Introductory Graph Theory with Applications [1 ed.] 1478611758, 9781478611752
Graph theory's practical applications extend not only across multiple areas of mathematics and computer science but
198 69 9MB
English Pages 365 [382] Year 2013
Contents
Preface
Notation
1. Introductory Concepts
1.1 Mathematical Preliminaries
1.2 Mathematical Induction
1.3 Permutations and Combinations
1.4 Pascal's Triangle and Combinatorial Identities
2. Introduction to Graphs and Their Uses
2.1 Graphs as Models
2.2 Subgraphs and Types of Graphs
2.3 Isomorphic Graphs
2.4 Graph Operations
3. Trees and Bipartite Graphs
3.1 Properties of Trees
3.2 Minimum Spanning Trees
3.3 A Characterization of Bipartite Graphs
3.4 Matchings and Job Assignments
4. Distance and Connectivity
4.1 Distance in Graphs
4.2 Connectivity Concepts
4.3 Applications
5. Eulerian and Hamiltonian Graphs
5.1 Characterization of Eulerian Graphs
5.2 Hamiltonicity
5.3 Applications
6. Graph Coloring
6.1 Vertex Coloring and Independent Sets
6.2 Edge Coloring
6.3 Applications of Graph Coloring
7. Matrices
7.1 Review of Matrix Concepts
7.2 The Adjacency Matrix
7.3 The Distance Matrix
8. Graph Algorithms
8.1 Graph Searching
8.2 Graph Coloring Algorithms
8.3 Tree Codes
9. Planar Graphs
9.1 Planarity
9.2 Planar Graphs, Graph Coloring, and Embedding
9.3 Graph Duals and a Planar Graph Application
10. Digraphs and Networks
10.1 Directed Graphs
10.2 Networks
10.3 The Critical Path Method
11. Special Topics
11.1 Ramsey Theory
11.2 Domination in Graphs
Answers/Solutions to Selected Exercises
Index
Blank Page
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Contents Preface
IX
Notation
Xlll
1. Introductory Concepts
1
Mathematical Preliminaries Mathematical Induction Permutations and Combinations Pascal's Triangle and Combinatorial Identities
1 15 25 36
2. Introduction to Graphs and their Uses
47
1.1 1.2 1.3 1.4
2.1 Graphs as Models 2.2 Subgraphs and Types of Graphs 2.3 Isomorphic Graphs 2.4 Graph Operations
3. Trees and Bipartite Graphs 3.1 3.2 3.3 3.4
Properties of Trees Minimum Spanning Trees A Characterization of Bipartite Graphs Matchings and Job Assignments
47
57 64
74
85
85 93 102 107
4. Distance and Connectivity
119
5. Eulerian and Hamiltonian Graphs
143
4.1 Distance in Graphs 4.2 Connectivity Concepts 4.3 Applications
5.1 Characterization of Eulerian Graphs 5.2 Hamiltonicity 5.3 Applications vii
119 129 136 143 151 159