0.4,−0.7)

has T p , I p , F p , T n , I n , and F n values of different psychological behaviors in Tables 10.14, 10.15, 10.16, 10.17, 10.18, 10.19, 10.20, 10.21, 10.22. Each pair of patients T p , T n of any particular Psychological behavior indicates the percentage of strength of that behavior of the first patient and the second patient, respectively. Similarly, F p , F n represent the percentage of weakness and I p , I n demonstrate the percentage of ambiguity of that behavior of first and second patients, respectively. Consider seven relations on set X as X 1 = Violent, X 2 = Friendly, X 3 = Jealousy, X 4 = Helping, X 5 = Forgiving, X 6 = Problem sharing, X 7 = Communicative. Define the relations with those elements such that (X, X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 ) is a graph structure. Every element of the relation denotes a conspicuous psychological behavior of that pair of patients. Since (X, X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 ) is a graph structure, every relation has unique elements, that is, they are present in that relation only. Hence, any pair of patients

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures

435

Table 10.17 Psychological behavior of Calvert with other patients Psychological behaviors

(Calvert, Christopher)

(Calvert, David)

(Calvert, Chapman)

Violent

(0.1,0.7,0.4,−0.1,−0.2,−0.7)

(0.4,0.8,0.5,−0.1,−0.4,−0.8)

(0.2,0.7,0.4,−0.1,−0.6,−0.8)

Friendly

(0.4,0.4,0.3,−0.4,−0.2,−0.3)

(0.3,0.6,0.4,−0.4,−0.3,−0.7)

(0.4,0.6,0.4,−0.2,−0.5,−0.7)

Jealousy

(0.2,0.6,0.3,−0.2,−0.4,−0.6)

(0.4,0.7,0.3,−0.5,−0.2,−0.6)

(0.2,0.5,0.5,−0.3,−0.4,−0.6)

Helping

(0.3,0.5,0.4,−0.1,−0.2,−0.3)

(0.5,0.5,0.2,−0.5,−0.5,−0.5)

(0.4,0.4,0.6,−0.3,−0.3,−0.5)

Forgiving

(0.4,0.4,0.5,−0.3,−0.4,−0.5)

(0.4,0.4,0.1,v4,−0.4,−0.4)

(0.4,0.2,0.3,−0.4,−0.1,−0.1)

Problem sharing

(0.1,0.7,0.6,−0.1,−0.5,−0.3)

(0.3,0.6,0.6,−0.3,−0.3,−0.2)

(0.3,0.3,0.6,−0.1,−0.2,−0.4)

Communicative

(0.2,0.6,0.7,−0.2,−0.2,−0.4)

(0.2,0.7,0.7,−0.1,−0.2,−0.3)

(0.2,0.2,0.7,−0.2,−0.3,−0.3)

Table 10.18 Psychological behavior of Christopher with other patients Psychological behaviors

(Christopher, David)

(Christopher, Chapman)

Violent

(0.1,0.2,0.4,−0.1,−0.2,−0.3)

(0.4,0.2,0.1,−0.1,−0.2,−0.3)

(Christopher, Joseph) (0.2,0.1,0.4,−0.1,−0.2,−0.7)

Friendly

(0.2,0.2,0.3,−0.2,−0.3,−0.4)

(0.2,0.7,0.2,−0.2,−0.4,−0.4)

(0.5,0.6,0.3,−0.2,−0.5,−0.6)

Jealousy

(0.1,0.4,0.4,−0.3,−0.4,−0.3)

(0.4,0.6,0.3,−0.1,−0.3,−0.5)

(0.2,0.1,0.5,−0.1,−0.2,−0.5)

Helping

(0.3,0.5,0.6,−0.4,−0.2,−0.5)

(0.3,0.5,0.4,−0.3,−0.2,−0.6)

(0.3,0.5,0.4,−0.3,−0.3,−0.3)

Forgiving

(0.1,0.6,0.5,−0.3,−0.2,−0.6)

(0.4,0.4,0.5,−0.1,−0.4,−0.5)

(0.2,0.1,0.4,−0.1,−0.4,−0.4)

Problem sharing

(0.4,0.7,0.4,−0.4,−0.3,−0.7)

(0.3,0.3,0.6,−0.2,−0.3,−0.5)

(0.4,0.4,0.4,−0.2,−0.5,−0.5)

Communicative

(0.5,0.2,0.3,−0.1,−0.2,−0.3)

(0.4,0.2,0.7,−0.3,−0.2,−0.3)

(0.2,0.1,0.6,−0.4,−0.2,−0.3)

Table 10.19 Psychological behavior of David with other patients Psychological behaviors

(David, Chapman)

(David, Joseph)

(David, Albert)

Violent

(0.1,0.7,0.6,−0.2,−0.3,−0.6)

(0.4,0.6,0.1,−0.1,−0.2,−0.2)

(0.4,0.2,0.1,−0.3,−0.1,−0.3)

Friendly

(0.3,0.6,0.7,−0.1,−0.4,−0.7)

(0.3,0.2,0.2,−0.2,−0.3,−0.3)

(0.2,0.2,0.9,−0.1,−0.5,−0.8)

Jealousy

(0.1,0.5,0.6,−0.3,−0.5,−0.8)

(0.4,0.5,0.3,−0.3,−0.4,−0.4)

(0.3,0.3,0.4,−0.1,−0.4,−0.7)

Helping

(0.2,0.7,0.7,−0.1,−0.3,−0.6)

(0.5,0.4,0.4,−0.4,−0.5,−0.5)

(0.2,0.4,0.5,−0.1,−0.3,−0.6)

Forgiving

(0.1,0.6,0.6,−0.2,−0.4,−0.7)

(0.4,0.2,0.5,−0.5,−0.2,−0.6)

(0.1,0.5,0.6,−0.1,−0.2,−0.5)

Problem sharing

(0.1,0.7,0.3,−0.1,−0.5,−0.8)

(0.3,0.6,0.1,−0.1,−0.3,−0.7)

(0.2,0.6,0.7,−0.2,−0.4,−0.4)

Communicative

(0.3,0.5,0.6,−0.4,−0.3,−0.6)

(0.2,0.7,0.1,−0.2,−0.4,−0.8)

(0.3,0.7,0.8,−0.1,−0.2,−0.3)

Table 10.20 Psychological behavior of Chapman with other patients Psychological behaviors

(Albert, Chapman)

(Chapman, Charles)

(Chapman, Burton)

Violent

(0.4,0.1,0.1,−0.2,−0.3,−0.5)

(0.2,0.7,0.5,−0.1,−0.4,−0.7)

(0.2,0.1,0.7,−0.1,−0.2,−0.4)

Friendly

(0.2,0.7,0.4,−0.3,−0.5,−0.9)

(0.1,0.6,0.4,−0.2,−0.3,−0.8)

(0.3,0.2,0.6,−0.3,−0.3,−0.5)

Jealousy

(0.1,0.6,0.5,−0.4,−0.6,−0.8)

(0.3,0.1,0.2,−0.3,−0.2,−0.4)

(0.4,0.3,0.5,−0.1,−0.2,−0.6)

Helping

(0.4,0.5,0.6,−0.2,−0.3,−0.7)

(0.2,0.5,0.5,−0.1,−0.4,−0.5)

(0.2,0.4,0.4,−0.4,−0.4,−0.3)

Forgiving

(0.3,0.4,0.7,−0.3,−0.4,−0.6)

(0.1,0.4,0.6,−0.−0.2,−0.5,−0.6)(0.3,0.5,0.3,−0.1,−0.2,−0.7)

Problem sharing

(0.2,0.3,0.8,−0.−0.4,−0.5,−0.5)(0.2,0.3,0.7,−0.3,−0.6,−0.7)

(0.4,0.2,0.4,−0.3,−0.6,−0.3)

Communicative

(0.1,0.2,0.9,−0.2,−0.6,−0.6)

(0.2,0.7,0.4,−0.1,−0.2,−0.8)

(0.1,0.1,0.8,−0.1,−0.2,−0.8)

436

10 Bipolar Neutrosophic Graph Structures

Table 10.21 Psychological behavior of Joseph with other patients Psychological behaviors

(Joseph, Albert)

(Joseph, Charles)

(Joseph, Calvert)

Violent

(0.1,0.5,0.9,−0.4,−0.2,−0.8)

(0.4,0.3,0.8,−0.3,−0.2,−0.3)

(0.7,0.3,0.4,−0.3,−0.2,−0.1)

Friendly

(0.4,0.2,0.8,−0.2,−0.4,−0.7)

(0.3,0.2,0.7,−0.2,−0.3,−0.5)

(0.2,0.1,0.5,−0.2,−0.3,−0.3)

Jealousy

(0.1,0.6,0.7,−0.1,−0.2,−0.3)

(0.4,0.6,0.6,−0.1,−0.4,−0.6)

(0.5,0.5,0.6,−0.1,−0.2,−0.2)

Helping

(0.2,0.2,0.3,−0.4,−0.3,−0.6)

(0.2,0.5,0.5,−0.4,−0.2,−0.3)

(0.2,0.4,0.4,−0.4,−0.4,−0.3)

Forgiving

(0.1,0.5,0.6,−0.3,−0.2,−0.5)

(0.5,0.4,0.4,−0.3,−0.3,−0.7)

(0.4,0.3,0.5,−0.3,−0.2,−0.4)

Problem sharing

(0.3,0.4,0.3,−0.2,−0.5,−0.4)

(0.3,0.3,0.1,−0.2,−0.2,−0.3)

(0.6,0.2,0.6,−0.2,−0.4,−0.5)

Communicative

(0.1,0.2,0.5,−0.1,−0.2,−0.3)

(0.4,0.2,0.3,−0.1,−0.4,−0.8)

(0.7,0.1,0.7,−0.1,−0.5,−0.6)

Table 10.22 Various psychological behaviors of patients Psychological behaviors

(Albert, Christopher)

(Charles, David)

(Joseph, Burton)

Violent

(0.4,0.1,0.0,−0.3,−0.2,−0.3)

(0.5,0.1,0.1,−0.6,−0.1,−0.0)

(0.2,0.6,0.4,−0.1,−0.4,−0.7)

Friendly

(0.1,0.2,0.3,−0.2,−0.3,−0.4)

(0.4,0.2,0.2,−0.5,−0.2,−0.6)

(0.4,0.1,0.7,−0.2,−0.2,−0.3)

Jealousy

(0.3,0.4,0.4,−0.3,−0.2,−0.3)

(0.5,0.5,0.5,−0.4,−0.3,−0.3)

(0.5,0.5,0.4,−0.1,−0.5,−0.4)

Helping

(0.2,0.2,0.5,−0.2,−0.5,−0.5)

(0.4,0.4,0.4,−0.3,−0.4,−0.5)

(0.4,0.1,0.6,−0.3,−0.2,−0.3)

Forgiving

(0.1,0.5,0.6,−0.1,−0.4,−0.4)

(0.3,0.6,0.3,−0.2,−0.2,−0.3)

(0.3,0.4,0.5,−0.1,−0.4,−0.5)

Problem sharing

(0.1,0.2,0.7,−0.2,−0.3,−0.5)

(0.5,0.7,0.2,−0.1,−0.5,−0.4)

(0.2,0.6,0.4,−0.3,−0.2,−0.3)

Communicative

(0.1,0.6,0.3,−0.1,−0.2,−0.4)

(0.4,0.2,0.1,−0.1,−0.2,−0.3)

(0.6,0.2,0.3,−0.4,−0.1,−0.2)

will be an element of that particular relation, for which its values of T p , I n , and F n are high, and T n , I p , and F p values are comparatively low, according to the data of Tables 10.14, 10.15, 10.16, 10.17, 10.18, 10.19, 10.20, 10.21, 10.22. Write down the T p , I p , F p , T n , I n , and F n values of all elements in the relations using Tables 10.14, 10.15, 10.16, 10.17, 10.18, 10.19, 10.20, 10.21, 10.22 such that B1 , B2 , B3 , B4 , B5 , B6 , B7 are bipolar single-valued neutrosophic sets on the relations X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 , respectively. Let X 1 = {(Albert, Chapman), (Albert, Christopher), (Calvert, Albert), (David, Albert)}, X 2 = {(Calvert, Christopher)}, X 3 = {(Charles, Chapman), (Burton, Albert)}, X 4 = {(Charles, Bur ton), (Bur ton, Christopher )}, X 5 = {(Charles, Albert), (Calvert, Chapman)}, X 6 = {(Burton, David), (Chapman, J oseph), (Calver t, Charles)}, X 7 = {(Joseph, Burton), (Chapman, David)}. Corresponding bipolar single-valued neutrosophic sets B1 , B2 , B3 , B4 , B5 , B6 , B7 , respectively, are B1 = {((Albert, Chapman), 0.4, 0.1, 0.1, −0.2, −0.3, −0.5), ((Albert, Christopher), 0.4, 0.1, 0.0, −0.3, −0.2, −0.3), ((Calver t, Alber t), 0.5, 0.1, 0.1, −0.4, −0.2, −0.1), ((David, Alber t), 0.4, 0.2, 0.1, −0.3, −0.1, −0.3)}, B2 = {((Calvert, Christopher), 0.4, 0.4, 0.3, −0.4, −0.2, −0.3)}, B3 = {((Charles, Chapman), 0.3, 0.1, 0.2, −0.3, −0.2, −0.4), ((Burton, Albert), 0.4, 0.5, 0.5, −0.3, −0.4, −0.6)},

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures P roblem sharing

8) −0. 0.5, .4, − 0 − , , 0.7 , 0.6 (0.4

−0 .5)

ng

−0 .4,

F or givi

−0 .4, 0.4 ,

Chapman

(0.4, 0. 1, 0.

0, −0.3,

−0.2, − 0.

m

−0 .3,

un

ic

at

−0 .5,

iv

e

−0 .6)

t

David

Vio

len

al o us y

ng l pi

Je

He

icat

len

m

J ealousy

ive

(0.6, 0.2, 0.3, −0.4, −0.1, −0.2)

−0 .6) −0 .3,

Calvert

(0 .4,

0.5 ,

0.5 ,

t

1) 0. − 1, . 0 − 4, 0. − 3, . ,0 .2 ,0 .4 (0

−0 .4,

Pr obl em 0.3 , 0. 4, − shari ng 0.5 ,− 0.2 ,− 0.4 )

6) 0. − 4, 0. − 4, 0. ,− .5 ,0 .4 ,0 .5 (0

mun

0.6 ,

3) Vi o

(0.3, 0.1, 0.2, −0.3, −0.2, −0.4)

Joseph

0.3 ,

(0.5 , 0.1 , 0.1 , −0 .6, − 0.1, 0.0)

Charles

Com

(0. 4,

(0.4, 0.2, 0.1, −0.3, −0.1, −0.3) Violent

0.3 , (0. 4,

Co

Violent (0.4, 0.1, 0.1, −0.2, −0.3, −0.5)

Albert

(0.5 ,

437

Burton H elpin g

Fr

i en dl y

4, 0. (0.4, 0.

0.2, (0.5, 0.2,

3,

0. −0.4, −

3) 0.2, −0. −0.4, −

3, −0.3)

Christopher 6) 0. − 3, . 0 − 4, 0. − , 3 0. 1, 0. 5, . (0

P roblem sharing

Fig. 10.21 Psychological behaviors of patients in a Mental Hospital

B4 = {((Charles, Burton), 0.5, 0.4, 0.5, −0.4, −0.4, −0.6), ((Burton, Christopher), 0.5, 0.2, 0.2, −0.4, −0.3, −0.3)}, B5 = {((Charles, Albert), 0.4, 0.3, 0.4, −0.4, −0.4, −0.5), ((Calvert, Chapman), 0.4, 0.2, 0.3, −0.4, −0.1, −0.1)}, B6 = {((Burton, David), 0.5, 0.1, 0.3, −0.4, −0.3, −0.6)((Chapman, Joseph), 0.4, 0.6, 0.7, −0.4, −0.5, −0.8), ((Calvert, Charles), 0.5, 0.3, 0.4, −0.5, −0.2, −0.4)}.

It is easy to check that (B, B1 , B2 , B3 , B4 , B5 , B6 ) is a bipolar single-valued neutrosophic graph structure as depicted in Fig.10.21. In a bipolar single-valued neutrosophic graph structure, depicted in Fig.10.21, every edge depicts the most conspicuous behavioral property of adjacent vertices(patients). For instance, the most conspicuous behavioral property between Charles and Burton is helping, Charles’ participation in its strength, weakness, and indeterminacy is 50%, 40%, and 50%, respectively, and Burton’s participation in its strength, weakness, and indeterminacy is 40%, 40%, and 60%, respectively. This bipolar single-valued neutrosophic graph structure also accentuates that vertex Albert has the highest degree for violent behaviour. This points out that Albert is the most violent mental patient among these eight patients. So, his psychiatrist must focus on Albert’s violent behavior, as Albert’s most violent behavior is with Chapman and his

438

10 Bipolar Neutrosophic Graph Structures

positive behavior is with just one person, that is, Charles. So the psychiatrist should try to minimize his interaction with Chapman to avoid his most violent behavior and should maximize his interaction with Charles to improve his good psychological behavior. Similarly, a psychiatrist can judge the behavior of all his patients and can take better decisions for their well-being. He can make two bipolar single-valued neutrosophic graph structures with a gap of some time period and can estimate their behavioral improvement by comparing those two bipolar single-valued neutrosophic graph structures; it also helps him in continuing or canceling his decisions for his patients. The general procedure regarding the psychological behavior of patients is explained in Algorithm 10.2. Algorithm 10.2 Computing psychological behavior of patients 1. Begin 2. Input membership values B(u i ) of n number of patients u 1 , u 2 , . . . , u n in a hospital. 3. Input the adjacency matrix of countries with respect to X 1 , X 2 , . . . , X m mutually disjoint, irreflexive, and symmetric psychological behaviors. 4. Consider two patients u k and u l , 1 ≤ k ≤ n, k ≤ l ≤ n. p p p p p p 5. If TX i (u k u l ) > TX j (u k u l ), I X i (u k u l ) < I X j (u k u l ), FX i (u k u l ) < FX j (u k u l ), TXni (u k u l ) < TXn j (u k u l ), I Xn i (u k u l ) > I Xn j (u k u l ), FXn i (u k u l ) > FXn j (u k u l ), i = j, 1 ≤ i, j ≤ m then label u k u l as X i . p p p 6. TX i , FX i , I X i values of an edge between two different vertices(patients) u k and u l show the participation of vertex(patient) u k in strength, weakness, and indeterminacy of the most conspicuous psychological behavior X i between them, whereas TXni , FXn i , I Xn i are corresponding values of vertex(patient) u l . 7. End

10.4.3 Uncovering the Undercover Reasons of Global Terrorism Terrorism is the process of using violent or threat of violent to purport a religious, ideological, or political change. It is committed by some non-state actors and may be undercover personnel servings on the behalf of respective governments of non-state actors. Terrorism achieves more than immediate target victims and is also directed at targets covering a large spectrum of society. It is considered as fourth-generation warfare and violent crime. Nowadays, terrorism is a major threat to the society and is illegal under antiterrorism laws in all jurisdictions. It is considered as a war crime under laws of war, when it is used to target the non-combatants, including neutral military personnel, civilians, or enemy prisoners of the war. Many political organizations have practiced terrorism to achieve their objectives. It is being practiced by both left-wing and rightwing political organizations, religious groups, nationalist groups, revolutionaries, and

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures Table 10.23 Bipolar neutrosophic information of terrorism in various countries Country Tp Ip Fp Tn In Pakistan India Afghanistan USA Iraq Israel Palestine Kashmir

0.7 0.7 0.8 0.4 0.5 0.8 0.4 0.6

0.6 0.6 0.4 0.4 0.4 0.4 0.5 0.4

0.5 0.5 0.3 0.6 0.5 0.3 0.7 0.8

−0.8 −0.6 −0.9 −0.3 −0.6 −0.7 −0.5 −0.7

−0.4 −0.5 −0.3 −0.2 −0.3 −0.5 −0.6 −0.6

439

Fn −0.3 −0.6 −0.2 −0.7 −0.4 −0.5 −0.6 −0.6

ruling governments. Terrorism can be used to exploit human fear to achieve these goals. According to Global Terrorism Database, non-state terrorism incidents are more than 61,000, claiming more than 140,000 lives from 2000 to 2014. Nowadays, many countries are blamed to be responsible for global terrorism. That’s why many countries are doing war against terror. When a country blames other country to be responsible for terrorism in its territory, many factors are responsible for it. But there are some reasons or factors which are strongly responsible for that particular incident and can not be ignored. A bipolar single-valued neutrosophic graph structure can be used to highlight those factors which are highly responsible for terrorism between any two countries or global terrorism due to those countries. There are many undercover reasons highly responsible for global terrorism, including religious factors, military interference/drone attacks, political benefits, to pressurize other governments, to occupy other countries, to obtain benefits of powerful countries, and to revenge past political occupation. Let X be the set of eight countries/states on the globe which are either victims of terrorism or are said to be responsible for global terrorism: X = {Pakistan, India, Afghanistan, USA, Iraq, Israel, Palestine, Kashmir}. Consider a bipolar single-valued neutrosophic set B on X as defined in Table 10.23. In Table 10.23, T p , T n of a country depict the percentage of its actual involvement in terrorism and the percentage of its involvement on the basis of other countries’ point of view, respectively, whereas F p , F n of a country represent the percentage of its actual non-involvement and of reputation of its non-involvement, respectively. Similarly, I p , I n indicate ambiguity in its involvement and reputation with respect to terrorism. Every pair of countries in set X has T p , I p , F p , T n , I n , and F n values of different undercover reasons in Tables 10.24, 10.25, 10.26, 10.27, 10.28, 10.29, 10.30, 10.31, 10.32. Each pair of countries T p , T n of any terrorism reason shows the percentage of responsibility from the first country and the second country, respectively. Similarly F p , F n indicate the percentage of being not responsible and I p , I n denote the percentage of ambiguity of being responsible for that terrorism reason from the first and the second countries, respectively. There are many relations on the set X ; define six relations on set X as X 1 = Religious factors, X 2 = Political benefits, X 3 = To pres-

440

10 Bipolar Neutrosophic Graph Structures

Table 10.24 Reasons of terrorism due to relationships of Pakistan with other countries Terrorism Reasons

(India, Pakistan)

(Pakistan, Afghanistan)

(USA, Pakistan)

Religious factors

(0.5,0.3,0.3,−0.4,−0.4,−0.4)

(0.3,0.2,0.2,−0.4,−0.2,−0.3)

(0.2,0.2,0.4,−0.2,−0.2,−0.6)

Political benefits

(0.6,0.3,0.3,−0.4,−0.3,−0.4)

(0.3,0.3,0.3,−0.4,−0.3,−0.3)

(0.3,0.3,0.3,−0.1,−0.1,−0.6)

To pressurize other govt.

(0.7,0.2,0.3,−0.5,−0.1,−0.4)

(0.1,0.3,0.5,−0.3,−0.2,−0.2)

(0.3,0.3,0.2,0.0,−0.2,−0.7)

To occupy other country

(0.5,0.2,0.3,−0.2,−0.2,−0.6)

(0.1,0.2,0.4,−0.1,−0.3,−0.3)

(0.3,0.4,0.4,0.0,0.0,−0.7)

Benefits of powerful countries (0.4,0.4,0.4,−0.4,−0.4,−0.4)

(0.6,0.2,0.3,−0.5,−0.2,−0.2)

(0.2,0.2,0.3,−0.1,−0.2,−0.6)

Military interference/Drones

(0.4,0.1,0.5,−0.4,−0.2,−0.3)

(0.4,0.2,0.2,−0.2,−0.1,−0.6)

(0.3,0.4,0.3,−0.1,−0.1,−0.5)

Table 10.25 Reasons of terrorism due to relationships of India with other countries Terrorism Reasons

(India, Afghanistan)

(India, USA)

(India, Iraq)

Religious factors

(0.4,0.3,0.4,−0.3,−0.2,−0.3)

(0.2,0.2,0.5,−0.4,−0.2,−0.5)

(0.4,0.1,0.4,−0.5,−0.2,−0.5)

Political benefits

(0.7,0.3,0.2,−0.4,−0.2,−0.2)

(0.3,0.4,0.3,−0.5,−0.2,−0.4)

(0.2,0.4,0.3,−0.1,−0.3,−0.5)

To pressurize other govt.

(0.3,0.4,0.4,−0.1,−0.3,−0.3)

(0.1,0.2,0.4,−0.3,−0.2,−0.4)

(0.2,0.4,0.5,−0.3,−0.2,−0.4)

To occupy other country

(0.4,0.3,0.4,−0.2,−0.2,−0.3)

(0.1,0.1,0.5,−0.3,−0.1,−0.4)

(0.2,0.1,0.4,−0.1,−0.3,−0.5)

Benefits of powerful countries (0.5,0.4,0.4,−0.3,−0.2,−0.3)

(0.4,0.3,0.4,−0.2,−0.1,−0.5)

(0.2,0.3,0.5,−0.3,−0.2,−0.3)

Military interference/Drones

(0.2,0.2,0.1,−0.3,−0.2,−0.4)

(0.2,0.1,0.4,−0.2,−0.3,−0.5)

(0.3,0.3,0.5,−0.3,−0.2,−0.2)

Table 10.26 Reasons of terrorism due to relationships of Afghanistan with other countries Terrorism Reasons

(USA, Afghanistan)

(Iraq, Afghanistan)

(Afghanistan, Israel)

Religious factors

(0.1,0.4,0.6,−0.2,−0.2,−0.5)

(0.3,0.4,0.5,−0.1,−0.3,−0.4)

(0.5,0.1,0.5,−0.3,−0.2,−0.3)

Political benefits

(0.3,0.3,0.3,−0.1,−0.2,−0.6)

(0.4,0.2,0.4,−0.2,−0.2,−0.4)

(0.3,0.2,0.4,−0.2,−0.3,−0.3)

To pressurize other govt.

(0.2,0.3,0.3,−0.1,−0.2,−0.6)

(0.3,0.3,0.4,−0.1,−0.3,−0.3)

(0.4,0.3,0.3,−0.4,−0.3,−0.5)

To occupy other country

(0.3,0.4,0.2,0.0,−0.2,−0.5)

(0.2,0.2,0.4,−0.1,−0.3,−0.4)

(0.3,0.4,0.4,−0.3,−0.2,−0.4)

Benefits of powerful countries (0.2,0.3,0.4,−0.1,−0.2,−0.5)

(0.5,0.2,0.3,−0.6,−0.2,−0.2)

(0.2,0.1,0.4,−0.3,−0.3,−0.3)

Military interference/Drones

(0.1,0.3,0.4,−0.1,−0.3,−0.4)

(0.4,0.2,0.4,−0.2,−0.3,−0.3)

(0.4,0.2,0.1,−0.3,−0.2,−0.5)

Table 10.27 Reasons of terrorism due to relationships of USA with other countries Terrorism Reasons

(USA, Kashmir)

(USA, Iraq)

(USA, Palestine)

Religious factors

(0.3,0.2,0.5,−0.3,−0.2,−0.5)

(0.1,0.4,0.6,−0.1,−0.1,−0.5)

(0.4,0.3,0.5,−0.2,−0.2,−0.6)

Political benefits

(0.4,0.2,0.4,−0.3,−0.1,−0.5)

(0.4,0.3,0.3,−0.2,−0.1,−0.5)

(0.4,0.1,0.3,−0.3,−0.1,−0.5)

To pressurize other govt.

(0.1,0.3,0.5,−0.1,−0.2,−0.6)

(0.4,0.4,0.4,−0.2,−0.2,−0.6)

(0.3,0.1,0.3,−0.3,−0.2,−0.6)

To occupy other country

(0.2,0.3,0.5,−0.1,−0.2,−0.6)

(0.2,0.3,0.4,−0.1,−0.1,−0.7)

(0.4,0.1,0.4,−0.1,−0.2,−0.5)

Benefits of powerful countries (0.3,0.2,0.4,−0.2,−0.2,−0.5)

(0.3,0.4,0.4,−0.1,−0.2,−0.5)

(0.4,0.1,0.5,−0.3,−0.1,−0.1)

Military interference/Drones

(0.2,0.3,0.3,−0.1,−0.1,−0.6)

(0.2,0.1,0.4,−0.3,−0.2,−0.7)

(0.1,0.2,0.5,−0.2,−0.2,−0.7)

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures

441

Table 10.28 Reasons of terrorism due to relationship of Iraq with other countries Terrorism Reasons

(Iraq, Israel)

(Iraq, Palestine)

(Iraq, Kashmir)

Religious factors

(0.4,0.1,0.3,−0.5,−0.1,−0.1)

(0.4,0.2,0.5,−0.1,−0.2,−0.3)

(0.3,0.3,0.3,−0.1,−0.2,−0.3)

Political benefits

(0.3,0.2,0.4,−0.4,−0.3,−0.3)

(0.3,0.2,0.4,−0.2,−0.2,−0.3)

(0.2,0.1,0.1,−0.4,−0.1,−0.1)

To pressurize other govt.

(0.1,0.3,0.5,−0.1,−0.2,−0.5)

(0.2,0.1,0.6,−0.2,−0.2,−0.6)

(0.4,0.4,0.5,−0.1,−0.2,−0.6)

To occupy other country

(0.1,0.2,0.5,−0.5,−0.3,−0.2)

(0.1,0.1,0.5,0.0,−0.3,−0.5)

(0.4,0.1,0.6,−0.1,−0.2,−0.5)

Benefits of powerful countries (0.4,0.3,0.2,−0.3,−0.2,−0.4)

(0.1,0.2,0.5,−0.3,−0.2,−0.3)

(0.4,0.4,0.2,−0.4,−0.2,−0.2)

Military interference/Drones

(0.3,0.1,0.7,−0.3,−0.3,−0.6)

(0.2,0.1,0.7,−0.1,−0.2,−0.6)

(0.4,0.2,0.2,−0.4,−0.2,−0.2)

Table 10.29 Reasons of terrorism due to relationship of Israel with other countries Terrorism Reasons

(Israel, Palestine)

(Israel, Kashmir)

(Israel, Pakistan)

Religious factors

(0.3,0.2,0.5,−0.1,−0.2,−0.6)

(0.4,0.2,0.1,−0.4,−0.3,−0.3)

(0.2,0.1,0.3,−0.1,−0.2,−0.4)

Political benefits

(0.4,0.2,0.2,0.0,−0.2,−0.5)

(0.3,0.2,0.4,−0.3,−0.2,−0.3)

(0.3,0.3,0.3,−0.1,−0.2,−0.4)

To pressurize other govt.

(0.3,0.2,0.4,0.0,−0.1,−0.6)

(0.5,0.0,0.3,−0.3,−0.2,−0.3)

(0.2,0.1,0.3,−0.1,−0.2,−0.5)

To occupy other country

(0.4,0.0,0.1,−0.1,0.0,−0.5)

(0.4,0.2,0.4,−0.1,−0.2,−0.6)

(0.1,0.1,0.3,−0.1,−0.1,−0.4)

Benefits of powerful countries (0.3,0.2,0.3,−0.1,−0.2,−0.6)

(0.4,0.2,0.1,−0.3,−0.2,−0.3)

(0.2,0.4,0.3,−0.1,−0.2,−0.3)

Military interference/Drones

(0.4,0.2,0.5,−0.1,−0.2,−0.6)

(0.2,0.1,0.2,−0.1,−0.2,−0.5)

(0.3,0.2,0.1,−0.1,−0.2,−0.6)

Table 10.30 Reasons of terrorism due to relationship of Palestine with other countries Terrorism Reasons

(Palestine, India)

(Palestine, Pakistan)

(Palestine, Afghanistan)

Religious factors

(0.4,0.2,0.4,−0.4,−0.1,−0.6)

(0.4,0.1,0.2,−0.5,−0.1,−0.1)

(0.4,0.1,0.4,−0.5,−0.2,−0.3)

Political benefits

(0.4,0.2,0.5,−0.3,0.0,−0.6)

(0.4,0.2,0.4,−0.2,−0.2,−0.3)

(0.3,0.1,0.4,−0.2,−0.2,−0.3)

To pressurize other govt.

(0.0,0.0,0.4,0.0,0.0,−0.6)

(0.1,0.2,0.6,0.0,−0.1,−0.5)

(0.2,0.1,0.4,−0.1,−0.2,−0.3)

To occupy other country

(0.2,0.2,0.6,−0.1,−0.2,−0.4)

(0.1,0.1,0.5,−0.1,−0.2,−0.5)

(0.1,0.1,0.5,−0.1,−0.1,−0.6)

Benefits of powerful countries (0.3,0.1,0.4,−0.5,−0.1,−0.5)

(0.3,0.2,0.4,−0.5,−0.2,−0.3)

(0.4,0.1,0.3,−0.3,−0.2,−0.3)

Military interference/Drones

(0.3,0.1,0.7,−0.1,−0.1,−0.5)

(0.4,0.1,0.5,−0.2,−0.1,−0.5)

(0.2,0.2,0.5,−0.2,−0.2,−0.6)

Table 10.31 Reasons of terrorism due to relationship of Kashmir with other countries Terrorism Reasons

(Kashmir, India)

(Kashmir, Afghanistan)

(Kashmir, Pakistan)

Religious factors

(0.1,0.2,0.8,−0.1,−0.2,−0.4)

(0.4,0.1,0.6,−0.2,−0.2,−0.6)

(0.6,0.1,0.4,−0.7,−0.2,−0.3)

Political benefits

(0.0,0.3,0.7,−0.5,−0.2,−0.4)

(0.0,0.1,0.6,−0.3,−0.3,−0.3)

(0.2,0.1,0.4,−0.1,−0.3,−0.5)

To pressurize other govt.

(0.1,0.2,0.7,−0.1,−0.2,−0.5)

(0.2,0.2,0.5,−0.4,−0.2,−0.5)

(0.1,0.1,0.4,−0.1,−0.2,−0.6)

To occupy other country

(0.2,0.2,0.7,−0.6,−0.1,−0.−0.1)(0.0,0.0,0.6,0.0,−0.3,−0.5)

(0.0,0.1,0.5,−0.1,−0.2,−0.5)

Benefits of powerful countries (0.1,0.4,0.8,−0.3,−0.2,−0.3)

(0.5,0.3,0.4,−0.6,−0.2,−0.3)

(0.2,0.1,0.6,−0.1,−0.2,−0.4)

Military interference/Drones

(0.0,0.2,0.4,−0.1,−0.2,−0.5)

(0.2,0.1,0.6,−0.4,−0.2,−0.5)

(0.1,0.2,0.7,−0.5,−0.2,−0.3)

442

10 Bipolar Neutrosophic Graph Structures

Table 10.32 Reasons of terrorism due to relationship between countries Terrorism Reasons

(USA, Israel)

(India, Israel)

(Pakistan, Iraq)

Religious factors

(0.2,0.1,0.5,−0.1,−0.2,−0.6)

(0.4,0.2,0.5,−0.1,−0.2,−0.6)

(0.2,0.1,0.4,−0.1,−0.2,−0.3)

Political benefits

(0.1,0.1,0.4,−0.1,−0.2,−0.6)

(0.4,0.3,0.4,−0.3,−0.2,−0.5)

(0.2,0.1,0.4,−0.1,−0.2,−0.3)

To pressurize other govt.

(0.4,0.1,0.2,−0.3,−0.1,−0.5)

(0.5,0.2,0.5,−0.1,−0.2,−0.3)

(0.2,0.1,0.4,−0.1,−0.2,−0.3)

To occupy other country

(0.1,0.2,0.5,−0.1,−0.1,−0.6)

(0.4,0.1,0.5,−0.1,−0.2,−0.5)

(0.2,0.1,0.4,−0.1,−0.2,−0.3)

Benefits of powerful countries (0.3,0.1,0.5,−0.1,−0.1,−0.5)

(0.6,0.2,0.4,−0.4,−0.2,−0.4)

(0.5,0.1,0.2,−0.6,−0.2,−0.3)

Military interference/Drones

(0.4,0.2,0.5,−0.3,−0.2,−0.5)

(0.5,0.1,0.5,−0.3,−0.2,−0.4)

(0.1,0.3,0.5,−0.3,−0.1,−0.6)

surize other government, X 4 = To occupy other country, X 5 = Benefits of powerful countries, and X 6 = Military interference/Drone attacks. Define the relations with those elements such that (X, X 1 , X 2 , X 3 , X 4 , X 5 , X 6 ) is a graph structure. Every element in a relation indicates the most prominent undercover reason for global terrorism due to that pair of countries. Since (X, X 1 , X 2 , X 3 , X 4 , X 5 , X 6 ) is a graph structure, elements in every relation are unique, that is, they belong to just that relation. Consider any pair of countries as an element of that particular relation, for which its value of T p , I n , and F n are high, and T n , I p , and F p values are low as compared to other relations, according to the data of Tables 10.24, 10.25, 10.26, 10.27, 10.28, 10.29, 10.30, 10.31, 10.32. Write down the T p , I p , F p , T n , I n , and F n values of all elements in the relations, using the data of Tables10.24, 10.25, 10.26, 10.27, 10.28, 10.29, 10.30, 10.31, 10.32, such that B1 , B2 , B3 , B4 , B5 , B6 are bipolar single-valued neutrosophic sets on X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , respectively. Let X 1 = {(Kashmir, Pakistan)}, X 2 = {(India, Afghanistan), (USA, Iraq), (USA, Kashmir)}, X 3 = {(India, Pakistan)}, X 4 = {(India, Kashmir), (Palestine, Israel)}, X 5 = {(Pakistan, Afghanistan), (Kashmir, Afghanistan), (Iraq, Pakistan), (A f ghanistan, I raq)}, X 6 = {(USA, Pakistan), (USA, Afghanistan)}.

Corresponding bipolar single-valued neutrosophic sets B1 , B2 , B3 , B4 , B5 , B6 are as follows: B1 = {((Kashmir, Pakistan), 0.6, 0.1, 0.4, −0.7, −0.2, −0.3)}, {((India, Afghanistan), 0.7, 0.3, 0.2, −0.4, −0.2, −0.2), ((USA, Iraq), 0.4, 0.3, 0.3, = B2 −0.2, −0.1, −0.5), ((USA, Kashmir), 0.4, 0.2, 0.4, −0.3, −0.1, −0.5)}, B3 = {((India, Pakistan), 0.7, 0.2, 0.3, −0.5, −0.1, −0.4)}, = {((India, Kashmir), 0.6, 0.1, 0.1, −0.2, −0.2, −0.7), ((Palestine, Israel), 0.1, 0.0, 0.5, B4 −0.4, 0.0, −0.1)}, B5 = {((Pakistan, Afghanistan), 0.6, 0.2, 0.3, −0.5, −0.2, −0.2), ((Kashmir, Afghanistan), 0.5, 0.3, 0.4, -0.6,-0.2,-0.3), ((Pakistan, I raq), 0.5, 0.1, 0.2, −0.6, −0.2, −0.3), ((I raq, A f ghanistan), 0.5, 0.2, 0.3, −0.6, −0.2, −0.2)}, = {((USA, Pakistan), 0.4, 0.2, 0.2, −0.2, −0.1, −0.6), ((USA, Afghanistan), 0.4, 0.2, B6 0.1, −0.3, −0.2, −0.5)}.

Obviously, (B, B1 , B2 , B3 , B4 , B5 , B6 ) is a bipolar single-valued neutrosophic graph structure as shown in Fig.10.22.

10.4 Applications of Bipolar Single-Valued Neutrosophic Graph Structures Political benefits USA (0.4, 0.2, 0.2, −0.2, −0.1, −0.6)

443

Politi cal b enefi ts

Military interference

(0.4, 0.2, 0.1, −0.3, −0.2, −0.5)

4, (0.

Drone attacks

, 0.3

7, 0 .

3, 0 .

Benefits −0 .2,

of pow erful co untries 0.3, − 0.5, − 0.2, − 0.2)

Afghanistan

.5) −0

(0.6, 0.2,

−0 .2)

India

.1, −0

2, − 0.4 ,

.2, −0

(0.

, 0.3

Israel

Political benefits

Pakistan

, 0.1 , 0.1

(0.7, 0.2, 0.3, −0.5. − 0.1. − 0.4)

try oun er c oth upy occ .7) To −0 .2, −0

.2, −0

(0 .4 ,0 .2 ,0 .4 ,− 0. 3, − 0. 1, − 0. 5)

(0 .5 Be , 0. ne 1, 0 fits . of 2, − po 0.6 we rfu , −0 l c .2, ou ntr −0.3 ies ) To

Kashmir

) .1) 0.3 ors act −0 ,− us f 0.2 .0, o i − 0 g , , i .7 Rel 0.4 −0 ,− .4, 0.5 1, 0 , 0. .0, 6 0 . (0 .1, (0 Palestine

f pow fits o Bene

py cu oc

nt ou rc he ot

Iraq

ry

nt ou lc fu r we po

s rie

(0 .5 ,0 (0 .2 .5 ,0 ,0 .3 .3 ,− ,0 0. .4 6, ,− − 0. 0. 6, 2, − − 0. 0. 2, 2) − 0. 3)

6, (0.

To pressurize other Govt.

fit ne Be

f so

tries coun erful

Fig. 10.22 Highlighting undercover reasons of global terrorism

In a bipolar single-valued neutrosophic graph structure, shown in Fig.10.22, every edge demonstrates the most dominating and attention-getting undercover reason of global terrorism due to those contiguous countries. For instance, the attention-getting undercover reason of global terrorism due to Pakistan and USA relationship is the drone attacks of USA on Pakistan, the participation of USA in this reason’s strength, weakness, and indeterminacy is 40%, 20%, and 20%, respectively, and the participation of Pakistan in its strength, weakness, and indeterminacy is 20%, 10%, and 60%, respectively. This Pakistan–USA edge is not representing direct terrorism from contiguous countries but it points out the dominating reason of global terrorism due to Pakistan–USA relationships. This bipolar single-valued neutrosophic graph structure also highlights that sovereignty issues of some countries or states are also increasing global terrorism and tells its intensity, weakness, and uncertainty due to both occupying and occupied countries. For instance, the sovereignty issues of Kashmir and Palestine are also causing global terrorism. Furthermore, it also helps any country’s government that what is the main reason for terrorism due to its relationships with other countries and what the percentage of its strength, weakness, and ambiguity is from its side and other country’s side. Moreover, the most frequent relation in this bipolar single-valued neutrosophic graph structure is the benefits of powerful

444

10 Bipolar Neutrosophic Graph Structures

countries. It shows that powerful countries are destroying global peace to achieve their political benefits. A bipolar single-valued neutrosophic graph structure of all countries can be very helpful for anti terrorism organizations to find out as to which perspective is to be considered to control global terrorism. According to this bipolar single-valued neutrosophic graph structure, they should pressurize powerful countries and solve sovereignty issues to maintain global peace. The method to compute the reasons for terrorism is explained in Algorithm 10.3. Algorithm 10.3 Finding reasons of global terrorism 1. Begin 2. Input membership values B(u i ) of n number of countries u 1 , u 2 , . . . , u n which are either victims of terrorism or said to be responsible for global terrorism. 3. Input the adjacency matrix of countries with respect to X 1 , X 2 , . . . , X m terrorism reasons due to relationships among all pairs of vertices(i.e., pairs of countries). 4. Follow steps 3 and 4 of Algorithm 10.2. p p p 5. TX i , FX i , I X i values of an edge between two different vertices(countries) u k and u l show participation of vertex(country) u k in strength, weakness, and indeterminacy of most dominating and attention-getting terrorist reason X i due to relationships between them, whereas TXni , FXn i , I Xn i are corresponding values of vertex(country) ul . 6. End

10.5 Conclusions A bipolar neutrosophic graph structure is a generalization of a graph structure and is useful to study various domains of computer science and computational intelligence. In this chapter, we have discussed the concepts of bipolar neutrosophic graph structures, Bk −edges, bipolar neutrosophic subgraph structures, strong and complete bipolar neutrosophic graph structures, and studied their fundamental properties and operations. We have described isomorphism properties and relations of selfcomplementary, totally self-complementary, and totally strong self-complementary bipolar neutrosophic graph structures. We have studied the importance of bipolar neutrosophic graph structures with a number of real-world applications in international relations, psychology, and global terrorism.

Exercises 10

445

Exercises 10 1. Is a bipolar single-valued neutrosophic graph structure simply a bipolar singlevalued neutrosophic graph on the set X = {a, b}? 2. Let B : X → [0, 1]3 × [−1, 0]3 , B1 : X 1 × X 1 → [0, 1]3 × [−1, 0]3 , and B2 : X 2 × X 2 → [0, 1]3 × [−1, 0]3 be constant functions, where X 1 , X 2 ⊂ X and X = {a, b, c, d}. Draw three bipolar single-valued neutrosophic graph structures ˇ Also compute Hˇ Gˇ and ˇ Hˇ and Kˇ of (X, X 1 , X 2 ) such that Kˇ ⊆ Hˇ ⊆ G. G, Kˇ × Hˇ , where  and × represent Cartesian product and direct product, respectively. 3. Determine the conditions under which a bipolar single-valued neutrosophic graph structure is a bipolar single-valued neutrosophic graph. 4. Prove or disprove that every strong bipolar single-valued neutrosophic graph structure is the join of two strong bipolar single-valued neutrosophic graph structures. 5. Determine whether or not the lexicographic product of two strong bipolar singlevalued neutrosophic graph structures is a strong bipolar single-valued neutrosophic graph structure. 6. Is the union of two complete bipolar single-valued neutrosophic graph structures a complete bipolar single-valued neutrosophic graph structure? ˇ Hˇ , and Kˇ be three bipolar single-valued neutrosophic graph structures 7. Let G, such that their sets of vertices are disjoint. Determine whether or not the following statements are true or false. (a) (b)

(Gˇ ∪ Hˇ ) • Kˇ = (Gˇ • Kˇ ) ∪ ( Hˇ • Kˇ ), (Gˇ + Hˇ ) × Kˇ = (Gˇ × Kˇ ) + ( Hˇ × Kˇ ).

8. If Gˇ 1 and Gˇ 2 are two complete bipolar single-valued neutrosophic graph structures. Then show that Gˇs1 ∪ Gˇs2 is a complete graph structure. 9. If Gˇ 1 and Gˇ 2 are two strong bipolar single-valued neutrosophic graph structures p p p p with TB1 , I B1 , FB1 , TBn2 , I Bn2 , FBn2 as constant functions. Then show that TB1 ∪B2 , p p n n n I B1 ∪B2 , FB1 ∪B2 , TB1 ∪B2 , I B1 ∪B2 , FB1 ∪B2 are constant functions. 10. If Gˇ 1 and Gˇ 2 are two complete bipolar single-valued neutrosophic graph structures. Then prove that Gˇs1 • Gˇs2 is a complete graph structure. 11. If Gˇ 1 and Gˇ 2 are two strong bipolar single-valued neutrosophic graph structures p p p p with TB1 , I B1 , FB1 , TBn2 , I Bn2 , FBn2 as constant functions. Then prove that TB1 •B2 , p p I B1 •B2 , FB1 •B2 , TBn1 •B2 , I Bn1 •B2 , FBn1 •B2 are constant functions. 12. If Gˇ 1 and Gˇ 2 are two complete bipolar single-valued neutrosophic graph structures. Then prove that Gˇs1 ⊕ Gˇs2 is a complete graph structure. 13. If Gˇ 1 and Gˇ 2 are two strong bipolar single-valued neutrosophic graph structures p p p p with TB1 , I B1 , FB1 , TBn2 , I Bn2 , FBn2 as constant functions. Then prove that TB1 ⊕B2 , p p I B1 ⊕B2 , FB1 ⊕B2 , TBn1 ⊕B2 , I Bn1 ⊕B2 , FBn1 ⊕B2 are constant functions. 14. If Gˇ 1 and Gˇ 2 are two complete bipolar single-valued neutrosophic graph structures. Then prove that Gˇs1  Gˇs2 is a complete graph structure.

446

10 Bipolar Neutrosophic Graph Structures

15. If Gˇ 1 and Gˇ 2 are two strong bipolar single-valued neutrosophic graph structures p p p p with TB1 , I B1 , FB1 , TBn2 , I Bn2 , FBn2 as constant functions. Then prove that TB1 B2 , p p I B1 B2 , FB1 B2 , TBn1 B2 , I Bn1 B2 , FBn1 B2 are constant functions.

References 1. Akram, M., Akmal, R.: Application of bipolar fuzzy sets in graph structures. Appl. Comput. Intell. Soft Comput. 2016, Article ID 5859080, 13 (2016) 2. Akram, M., Akmal, R.: Certain operations on bipolar fuzzy graph structures. Appl. Appl. Math. 11(1), 443–468 (2016) 3. Akram, M., Sarwar, M.: Novel multiple criteria decision making methods based on bipolar neutrosophic sets and bipolar neutrosophic graphs. Ital. J. Pure Appl. Math. 38, 368–389 (2017) 4. Akram, M., Sitara, M.: Bipolar neutrosophic graph structures. J. Indones. Math. Soc. 23(1), 50–80 (2017) 5. Akram, M., Sitara, M.: Certain fuzzy graph structures. J. Appl. Math. Comput. 61, 25–56, (2019) 6. Akram, M., Sitara, M.: Graph structures in bipolar neutrosophic environment, Mathematics, 5(4), 60, (2017) 7. Akram, M., Sitara, M., Saeid, A. B.: Residue product of fuzzy graph structures, J. Mult.-Valued Log. & Soft Comput. 34 365–399, (2020) 8. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 9. Deli, I., Ali, M., Smarandache, F.: Bipolar neutrosophic sets and their application based on multi-criteria decision making problems. In: IEEE International Conference on Advanced Mechatronic Systems, pp. 249–254 (2015) 10. Dinesh, T., Ramakrishnan, R.V.: On generalised fuzzy graph structures. Appl. Math. Sci. 5(4), 173–180 (2011) 11. Sampathkumar, E.: Generalized graph structures. Bull. Kerala Math. Assoc. 3(2), 65–123 (2006) 12. Sitara, M., Akram, M., Bhatti, M.Y.: Fuzzy graph structures with application. Mathematics 7, 63 (2019). https://doi.org/10.3390/math7010063 13. Smarandache, F.: Neutrosophic set a generalization of the intuitionistic fuzzy set. J. Def. Resour. Manag. 1(1), 107–116 (2010) 14. Zhan, J., Akram, M., Sitara, M.: Novel decision-making method based on bipolar neutrosophic information. Soft Comput. 23, 9955–9977 (2019) 15. Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of IEEE Conference, pp. 305–309 (1994) 16. Zhang, W.-R.: (YinYang) Bipolar fuzzy sets. In: IEEE International Conference on Fuzzy Systems, pp. 835–840 (1998) 17. Zhang, W.-R., Wang, P., Peace, K., Zhan, J., Zhang, Y.: On truth, uncertainty, equilibrium, and harmony-A taxonomy for yinyang scientific computing. New Math. Nat. Comput. 4(2), 207–229 (2008)

Index

Symbols (α, β)-cut, 8, 284 (α, β)-level graph, 8, 284 α-cut, 8, 283 Bk -edge, 399 Bk -strong, 399 L-subset, 5 μ-distance, 82 μn -bridge, 235 μn -cut vertex, 238 μn −length, 103 μn -spectrum, 312, 329 μn -status, 120 μn -strength, 234 μ p -bridge, 235 μ p -cut vertex, 238 μ p −length, 103 μ p -spectrum, 312, 329 μ p -status, 120 μ p -strength, 234 t-cut, 297 t-level soft graph, 297 k-dominating set, 265 k-domination number, 265 k-independent dominating set, 266 k-independent domination number, 266 m-step bipolar single-valued neutrosophic competition graph, 359 m-step bipolar single-valued neutrosophic in-neighborhood, 358 m-step bipolar single-valued neutrosophic neighborhood, 366 m-step bipolar single-valued neutrosophic neighborhood graph, 366

m-step bipolar single-valued neutrosophic out-neighborhood, 358 m-step prey, 361 m-totally regular, 129 nn-partite, 23 ϕ-complement, 420

A A bipolar fuzzy multigraph, 225 Adjacency matrix, 312 Adjacent, 52 Amplitude, 43 Antipodal bipolar fuzzy graph, 112 Automorphism, 27

B Biconnected, 234, 239 Bipartite bipolar fuzzy graph, 22 Bipolar fuzzy approximation function, 296 Bipolar fuzzy basis, 286 Bipolar fuzzy block, 239 Bipolar fuzzy bridge, 39, 109, 235 Bipolar fuzzy circuit, 287 Bipolar fuzzy closed neighborhood, 188 Bipolar fuzzy common enemy graph, 199 Bipolar fuzzy competition common enemy graph, 201 Bipolar fuzzy competition graph, 163 Bipolar fuzzy conflict graph, 213, 216 Bipolar fuzzy confusion graph, 217, 219 Bipolar fuzzy cut vertex, 238 Bipolar fuzzy cycle, 21, 85, 241

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Akram et al., Graphs for the Analysis of Bipolar Fuzzy Information, Studies in Fuzziness and Soft Computing 401, https://doi.org/10.1007/978-981-15-8756-6

447

448 Bipolar fuzzy digraph, 52 Bipolar fuzzy directed cycle, 53 Bipolar fuzzy directed path, 53 Bipolar fuzzy directed walk, 53 Bipolar fuzzy edge graph, 235 Bipolar fuzzy equivalence relation, 6, 147 Bipolar fuzzy face, 230 Bipolar fuzzy forest, 243 Bipolar fuzzy influence graph, 214 Bipolar fuzzy in neighborhood, 163 Bipolar fuzzy (k)−competition graph, 189 Bipolar fuzzy [k]−competition graph, 189 Bipolar fuzzy k−competition graph, 187 Bipolar fuzzy linearly independent, 286 Bipolar fuzzy linear order relation, 7, 147 Bipolar fuzzy line graph, 141 Bipolar fuzzy marketing digraph, 206 Bipolar fuzzy matroid, 287 Bipolar fuzzy multiedge set, 225 Bipolar fuzzy multiset, 225 Bipolar fuzzy niche graph, 202 Bipolar fuzzy open neighborhood, 188 Bipolar fuzzy open neighborhood graph, 188 Bipolar fuzzy out neighborhood, 163 Bipolar fuzzy partial order relation, 7, 147 Bipolar fuzzy path, 21, 85 Bipolar fuzzy path cover, 107 Bipolar fuzzy planar graph, 228 Bipolar fuzzy preference relation, 335 Bipolar fuzzy relation, 6, 8, 147, 283 Bipolar fuzzy set, 5, 283 Bipolar fuzzy soft circuit, 298 Bipolar fuzzy soft class, 296 Bipolar fuzzy soft edge set, 296 Bipolar fuzzy soft graph, 296 Bipolar fuzzy soft matroid, 297 Bipolar fuzzy soft relation, 296 Bipolar fuzzy soft set, 296 Bipolar fuzzy soft vertex set, 296 Bipolar fuzzy subdigraph, 58 Bipolar fuzzy subgraph, 23 Bipolar fuzzy subgroup, 35 Bipolar fuzzy subsemigroup, 35 Bipolar fuzzy tree, 242, 244 Bipolar fuzzy vector space, 285 Bipolar fuzzy walk, 21 Bipolar polar fuzzy graph, 8, 283 Bipolar single-valued neutrosophic Bk edge, 399 Bipolar single-valued neutrosophic Bk -path, 399 Bipolar single-valued neutrosophic closed neighborhood, 369

Index Bipolar single-valued neutrosophic closed neighborhood graph, 369 Bipolar single-valued neutrosophic competition graph, 354 Bipolar single-valued neutrosophic digraph, 352 Bipolar single-valued neutrosophic graph, 352 Bipolar single-valued neutrosophic graph structure, 394 Bipolar single-valued neutrosophic induced subgraph structure, 398 Bipolar single-valued neutrosophic inneighborhood, 353 Bipolar single-valued neutrosophic open neighborhood, 369 Bipolar single-valued neutrosophic open neighborhood graph, 369 Bipolar single-valued neutrosophic outneighborhood, 353 Bipolar single-valued neutrosophic pcompetition graph, 355 Bipolar single-valued neutrosophic relation, 352 Bipolar single-valued neutrosophic set, 351 Bipolar single-valued neutrosophic spanning subgraph structure, 398 Bipolar single-valued neutrosophic subgraph structure, 397 Block, 234 Bridge, 234

C Cardinality, 187, 193 Cartesian product, 16, 49, 55, 82, 371 Cayley bipolar fuzzy graph, 148 Cayley graph, 147 Central vertex, 97 Circuit function, 294 Circuit rectangle, 295 Closed bipolar fuzzy matroid, 291 Closed bipolar fuzzy soft matroid, 300 Closed neighborhood degree, 131 Closed neighborhood graph, 188 Closure, 291 Competition graph, 163 Complement, 24, 51, 352 Complete, 227, 400 Complete bipartite bipolar fuzzy graph, 23 Complete bipolar fuzzy graph, 22, 132 Complex bipolar fuzzy closed neighborhood, 194

Index Complex bipolar fuzzy closed neighborhood graph, 195 Complex bipolar fuzzy competition graph, 192 Complex bipolar fuzzy digraph, 191 Complex bipolar fuzzy graph, 45, 191 Complex bipolar fuzzy in neighborhood, 192 Complex bipolar fuzzy k−competition graph, 194 Complex bipolar fuzzy open neighborhood, 194 Complex bipolar fuzzy open neighborhood graph, 194 Complex bipolar fuzzy out neighborhood, 192 Complex bipolar fuzzy relation, 45 Complex bipolar fuzzy set, 44 Complex fuzzy set, 43 Composition, 15, 48, 55 Connected, 21, 85, 234 Connected equitable dominating set, 262 Count negative membership, 225 Count positive membership, 225 Co-weak isomorphism, 27, 144 Crisp graph, 7 Cut vertex, 234 Cycle, 234 Cycle bipolar fuzzy matroid, 289 Cycle bipolar fuzzy soft matroid, 298

D Degree, 35, 128, 226, 254, 311 Degree equitable, 261 Degree matrix, 317 Degree sequence, 76 Depth, 8, 163, 283 Diameter, 97 Digraph, 51 Directed graph, 51 Direct product, 18, 49, 56, 83, 372 Distance, 86, 104 Dominates, 255, 328 Dominating adjacency matrix, 329 Dominating energy, 329 Dominating set, 254, 255, 328 Domination number, 254 Double dominating adjacency matrix, 333 Double dominating set, 333 Double out-dominating energy, 335 Double out-dominating set, 334 Dual, 231

449 E Eccentricity, 97 Eccentric vertex, 97 Edge cardinality, 254 Edge cover, 107 ED number, 261 ED set, 260 Effective edge, 267 EI number, 262 EI set, 262 EN degree, 259 Endomorphism, 27 End vertex, 39 Energy, 311, 312, 316 Equitable dominating set, 260 Equitable domination number, 261 Equitable independence number, 262 Equitable independent set, 262 Equitable isolated vertex, 259 Equitable neighborhood, 259 F Firm, 240 Food web, 196 Forest, 234 Full bipolar fuzzy block, 239 Full bipolar fuzzy bridge, 235 Full bipolar fuzzy cycle, 241 Full bipolar fuzzy forest, 243 Full bipolar fuzzy tree, 244 Fully connected, 245 Fundamental sequence, 291, 300 Fuzzy multiset, 225 Fuzzy set, 1 G Geodesic distance, 81 Global restrained dominating set, 266 Global restrained domination number, 266 Graph, 7 Graph structure, 394 Group decision-making, 60 H Height, 8, 163, 192, 283, 351 Highly irregular, 134 Homomorphism, 26, 144 I Identical, 418 Independent, 256

450 Independent bipolar fuzzy soft subsets, 298 Independent bipolar fuzzy subsets, 287 Independent number, 254 Independent set, 254, 256 Independent soft subsets, 296 Independent strong, 357 Independent subsets, 282 Induced matroid sequence, 291 Inner bipolar fuzzy faces, 230 Intersecting value, 227 Intersection, 6, 12, 47, 53, 351 Intuitionistic fuzzy set, 5 Irredundant set, 272 Irredundant vertex, 272 Irregular, 131 Isolated vertex, 255 Isomorphic, 417 Isomorphism, 27, 144

Index Minimal double dominating set, 333 Minimal double out-dominating set, 334 Minimal ED set, 261 Minimal equitable dominating set, 261 Minimal out-dominating set, 332 Minimal total dominating set, 257 Minimal total ED set, 265 Minimum degree, 254 Minimum dominating energy, 329 Minimum dominating set, 328 Minimum double dominating set, 333 Minimum double out-dominating set, 334 Minimum EN degree, 259 Minimum out-dominating set, 332 Minimum status, 120 Multigraph, 225 Multipartite, 23 Multi-person decision-making, 60 Multiplicity, 225 Multiset, 225

J Join, 13, 47, 54, 85, 416

L Laplacian energy, 320, 323 Laplacian matrix, 317, 322 Laplacian spectrum, 317 Lattice, 4 Length, 86, 103, 234 Lexicographic product, 21, 57, 84, 401 Linear bipolar fuzzy linear matroid, 288 Lower domination number, 256 Lower independent number, 256 Lower irredundance number, 272 Lower total dominating number, 257

M Market competition, 384 Matroid, 282 Maximal connected, 107 Maximal EI set, 262 Maximal equitable independent set, 262 Maximal independent set, 254, 256 Maximal irredundant set, 272 Maximum degree, 254 Maximum EN degree, 259 Maximum status, 120 Max-min-max composition, 59 Max-product−min-product composition, 59 Median, 121 Metric, 96 Minimal dominating set, 255, 328

N Negative support, 8, 283 Neighborhood, 131 Neighborhood degree, 131 Neighborly irregular, 134 Neighborly totally irregular, 134 Neighbors, 255 Network analysis, 303 Neutrosophic set, 350 Noisy channel, 215, 218 Noneffective edge, 267 Normal, 8 Null bipolar fuzzy graph, 76

O Order, 33, 132 Out-degree matrix, 322 Out-dominating adjacency matrix, 332 Out-dominating energy, 333 Out-dominating set, 332 Outer bipolar fuzzy face, 230 Out-regular digraph, 148

P Partial bipolar fuzzy block, 239 Partial bipolar fuzzy bridge, 235 Partial bipolar fuzzy cut vertex, 238 Partial bipolar fuzzy cycle, 241 Partial bipolar fuzzy forest, 243 Partial bipolar fuzzy tree, 244

Index Partially connected, 245 Partition bipolar fuzzy matroid, 289 Path, 234 Pendant edge, 231 Peripheral vertex, 97 Phase, 43 Planar graph, 228 Planarity, 227 Planarity value, 228 Political competition, 206 Positive support, 8, 283 Private neighbor, 272 Projection, 87

R Radius, 97 Rank function , 282 Redundant vertex, 272 Reflexive, 6, 147 Regular, 129, 305 Regular bipolar fuzzy graph, 311 Restrained dominating set, 266 Restrained domination number, 266

S Self-centered bipolar fuzzy graph, 97 Self-complementary, 27, 424 Self-median, 121 Signless Laplacian energy, 326, 327 Signless Laplacian matrix, 324, 326 Single-valued neutrosophic set, 351 Size, 33, 132 Social network, 212 Soft fundamental sequence, 300 Soft graph, 295 Soft matroid, 295 Soft set, 295 Soft universe, 295 Spanning subgraph, 23 Spanning tree, 234 Spectrum, 311, 312, 322, 324, 326, 329, 332 Status, 120 Strength, 39, 227 Strength of connectedness, 39 Strong, 39, 58, 187, 193, 226, 229, 399 Strong bipolar fuzzy cycle, 110 Strong bipolar fuzzy edge, 227 Strong bipolar fuzzy face, 230 Strong bipolar fuzzy graph, 21 Strong bipolar fuzzy planar graph, 230 Strong complex bipolar fuzzy graph, 51

451 Strong edge, 255 Strong equitable dominating set, 262 Strong equitable domination number, 262 Strongest bipolar fuzzy path, 39 Strongly dominates, 262 Strong neighbors, 39 Strong prey, 361 Strong product, 19, 50, 57, 83, 405 Strong self-complementary, 424 Sum distance, 82 Support, 8, 283, 355, 399 Symmetric, 7, 8, 46, 147, 192, 283

T Total degree, 129, 311 Total dominating set, 257 Total ED set, 265 Total equitable domination number, 265 Total k-dominating set, 266 Total k-domination number, 266 Totally irregular, 131 Totally self-complementary, 424 Totally strong self-complementary, 424 Total status, 121 Transitive, 7, 147 Tree, 234 Truncation, 294

U Underlying bipolar fuzzy graph, 189 Underlying complex bipolar fuzzy graph, 195 Uniform bipolar fuzzy matroid, 288 Uniform bipolar fuzzy soft matroid, 298 Union, 6, 9, 46, 53, 84, 351, 410 Upper domination number, 256 Upper independent number, 256 Upper irredundance number, 272 Upper total dominating number, 257

V Vertex cardinality, 254

W Weak bipolar fuzzy block, 239 Weak bipolar fuzzy bridge, 235 Weak bipolar fuzzy cut vertex, 238 Weak bipolar fuzzy cycle, 241 Weak bipolar fuzzy edge, 227 Weak bipolar fuzzy face, 230

452 Weak bipolar fuzzy forest, 243 Weak bipolar fuzzy tree, 244 Weak edge, 357 Weak equitable dominating set, 262 Weak equitable domination number, 262

Index Weakest bipolar fuzzy edge, 39 Weak isomorphism, 27, 144 Weakly connected, 245 Weakly dominates, 262