Regular Graphs: A Spectral Approach 9783110351347, 9783110351286

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Zoran Stanić Regular Graphs

De Gruyter Series in Discrete Mathematics and Applications

Edited by Michael Drmota, Vienna, Austria János Pach, Lausanne, Switzerland Martin Skutella, Berlin, Germany

Volume 4

Zoran Stanić

Regular Graphs

A Spectral Approach

Mathematics Subject Classification 2010 05C50, 05C12, 05C76, 05C81, 05B05, 05B20, 65F15, 68R10, 94B05 Author Prof. Dr. Zoran Stanić University of Belgrade Faculty of Mathematics Studentski trg 16 11000 Belgrade Serbia [email protected]

ISBN 978-3-11-035128-6 e-ISBN (PDF) 978-3-11-035134-7 e-ISBN (EPUB) 978-3-11-038336-2 Set-ISBN 978-3-11-035135-4 ISSN 2195-5557 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2017 Walter de Gruyter GmbH, Berlin/Boston Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface This book has been written to be of use to scientists working in the theory of graph spectra. It may also attract the attention of graduate students dealing with the same subject area as well as all those who use graph spectra in their research, in particular, computer scientists, chemists, physicists or electrical engineers. In the entire book we investigate exactly one class of graphs (regular) by using exactly one approach (spectral), that is we study eigenvalues and eigenvectors of a graph matrix and their interconnections with the structure and other invariants. Regular graphs appear in numerous sources. In particular, they can probably be encountered in all the books concerning graph theory. An intriguing phenomenon is that, in many situations, regular graphs are met as extremal, exceptional or unique graphs with a given property. More tangibly, an inequality is attained for a regular graph or a claim holds unless a graph under consideration is regular or it does hold exactly for regular graphs. In other words, depending on a given problem they fluctuate from one amplitude to another, which is making them probably the most investigated and the most important class of graphs, and simultaneously motivating us to consider the theory of graph spectra through the prism of regular graphs. There is also a relevance of regular graphs in the study of other discrete structures such as linear spaces or block designs. A final motivation for this book is a remarkable significance of regular graphs in branches of computer science, chemistry, physics and other disciplines that consider processes dealing with graph-like objects with high regularity and many symmetries. The rapid development of graph theory in last decades caused an inability to cover all relevant results even if we restrict ourselves to a singular approach, so we must say that this material represents a narrowed selection of (in most cases, known) results. Our intention was to include classical results concerning the spectra of regular graphs along with recent developments. Occasionally, possible applications are indicated. The terminology and notation are taken from our previous book [290]. This book was written to be as self-contained as possible, but we assume a decent mathematical knowledge, especially in algebra and the theory of graph spectra. Here is a brief outline of the contents. Chapter 1 is preparatory. In Chapter 2 we focus on basic properties of the spectra of regular graphs. In Chapter 3 we pay attention to some particular types of regular graph. Graph products, walk-regular graphs, different subclasses of distance-regular graphs, regular graphs of line systems and various block designs occupy the central place of this chapter. In Chapter 4 we consider the problem of determining regular graphs that satisfy fixed spectral constraints. There, we mostly deal with regular graphs with bounded least or second largest eigenvalue, those with integral spectrum or a comparatively small number of distinct eigenvalues. Cospectrality of regular graphs is also considered. Chapter 5 deals with a single subclass of regular graphs called expanders. Expanders have an enormous number DOI: 10.1515/9783110351347-202

VI | Preface of surprising properties making them relevant in mathematics and other contexts. We consider the three different approaches to these graphs, their constructions and applications in the coding theory. For almost all standard square matrices associated with graphs, the spectrum of one of them contains full information about the spectrum of the remaining ones whenever a graph under consideration is regular, which means that in the major part of the research there is no difference which matrix we are dealing with. An exception is the distance matrix which is the subject of Chapter 6. Observing the table of contents, the reader may notice that expanders explored in Chapter 5 can be considered as a part of Chapter 3. The reasons for separating them in a single chapter are the quite different approach they require and the size of the presented material. Many results presented in Chapter 4 are based on those of Chapter 3 which motivated us to set these two chapters next to one another. All chapters are divided into sections and some sections are divided into subsections. The reader will also notice the existence of paragraphs that occur irrespectively of the above hierarchy. Each of the Chapters 2–6 contains theoretical results, comments (including additional explanations or possible applications), examples and exercises. Some of the results presented in these chapters can be found in other books concerning the theory of graph spectra. In particular, the books [49, 87, 89] cover some parts of Chapter 2. Some results of Chapter 3 can be found in [48] (especially those related to distanceregular graphs), [145] (strongly regular graphs, Kneser graphs, line systems), [94] (line systems, star complements) or [62] (block designs). The book [94] also covers a smaller part of Chapter 4. Chapter 5 is partially covered by [113]. For details that are not presented here, we recommend some of these books. The author is grateful to Tamara Koledin who read parts of the manuscript and gave valuable suggestions, and to Sebastian Cioabă, Dragoš Cvetković, Edwin R. van Dam, Willem H. Haemers, Tamara Koledin again and Peter Rowlinson for permissions to reproduce some of their proofs without significant change. Finally, Nancy Christ, Apostolos Damialis and Nadja Schedensack on behalf of the publisher helped with editing the book by answering a lot of questions, which is much appreciated. Last but not least, the author apologizes in advance for possible misprints or computational errors as well as for the inconvenience these might cause to the reader.

Contents Preface | V List of Figures | XI 1 1.1 1.2 1.3

Introduction | 1 Terminology and notation | 1 Notes on regular graphs | 5 Elements of the theory of graph spectra | 7

2 2.1 2.2 2.3 2.3.1 2.3.2 2.4 2.5 2.6 2.6.1 2.6.2 2.6.3 2.7 2.8 2.9

Spectral properties | 15 Basic properties of the spectrum of a regular graph | 15 Operations on regular graphs and resulting spectra | 17 Other kinds of graph spectrum | 22 Laplacian and signless Laplacian spectrum | 22 Seidel spectrum | 26 Deducing regularity from the spectrum | 27 Spectral distances of regular graphs | 28 Inequalities for eigenvalues | 33 Second largest eigenvalue | 33 Toughness of regular graphs | 36 More inequalities | 37 Extremal regular graphs | 39 Eigenvalue estimation | 43 Eigenvalue distribution | 46

3 3.1 3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4 3.4.1 3.4.2 3.4.3 3.4.4

Particular types of regular graph | 48 Non-complete extended p-sums | 49 Walk-regular graphs | 51 Distance-regular graphs | 55 Basic properties | 56 Classical examples | 61 Inequalities for eigenvalues | 65 Geometric graphs | 67 Deducing distance-regularity from the spectrum | 68 Strongly regular graphs | 70 Basic properties | 71 Small examples | 75 Conference graphs | 76 Generalized Latin squares and partial geometries | 81

VIII | Contents 3.4.5 3.5 3.6 3.6.1 3.6.2 3.7 3.7.1 3.7.2 3.7.3 3.8 3.8.1 3.8.2

Spectral distances | 84 Kneser graphs | 86 Regular graphs of line systems | 88 Equidistant lines | 88 Root systems | 93 Star complements | 98 Reconstruction theorem and its consequences | 99 Star complement technique | 100 Star complements in regular graphs | 104 Partially balanced incomplete block designs | 105 Balanced incomplete block designs | 107 Partially balanced incomplete block designs with 2 associate classes | 113

4.5 4.5.1 4.5.2 4.5.3

Determinations of regular graphs | 120 Exceptional regular graphs | 120 Regular graphs with λ2 ≤ 1 | 121 Reflexive bipartite regular graphs | 122 Integral regular graphs | 126 Integral cubic graphs | 127 Integral 4-regular graphs | 129 Integral regular line graphs | 132 Regular graphs with a small number of distinct eigenvalues | 141 Strongly regular graphs of larger order | 141 Regular graphs with 4 distinct eigenvalues | 146 Bipartite regular graphs with 3 distinct non-negative eigenvalues | 152 Regular graphs determined by their spectra | 165 Line graphs | 166 Graphs with at most 4 distinct eigenvalues | 169 Some constructions | 170

5 5.1 5.2 5.3 5.4 5.4.1 5.4.2 5.4.3 5.5 5.5.1

Expanders | 173 Combinatorial approach | 174 Algebraic approach | 179 Random walks | 184 Constructions | 187 Covers of graphs | 188 Zig-zag product | 189 Examples of Cayley expanders | 191 Codes from graphs | 194 A probabilistic algorithm | 195

4 4.1 4.1.1 4.2 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.4.3

Contents

5.5.2 6 6.1 6.2 6.3 6.4

Error correcting codes | 196 Distance matrix of regular graphs | 199 Definition and a Hoffman-like polynomial | 199 D-spectrum of certain regular graphs | 200 D-spectrum of distance-regular graphs | 206 Regular graphs constrained by D-spectrum | 212

References | 217 Graph index | 231 Index | 233

| IX

List of Figures Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4

Two illustrations of the 4-dimensional cube. | 2 Cubic graph 2C3 and its 4-regular line graph. | 4 A cubic graph with 8 vertices. | 6 A pair of cospectral regular graphs with 10 vertices. | 10

Fig. 2.1 Fig. 2.2

Petersen graph. | 21 Cubic graphs with at least 10 vertices for which λ2 is maximal. | 42

Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6

C6 (top), its bipartite double (left) and extended bipartite double (right). | 51 A walk-regular graph with 13 vertices. | 52 8-crossed prism. | 54 Halved 4-dimensional cube. | 62 Shrikhande graph. | 63 Taylor graph with (r, s) = (5, 2) (icosahedron), Johnson graph J(4, 2) (octahedron) and Hamming graph H(3, 2) (lattice Line(K3,3 )). | 64 Hoffman graph. | 69 13-Paley graph and Clebsch graph. | 76 Generalized quadrangle GQ(2, 4). | 83 Desargues graph and its cospectral mate. | 87 Two unicyclic star complements for λ2 = 1 with unique maximal extensions (for Exercise 3.7.11). | 103

Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10

Cubic and 4-regular graphs with λ2 ≤ 1 (for Theorem 4.1.3). | 122 Integral bipartite cubic graphs (for Theorem 4.3.2). | 128 Integral non-bipartite cubic graphs (for Theorem 4.3.2). | 128 Integral bipartite 4-regular graphs (for Theorem 4.3.3). | 130 Integral non-bipartite 4-regular graphs (for Theorem 4.3.3). | 131 Q-integral bipartite (3, 4)-semiregular graphs (for Theorem 4.3.11). | 138 A 5-regular graph with a unique spectrum (for Theorem 4.3.15). | 141 Heawood graph and its bipartite complement. | 149 Franklin, Möbius–Kantor, F26A and Dyck graph. | 162 Graph cospectral with Line(K3,6 ). | 167

Fig. 5.1 Fig. 5.2

Multigraph G11 . | 179 Data transmission. | 196

Fig. 6.1

Three forbidden induced subgraphs (for Exercise 6.4.2). | 214

1 Introduction Here we give a survey of the main graph-theoretic terminology, notation and necessary results. The presentation is separated into three sections. In the first two we deal with graph structure and give some observations including statistical data on regular graphs. In the third section we focus on the adjacency matrix and the corresponding spectrum. Since all parts of this chapter can be found in numerous sources, by the assumption that the reader is familiar with most of the concepts presented, we orientate to a brief but very clear and intuitive exposition. More details are given in the introductory chapter of our previous book [290].

1.1 Terminology and notation Vertices and edges Let G be a finite undirected simple graph (so, without loops or multiple edges). We denote its set of vertices (resp. edges) by V or V(G) (resp. E or E(G)). In addition, we assume that |V| ≠ 0. The quantities n = |V| and m = |E| are called the order and the size of G, respectively. Two vertices u and v are adjacent (or neighbours) if they are joined by an edge. In this case we write u ∼ v and say that the edge uv is incident with vertices u and v. Similarly, two edges are adjacent if they are incident with a common vertex. The set of neighbours (or the neighbourhood) of a vertex v is denoted N(v). The closed neighbourhood of v is denoted N[v] (= {v} ∪ N(v)). Two graphs G and H are said to be isomorphic if there is a bijection between sets of their vertices which respects adjacencies. If so, then we write G ≅ H. Observe that the graphs illustrated in Figure 1.1 are isomorphic. In particular, an automorphism of a graph is an isomorphism to itself. The degree d v of a vertex v is the number of edges incident with it. The minimal and the maximal vertex degrees in a graph are denoted δ and ∆, respectively. A vertex of degree 1 is called an endvertex or a pendant vertex. We say that a graph G is regular of degree r if all its vertices have degree r. If so, then G is referred to as r-regular. In particular, a 3-regular graph is called a cubic graph. The complete graph with n vertices K n is the unique (n − 1)-regular graph. Similarly, the cocktail party CP(n) is the unique (2n − 2)-regular graph with 2n (n ≥ 1) vertices. The complete graph K1 is called the trivial graph. The r-dimensional cube Q r is an r-regular graph of order 2r with the vertex set V(Q r ) = {0, 1}r (all possible binary r-tuples) in which two vertices are adjacent if they differ in exactly one coordinate (see Figure 1.1).

DOI: 10.1515/9783110351347-001

2 | 1 Introduction

Fig. 1.1. Two illustrations of the 4-dimensional cube.

The edge degree of an edge uv is defined as equal to d u + d v − 2. In other words, it is the number of edges adjacent to uv. Similarly to the above, a graph is edge-regular of degree s if all its edges have degree s. A matching in a graph is a set of edges such that any vertex in a graph is incident with at most one edge of a matching. We say that a matching is perfect if every vertex is incident with an edge from the matching. Induced subgraphs and subgraphs An induced subgraph of a graph G is any graph H obtained by deleting some vertices (together with their edges incident). A graph G is H-free if it does not contain H as an induced subgraph. A subgraph of G is any graph H satisfying V(H) ⊆ V(G) and E(H) ⊆ E(G). In particular, if V(H) = V(G) then H is called a spanning subgraph of G. If S ⊂ V(G), then we write G[S] to denote the induced subgraph of G with vertex set S in which two vertices are adjacent if and only if they are adjacent in G. For short, G − v (resp. G − e) designates the induced subgraph (resp. subgraph) obtained by deleting a single vertex v (resp. edge e). We call it a vertex-deleted (resp. edge-deleted) subgraph. If S and T are disjoint subsets of V(G), then m(S) and m(S, T) stand for the number of edges in G[S] and the number of edges with one end in S and the other in T, respectively. A clique is any complete induced subgraph of a graph G. The clique number ω is the number of vertices in the largest clique of G. Similarly, a co-clique is any induced subgraph without edges. The vertices of a co-clique make an independent set of vertices of G, while the number of vertices in the largest independent set is called the independence number and denoted α.

1.1 Terminology and notation | 3

Walks and connectivity A k-walk (or simply, walk) in a graph G is a sequence of alternate vertices and edges v1 , e1 , v2 , e2 , . . . , e k−1 , v k such that, for 1 ≤ i ≤ k − 1, the vertices v i and v i+1 are distinct ends of the edge e i . A walk is closed if v1 coincides with v k . The number of k-walks is denoted w k , while the number of closed k-walks starting at vertex v is denoted w k (v). The length of a walk is equal to the number of its edges. Remark 1.1.1. We suggest to the reader to remember that in this book the parameter k denotes the number of vertices in a k-walk, while its length is then equal to k − 1. A path is a walk in which all vertices are mutually distinct. A graph which is itself a path with n vertices is denoted P n . There is a special case of a path consisting of a single vertex (i.e., P1 ≅ K1 ). A vertex of minimal degree in P n is called an end of P n . By inserting an edge between the ends of P n (n ≥ 3), we get a cycle C n . The cycle C3 is known as a triangle, C4 is known as a quadrangle, and so on. Since all paths and cycles are particular walks, the length of a path or a cycle is equal to the number of its edges. A graph with n vertices is Hamiltonian if it contains C n as a spanning subgraph. Any such cycle is referred to as a Hamiltonian cycle. We say that a graph G is connected if every two (not necessary distinct) vertices are the ends of at least one path in G. Otherwise, we say that G is disconnected. Maximal connected induced subgraphs of a disconnected graph are referred to as its components. A component consisting of a single vertex is called an isolated vertex of a graph, while a graph consisting entirely of (at least two) isolated vertices is called totally disconnected. A tree is a connected graph that does not contain any cycle (as a subgraph). Obviously, the number of edges m of any tree equals n − 1. The vertex (resp. edge) connectivity c v (resp. c e ) of a connected graph is the minimal number of vertices (resp. edges) whose removal results in a trivial or disconnected graph. The distance d(u, v) between (not necessary distinct) vertices u and v belonging to the same component is the length of the shortest path between u and v. The diameter of a connected graph G is defined by D = max{d(u, v) : u, v ∈ V(G)} (i.e., it is the longest distance between two vertices of G). The girth g is the length of the shortest cycle induced in G. Colouring and bipartite graphs A graph G is k-colourable if its vertices can be properly coloured (i.e., in such a way that two adjacent vertices are coloured by different colours) by k colours. The chromatic number χ is the minimal value of k such that G is k-colourable. A graph G is called bipartite if its chromatic number is 1 or 2. Obviously, χ(G) = 1 holds if and only if E(G) = 0. The vertex set of a bipartite graph can be partitioned into two parts (or colour classes) X and Y in such a way that every edge of G joins a

4 | 1 Introduction

Fig. 1.2. Cubic graph 2C3 and its 4-regular line graph.

vertex in X with a vertex in Y. Observe that there is a unique partition whenever G is connected (on up to interchanging the roles of X and Y). A graph is called complete bipartite if every vertex in one part is adjacent to every vertex in the other part. If |X| = n1 and |Y| = n2 , the complete bipartite graph is denoted K n1 ,n2 . In particular, if at least one of the parameters is equal to 1, it is called a star. A vertex with maximal degree in the star K1,n−1 is called the centre. We use K −n,n to denote the graph obtained by removing a perfect matching from K n,n . More general, a k-partite graph is a graph whose set of vertices can be partitioned into k parts such that no two vertices in the same part are adjacent. If there is an edge between every pair of vertices belonging to different parts, the graph is referred to as a complete k-partite or, if k is suppressed, a complete multipartite. A graph is called bipartite semiregular if it is bipartite and the vertices belonging to the same part have equal degree. If the corresponding vertex degrees are r and s, then the graph is referred to as a bipartite (r, s)-semiregular. Certain operations For two graphs G and H we define G ∪ H to be their disjoint union. We also use kG to denote the disjoint union of k copies of G. The join G∇H is the graph obtained by inserting an edge between every vertex of G and every vertex of H. This operation may be applied to a sequence of graphs G1 , G2 , . . . , G k . In that case we usually write G1 ∇G2 ∇ ⋅ ⋅ ⋅ ∇G k . The complement of a graph G is the graph G with the same vertex set as G, in which any two distinct vertices are adjacent if and only if they are non-adjacent in G. Observe that if G is disconnected, then G must be connected. If G is bipartite with parts X and Y, then its bipartite complement is the bipartite graph G with the same parts having the edge between X and Y exactly where G does not. A multigraph includes the possible existence of loops or multiple edges. All previous numerical invariants can be defined for multigraphs in a similar way.

1.2 Notes on regular graphs | 5

The line graph Line(G) of a multigraph G is the graph whose vertices are identified with the edges of G, with two vertices adjacent whenever the corresponding edges have exactly one common vertex. Clearly, if G is a (simple) graph, then any two edges cannot have more than one common vertex. We say that G is the root graph (of Line(G)). An example is illustrated in Figure 1.2. We say that a petal is added to a multigraph when we add a pendant vertex and then duplicate the edge incident with it. Given a graph G with vertex set V = {v1 , v2 , . . . , v n }, let a1 , a2 , . . . , a n be non-negative integers. The generalized line graph Line(G; a1 , a2 , . . . , a n ) is the graph Line(̂ G), where ̂ G is the multigraph obtained from G by adding a i petals at vertex v i (1 ≤ i ≤ n).

1.2 Notes on regular graphs Regular graphs of degree 0 consist of isolated vertices, all components of those of degree 1 are isomorphic to the complete graph K2 , while those of degree 2 are disjoint unions of cycles. Observing that the complement of an r-regular graph with n vertices is regular of degree n − r − 1, we conclude that the complete graph K n and, for n even, the cocktail party graph CP(n) are obtained by complementing regular graphs of degree 0 and 1, respectively. Similarly, every (n − 3)-regular graph is the complement of a disjoint union of cycles. Therefore, r-regular graphs with r ∈ {0, 1, 2, n − 1, n − 2, n − 3} are easy to describe. By considering the remaining possible values for r, we are stepping into a rich world of regular graphs having various structural properties. Of course, since for an r-regular graph it holds nr = 2m, we have that there is no regular graph of odd order and odd degree. Moreover, concerning bipartite regular graphs that are not totally disconnected, we easily conclude that two parts must be of the same size, and so there is no bipartite regular graph of odd order unless it is totally disconnected. All r-regular graphs with r ≥ 2n are obtained by complementing those with r < 2n . In addition, every regular graph whose degree is at least half the number of its vertices must be connected. Considering small values for n, we deduce that all connected regular graphs with at most 7 vertices are cycles, cocktail party graphs, complete graphs or complements of (unions of) cycles. This gives 16 graphs in total.¹ For n = 8, the only critical case is a determination of cubic graphs. One of them is illustrated in Figure 1.3, while the remaining are left as an exercise. Exercise 1.2.1. Show that all connected cubic graphs with 8 vertices are Hamiltonian. Show next that exactly 4 additional non-isomorphic cubic graphs can be obtained by

1 In the entire book, whenever counting graphs we assume that isomorphic graphs are identified.

6 | 1 Introduction

Fig. 1.3. A cubic graph with 8 vertices.

relocating the 4 edges that do not belong to a Hamiltonian cycle of the graph illustrated in Figure 1.3. The 12 remaining connected regular graphs with 8 vertices are easily determined, giving 17 in total. The situation is similar for n = 9. There are 16 connected 4-regular graphs, giving 22 in total. The remaining numbers for 16 or fewer vertices are as follows (regular graphs of a comparatively small order can efficiently be generated by the publicly available program GENREG described in [228]): Order #

10 167

11 539

12 18 979

13 389 436

14 50 314 796

15 2 942 198 440

16 1 698 517 036 411

According to the Pólya enumeration theorem [250], the number of non-isomorphic graphs with n vertices is asymptotically 2(2) , n! n

while according to Bollobás [38] (see also Bender and Canfield [32]), the number of non-isomorphic regular graphs of degree at least 3 is asymptotically n−1

∑

r=3

(2m)! 2m (r!)n m!n!

e−

r2 −1 4

,

where m = nr 2 . It immediately follows that the share of regular graphs in the set of all graphs with a given order is comparatively small. For example, less than 2.7 ⋅ 10−9 % out of all connected graphs with 16 vertices are regular.

1.3 Elements of the theory of graph spectra | 7

1.3 Elements of the theory of graph spectra Here we introduce certain basic results concerning the spectrum of the adjacency matrix of a graph.² We start with the following, general, consideration. If the eigenvalues of a real n × n matrix M are real, then they can be arranged in non-increasing order as ν1 ≥ ν2 ≥ ⋅ ⋅ ⋅ ≥ ν n . If M is symmetric then it is said to be reducible if there exists a permutation matrix P such that P−1 MP = (

N1 0

0 ), N2

where N1 and N2 are square matrices. Otherwise, M is irreducible. It is well known from the Perron–Frobenius theory for non-negative matrices (see, for example, [136, Chapter 10] or [231, Chapter 8]) that the largest eigenvalue ν1 of an irreducible real symmetric matrix³ with non-negative entries is simple, while the coordinates of any eigenvector associated with ν1 are non-zero and of the same sign. In addition, |ν| ≤ ν1 holds for all remaining eigenvalues ν. Moreover, the largest eigenvalue of any principal submatrix of M is less than ν1 . Throughout the book we use I, J and O to denote a unit, an all-1 and an all-0 matrix, respectively. We also write j for an all-1 vector, O for an all-0 vector and ei for the ith vector of the canonical basis of ℝn (1 ≤ i ≤ n). Occasionally, the size of any of these matrices or the length of vectors will be given in a subscript. Rayleigh principle The Rayleigh quotient for a real symmetric matrix M is a scalar yT My , yT y where y is a non-zero vector in ℝn . The Rayleigh principle reads as follows. If {x1 , x2 , . . . , xn } is an orthonormal basis of the eigenvectors of M (that correspond to ν1 , ν2 , . . . , ν n , respectively) and if a non-zero vector y lies in the space spanned by the vectors xi , xi+1 , . . . , xn , then νi ≥

yT My , yT y

2 The whole contents can be extended to multigraphs in a very intuitive way. Multigraphs appear sporadically in Chapter 5. Occasionally, they are abbreviated to ‘graphs’. 3 Here we restrict ourselves to symmetric matrices, although a large part of the Perron–Frobenius theory covers non-symmetric ones.

8 | 1 Introduction with equality if and only if My = ν i y. Similarly, if a non-zero vector y lies in the space spanned by the vectors x1 , x2 , . . . , xi , then νi ≤

yT My , yT y

with equality if and only if My = ν i y. Simultaneously diagonalizable matrices We say that the n × n matrices N1 , N2 , . . . , N k are simultaneously diagonalizable if there exists a non-singular matrix P such that D i = P−1 N i P are diagonal for all i’s. If it exists, then P can be chosen in such a way that its columns have unit length. In that case, the main diagonal of D i consists entirely of the eigenvalues of N i . The identity ∑ki=1 P−1 N i P = ∑ki=1 D i gives ∑ki=1 N i = P (∑ki=1 D i ) P−1 , and so we have P−1 (∑ki=1 N i ) P = ∑ki=1 D i . Therefore, each eigenvalue of the sum ∑ki=1 N i is the sum of eigenvalues of summands. If, in addition, all N i ’s satisfy assumptions of the Perron–Frobenius theory, then ν1 (N i ) (1 ≤ i ≤ k) corresponds to a unique column of P whose entries are non-zero and of the same sign, giving k

k

i=1

i=1

ν1 ( ∑ N i ) = ∑ ν1 (N i ). By [178, Theorem 1.3.19], the matrices N1 , N2 , . . . , N k are simultaneously diagonalizable if and only if they pairwise commute. Incidence matrix, adjacency matrix and spectrum of a graph The vertex-edge incidence matrix R (= R(G)) of a graph G with n vertices and m edges is an n × m matrix whose rows and columns are indexed by the vertices and edges of G, that is the (i, e)-entry is r ie = {

1, 0,

if i is incident with e, otherwise.

The adjacency matrix A (or A(G)) of G is defined by A = (a ij ), where a ij = {

1, 0,

if i ∼ j, otherwise.

Remark 1.3.1. In a directed graph, every edge has a direction assigned with it. If such a graph is simple, then its adjacency matrix is also a (0, 1)-matrix in which the entry (i, j) is 1 if there is an edge directed from i to j, and 0 otherwise. Directed graphs appear in only a few pages of this book. The characteristic polynomial of A, denoted P G = det(xI − A), is simultaneously the characteristic polynomial of G, while its roots are the eigenvalues of G. Observe that

1.3 Elements of the theory of graph spectra | 9

the characteristic polynomial of a graph is invariant under vertex labellings since all adjacency matrices are mutually similar. The collection of eigenvalues of G (with repetition) is called the spectrum of G. These eigenvalues are denoted λ1 , λ2 , . . . , λ n . In addition, we assume that λ i ≥ λ j holds whenever i < j. We simultaneously use θ1 , θ2 , . . . , θ k to denote all distinct eigenvalues of G ordered decreasingly. We denote the multiplicity of the eigenvalue θ i by m i . Sporadically, we also use m λ for the multiplicity of an arbitrary eigenvalue λ. The spectrum of G may be denoted [λ1 , λ2 , . . . , λ n ], but we rather use the short expression [θ1 m1 , θ2 m2 , . . . , θ k m k ], where the multiplicities of distinct eigenvalues are given in the exponents. Remark 1.3.2. The λ-notation for eigenvalues dominates in this book. The θ-notation is convenient when we deal with distinct eigenvalues of a graph. It is used in Theorem 2.1.6, Sections 3.3, 3.4, 4.3–4.5 and Chapter 6. Also, the spectrum of a graph is always given in this notation. Throughout the book, when we say that a graph has k distinct eigenvalues, we always mean that it has exactly k distinct eigenvalues. The largest eigenvalue λ1 is called the spectral radius of G. For a non-negative integer k, the (i, j)-entry of A k is equal to the number of walks that contain k edges, start at vertex i and terminate at vertex j. In addition, for k ≥ 0 the kth spectral moment of G is defined by M k = ∑ni=1 λ ki . Note that M k is equal to the trace tr(A k ). The spectral gap η is the difference between the two largest eigenvalues, that is η = λ1 − λ2 . If a graph under consideration contains at least one edge, then its spectrum contains both positive and negative eigenvalues. By Smith [283], a graph has exactly one positive eigenvalues if and only if it has at least one edge and its non-isolated vertices form a complete multipartite graph. Since the adjacency matrix A of a graph G is symmetric, we can determine its minimal polynomial on the basis of the spectrum. It is given by ∏ki=1 (x − θ i ), where θ i ’s are all distinct eigenvalues of G. The identity ∏ki=1 (A l − θ i I) = O (l ≥ 0) follows directly. Using this identity, we obtain the result concerning an interplay between the number of distinct eigenvalues and the diameter. Theorem 1.3.3 ( cf. [87, Theorem 3.13]). If a connected graph has k distinct eigenvalues, then its diameter satisfies the inequality D ≤ k − 1. Since the characteristic polynomial P G is monic with integral coefficients, any of its monic divisors with rational coefficients cannot have non-integral coefficients (see [208, Chapter 4]). Consequently, every rational eigenvalue of a graph is integral.

10 | 1 Introduction

Fig. 1.4. A pair of cospectral regular graphs with 10 vertices.

We say that a graph is determined by the spectrum if there is no another graph with the same spectrum. Conversely, non-isomorphic graphs are said to be cospectral if they share the same spectrum. The smallest (with respect to the order) cospectral graphs are K1,4 and C4 ∪ K1 with common spectrum [2, 03 , −2]. The smallest connected cospectral graphs have 6 vertices. It can be confirmed by a computer search that all regular graphs with at most 9 vertices are determined by the spectrum. A pair of cospectral regular graphs with 10 vertices is illustrated in Figure 1.4. Their common spectrum⁴ is [4, 2.24, 1.56, 1, (−1)4 , −2.24, −2.56] . Eigenvectors Any eigenvalue λ of G satisfies the identity Ax = λx,

(1.1)

for some non-zero vector x ∈ ℝn . Every vector that satisfies this identity is called an eigenvector of G. A subspace of ℝn spanned by eigenvectors associated with λ is referred to as the eigenspace of λ. We denote it E(λ). If the vertices of G are labelled 1, 2, . . . , n and x = (x1 , x2 , . . . , x n )T is an eigenvector of G, then it is usually assumed that the coordinate x i corresponds to the ith vertex, where 1 ≤ i ≤ n. The equation λx i = ∑ x j (1 ≤ i ≤ n). j∼i

4 In the entire book, irrational eigenvalues are rounded to two decimal places.

(1.2)

1.3 Elements of the theory of graph spectra | 11

follows directly from the identity (1.1). It is known as the eigenvalue equation of G. In fact, this is a system of n equations involving an eigenvalue and the coordinates of an associated eigenvector. Given a complete system of orthonormal eigenvectors x1 , x2 , . . . , xn of A corresponding to the eigenvalues λ1 , λ2 , . . . , λ n , let X = (x1 | x2 | ⋅ ⋅ ⋅ | xn ) and let B be the diagonal matrix diag(λ1 , λ2 , . . . , λ n ). Then X is orthogonal matrix (i.e., X −1 = X T ) and it holds A = XBX T .

(1.3)

Let now x1 , x2 , . . . , xi be a complete set of orthonormal eigenvectors that correspond to an eigenvalue λ. If X λ = (x1 | x2 | ⋅ ⋅ ⋅ | xi ), then it follows that A k = ∑ λ k X λ X λT (k ≥ 0),

(1.4)

where the summation goes over all distinct eigenvalues. Moreover, if the orthonormal basis of ℝn formed by columns of X is constructed by stringing together the orthonormal basis of the eigenspaces of all distinct eigenvalues θ1 , θ2 , . . . , θ k , then the matrix B from (1.3) can be expressed as B = ∑ki=1 θ i E i , where for 1 ≤ i ≤ k, each matrix E i has a block diagonal form diag(O, . . . , O, I, O, . . . , O). Then the adjacency matrix has the spectral decomposition k

A = ∑ θi Pi ,

(1.5)

i=1

where P i = XE i X T (1 ≤ i ≤ k) is the orthogonal projection of ℝn onto E(θ i ). Using the eigenvalue equation, we get |λ||x i | ≤ ∑ |x j | ≤ ∆|x i |.

(1.6)

j∼i

In other words, the inequality |λ| ≤ ∆ holds for any eigenvalue λ of any graph G. Since the adjacency matrix of a connected graph is irreducible, the next results are immediate consequences of the Perron–Frobenius theory. Theorem 1.3.4 (cf. [89, Corollary 1.3.8]). A graph is connected if and only if its spectral radius is a simple eigenvalue with an eigenvector whose coordinates are non-zero and of the same sign. Unsurprisingly, the deletion of a vertex or an edge decreases the spectral radius of a connected graph. Corollary 1.3.5 (cf. [89, Proposition 1.3.10]). For any vertex i or any edge e of a connected graph G, we have λ1 (G − i) < λ1 (G) and λ1 (G − e) < λ1 (G). For a graph G with n vertices we have the following inequalities bounding its spectral radius

12 | 1 Introduction

λ1 (P n ) ≤ λ1 (G) ≤ λ1 (K n ),

(1.7)

where for the first one the connectedness is stipulated. Equalities hold if and only if G ≅ P n or G ≅ K n . The first inequality was obtained by Lovász and Pelikán [216], while the second is a direct consequence of the last corollary. Eigenvalue interlacing Consider the following general result. Theorem 1.3.6 (cf. [89, Theorem 1.3.11]). For a real n × n󸀠 matrix N such that N T N = I, let M be an n × n real symmetric matrix with eigenvalues ν1 ≥ ν2 ≥ ⋅ ⋅ ⋅ ≥ ν n . If the eigenvalues of N T MN are ξ1 ≥ ξ2 ≥ ⋅ ⋅ ⋅ ≥ ξ n󸀠 , then ν n−n󸀠 +i ≤ ξ i ≤ ν i (1 ≤ i ≤ n󸀠 ). Proof. If x1 , x2 , . . . , xn and y1 , y2 , . . . , yn󸀠 are the eigenvectors that respectively correspond to ordered eigenvalues of M and N, then let, for 1 ≤ i ≤ n󸀠 , zi denote a non-zero vector in the intersection of the subspace spanned by y1 , y2 , . . . , yi and the subspace that is orthogonal to the subspace spanned by N T x1 , N T x2 , . . . , N T xi−1 . Then Nzi is orthogonal to the subspace spanned by x1 , x2 , . . . , xi−1 . Using the Rayleigh principle, we get νi ≥

(Nzi )T M(Nzi ) zi T N T MNzi = ≥ ξi , (Nzi )T (Nzi ) zi T zi

which gives the second inequality. The first is obtained by replacing M with −M and N T MN with −N T MN in this proof. Now, if G is a graph with n vertices and H its induced subgraph with n󸀠 vertices, then A(H) = N T A(G)N, where N is the block matrix N = (I | O)T . Thus we get the following result. Theorem 1.3.7 ( Interlacing Theorem, cf. [89, Corollary 1.3.12]). Let G be a graph with n vertices and eigenvalues λ1 (G) ≥ λ2 (G) ≥ ⋅ ⋅ ⋅ ≥ λ n (G), and let H be an induced subgraph of G with n󸀠 vertices and eigenvalues λ1 (H) ≥ λ2 (H) ≥ ⋅ ⋅ ⋅ ≥ λ n󸀠 (H). Then λ n−n󸀠 +i (G) ≤ λ i (H) ≤ λ i (G) (1 ≤ i ≤ n󸀠 ). We say that the eigenvalues of H interlace the eigenvalues of G. The corresponding inequalities are known as the Cauchy inequalities. Here is another result concerning eigenvalue interlacing. Theorem 1.3.8 ( cf. [89, Corollary 1.3.13] or [158]). Let M be a real symmetric matrix with eigenvalues ν1 ≥ ν2 ≥ ⋅ ⋅ ⋅ ≥ ν n . Given a partition {1, 2, . . . , n} = P1 ⊔ P2 ⊔ ⋅ ⋅ ⋅ ⊔ P n󸀠 with |P i | = n i > 0, consider the blocking M = (M ij ) where M ij is an n i × n j block. Let e ij

1.3 Elements of the theory of graph spectra | 13

e

be the sum of entries in M ij and set B = ( niji ). Then the eigenvalues of B interlace those of M. The result follows by taking N=(

1 c1 √n1

󵄨󵄨 1 󵄨󵄨 󵄨󵄨 √n c2 2

󵄨 󵄨󵄨 󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨󵄨 1 cn󸀠 ) , 󵄨󵄨 √n 󸀠 󵄨󵄨 n

where the ci ’s are columns of the corresponding vertex-block incidence matrix. The matrices N, M and B = N T MN satisfy the assumptions of Theorem 1.3.6. The matrix B from the previous theorem is called the quotient matrix, while the corresponding vertex partition is said to be equitable if each block M ij has constant row sums. Divisor and main eigenvalues Given a graph G, let Π denote an equitable partition V(G) = P1 ⊔ P2 ⊔ ⋅ ⋅ ⋅ ⊔ P k such that each vertex in P i has b ij neighbours in P j . The directed multigraph with vertices P1 , P2 , . . . , P k and b ij directed edges from P i to P j is called the divisor of G (with respect to Π). The matrix B = (b ij ) is called the divisor matrix. We have the following result. Theorem 1.3.9 (Petersdorf and Sachs, cf. [87, Theorem 4.5]). The characteristic polynomial of any divisor of a graph divides its characteristic polynomial. An eigenvalue of a graph G is called a main eigenvalue if the corresponding eigenvector is not orthogonal to the all-1 vector j. Otherwise, the eigenvalue is non-main. The main part of the spectrum of G consists entirely of its main eigenvalues. Here is another result. Theorem 1.3.10 (Cvetković, cf. [90, Theorem 2.4.5]). The spectrum of any divisor of a graph G includes the main part of the spectrum of G. Remark 1.3.11. Theorems 1.3.9 and 1.3.10 remain valid for multigraphs [87, Chapter 4]. The spectral radius of a connected graph is always a main eigenvalue since the coordinates of a corresponding eigenvector are non-zero and of the same sign. The main angles of G are the cosines of angles between the eigenspaces of its eigenvalues and the vector j. Spectrum of a bipartite graph If G is bipartite, then its vertices can be labelled is such a way that its adjacency matrix A has the form (

O NT

N ). O

(1.8)

14 | 1 Introduction If G has n1 vertices in the first part and n2 vertices in the second, then the size of N is n1 × n2 . Let λ be an eigenvalue of A and x be a corresponding eigenvector. By setting x1 x1 x = ( ) and x󸀠 = ( ) x2 −x2 where the length of x i is n i (1 ≤ i ≤ 2), we obtain Ax󸀠 = (

O NT

N x1 −Nx2 −λx1 )( )=( T )=( ) = −λx󸀠 . −x2 N x1 λx2 O

In other words, the spectrum of G is symmetric with respect to the origin. The converse is also true. Indeed, if the spectrum of a graph is symmetric with respect to the origin, then its odd spectral moments are 0, and hence it does not contain odd cycles, which yields its bipartiteness. Moreover, it can be proved that the assumption of the converse state may be reduced to λ1 = −λ n (so, only the symmetry between extreme eigenvalues is required) [89, Theorem 3.2.4]. In this way we arrive at the following formulation. Theorem 1.3.12 (Cvetković, cf. [89, Theorems 3.2.3 and 3.2.4] or [100]). A graph G is bipartite if and only if its spectrum is symmetric with respect to the origin. If G is connected, then it is bipartite if and only if λ1 = −λ n . Remark 1.3.13. The form (1.8) of the adjacency matrix of a biparitite graph will be frequently used throughout the book. We suggest to the reader to remember the role of the submatrix N in this representation.

2 Spectral properties Basic properties of the spectra of regular graphs, including the Hoffman polynomial, are introduced in Section 2.1. In Section 2.2 we consider certain graph operations (like making the complement, the join or the line graph) applied to a regular graph and the spectrum of the resulting graph. Those who study graphs by means of their spectra usually work with different theories based on different matrices associated with graphs. In Section 2.3 we introduce the three frequently investigated matrices (precisely, the Laplacian matrix, the signless Laplacian matrix and the Seidel matrix) and give properties of their spectra. In a short Section 2.4 we inspect whether the regularity of a graph can be deduced from the spectrum of an associated matrix or not. The concept of distances of regular graphs, which is based on different kinds of the spectrum and includes reproduction of some of our computational results, is given in Section 2.5. Inequalities connecting spectral and structural invariants of regular graphs have received the attention in Section 2.6. There we meet the significance of the second largest and the least eigenvalue, and also consider the spectral radius of nearly regular graphs. There is a large amount of known inequalities that are attained exactly for regular graphs. Some of them are separated in Section 2.7. We include some classical results (like those of Stanley, Schwenk, Anderson and Morley or Merris) together with later and very recent developments. An extension of results concerning inequalities is given in Section 2.8 in which we estimate sets of largest or smallest eigenvalues by presenting the famous Serre theorem and also a variant of the Alon–Boppana theorem. In Section 2.9 we study the eigenvalue distribution for regular graphs. In particular, we consider the Erdős–Rényi random graphs and the corresponding semicircle eigenvalue distribution in relation to random regular graphs and the McKay distribution.

2.1 Basic properties of the spectrum of a regular graph If all-1 vector j is an eigenvector of a graph G with n vertices, then for the adjacency matrix A (of G) we have Aj = λj for some eigenvalue λ. In other words, it holds d i = λ (1 ≤ i ≤ n), that is G is a λ-regular graph. Conversely, for every r-regular graph G we deduce Aj = rj. Thus, we may formulate the following theorem. Theorem 2.1.1. A graph G is r-regular if and only if j is an eigenvector of G corresponding to the eigenvalue r. DOI: 10.1515/9783110351347-002

16 | 2 Spectral properties Even more, it can easily be concluded that the equality λ1 = ∆ (of (1.6)) is attained if and only if G is regular. A generalization of the last theorem is left as an exercise for the reader. T

Exercise 2.1.2 ([154]). Prove that if (√d1 , √d2 , . . . , √d n ) is an eigenvector of a graph G, then G is either regular or bipartite semiregular. Since tr(A2 ) = 2m, we immediately obtain the following result. Corollary 2.1.3 (cf. [87, Theorem 3.22] or [100]). A graph is r-regular if and only if n

r = λ1 and nλ1 = ∑ λ2i . i=1

In other words, regularity can be recognized from the spectrum. The next corollary is obvious. Corollary 2.1.4. The number of components of a regular graph is equal to the multiplicity of its spectral radius. Since in the case of a connected regular graph every eigenvector outside the space spanned by j is orthogonal to j, we have the following simple observation. Corollary 2.1.5. Connected graphs with exactly one main eigenvalue are precisely regular graphs. This eigenvalue is the largest one. Hoffman polynomial For any graph G with the adjacency matrix A, its adjacency algebra consists of all matrices P(A) where P is a polynomial with real coefficients. There is an interesting property of regular graphs. Theorem 2.1.6 (Hoffman Polynomial, [166], cf. [89, Theorem 3.5.5]). For a graph G with the adjacency matrix A there exists a polynomial P such that P(A) = J, where J is all-1 matrix, if and only if G is connected and regular. If G is r-regular, then we have k

P(x) = n ∏ i=2

x − θi , r − θi

(2.1)

where n is the number of vertices and r, θ2 , . . . , θ k are all distinct eigenvalues of G. Proof. Assume that the corresponding polynomial exists. In this case, J belongs to the adjacency algebra of G, which implies AJ = JA. If d1 , d2 , . . . , d n are vertex degrees, then we have AJ = (d1 , d2 , . . . , d n )T jT and JA = j(d1 , d2 , . . . , d n ), giving d1 = d2 = ⋅ ⋅ ⋅ = d n , and so G is regular. If G is disconnected, then there are vertices u and v belonging to different components of G. If so, then the (u, v)-entry of P(A) is 0 for any polynomial P, which contradicts the fact that J belongs to the adjacency algebra of G.

2.2 Operations on regular graphs and resulting spectra | 17

Assume now that G is connected and regular of degree r. Then the minimal polynomial of A has the form (x − r)W(x), for some polynomial W. From (A − rI)W(A) = 0, we deduce that AW(A) = rW(A). Hence, each column of W(A) is an eigenvector of A that corresponds to r. By Theorem 2.1.1, each column of W(A) must be a multiple of j. Since W(A) is symmetric it holds W(A) = cJ for some coefficient c, and the first part of the theorem follows. It remains to prove that the corresponding polynomial has the form (2.1). According to the above, we have P(x) = 1c W(x). Observe that the only non-zero eigenvalue of W(A) is ∏ki=2 (r − θ i ), and thus we have ∏ki=2 (r − θ i ) = cn, which gives the assertion. The cyclic structure of regular graphs can be considered by using the well known result which gives the coefficients of the characteristic polynomial on the basis of so-called elementary figures of a graph (i.e., the subgraphs isomorphic to K2 or C k (k ≥ 3)), see [87, Theorem 1.3]. We formulate the result in the form of an exercise, while the details can be found in [266] or [87, pp. 95–97]. Exercise 2.1.7. Let G be an r-regular graph with girth g and let k be a non-negative integer non-greater than min{n, 2g − 1}. Prove that the number of cycles of length k contained in G is determined by r and the first k coefficients of its characteristic polynomial. Consequently, the girth of a regular graph is determined by the spectrum.

2.2 Operations on regular graphs and resulting spectra Complement and bipartite complement Theorem 2.2.1 (cf. [23, Theorem 6.13] or [87, Theorem 2.6]). If G is an r-regular graph with n vertices and G its complement, then x−n+r+1 P G (−x − 1). x+r+1 That is, if the spectrum of G consists of r, λ2 , . . . , λ n , then the spectrum of G consists of n − 1 − r, −λ2 − 1, . . . , −λ n − 1. P G (x) = (−1)n

Proof. Let X be an orthogonal matrix whose first column is 1n j, such that X T A(G)X = diag(r, λ2 , . . . , λ n ) (i.e., the matrix X satisfies the identity (1.3)). On the basis of the identity A(G) + A(G) = J − I and the fact that the columns of X other than the first one are orthogonal to the first one, we compute X T A(G)X = X T (J − I − A(G))X = X T JX − I − X T A(G)X = diag(n − 1 − r, −λ2 − 1, . . . , −λ n − 1), which gives the assertion. Consequently, if two regular graphs are cospectral, so are their complements. By Theorem 2.1.1, j is an eigenvector corresponding to the spectral radius of both a regular

18 | 2 Spectral properties graph and its complement. We can see from the last proof that the remaining eigenvalues of G and G are paired in the same sense. Precisely, the following corollary holds. Corollary 2.2.2. An eigenvector that corresponds to the eigenvalue λ i (2 ≤ i ≤ n) of a regular graph G is simultaneously an eigenvector that corresponds to the eigenvalue −λ i − 1 of G. A graph is self-complementary if it is isomorphic to its complement. Observe that if G is a self-complementary r-regular graph, then G is connected and has n = 4k+1 vertices. In addition, r = 2k. There is a theorem. Theorem 2.2.3 (Sachs [267]). The characteristic polynomial of a self-complementary regular graph with 4k + 1 (k ≥ 1) vertices has the form 2k+1

P G (x) = (x − 2k) ∏ (x − λ i )(x + λ i + 1). i=2

Proof. According to Theorem 2.2.1, we have P G (x) = P G (x) = −

x − 2k P G (−x − 1). x + 2k + 1

Equivalently, P G (x) P G (−x − 1) = . (2.2) x − 2k −x − 1 − 2k For the eigenvalues of G that are different from 2k, we have ∏4k+1 i=2 (x − λ i ) = 4k+1 ∏j=2 (x + λ j + 1), that is each eigenvalue λ i corresponds to the eigenvalue λ j = −λ i − 1 where λ i ≠ λ j (otherwise, λ i = − 21 which is impossible since every rational eigenvalue of a graph must be integral). The last observation, together with (2.2), gives the assertion. Using the similar reasoning, the reader can prove the following result. Theorem 2.2.4 ([311]). If G is a connected bipartite r-regular graph with 2n vertices, then the characteristic polynomials of G and its bipartite complement G satisfy P G (x) P G (x) = . x2 − r2 x2 − (n − r)2 In other words, apart from the eigenvalues ±r of G and ±(n − r) of its bipartite complement, the spectra of these two graphs coincide. If G is disconnected, then its bipartite complement is not uniquely determined, but even then the above formula remains valid. Exercise 2.2.5 ([199]). Prove that the sum of two largest eigenvalues r and λ2 of a bipartite r-regular graph does not exceed 2n . Prove next that this bound is attained if and only if its bipartite complement is disconnected.

2.2 Operations on regular graphs and resulting spectra | 19

Join of regular graphs Theorem 2.2.6 (Finck and Grohmann [126]). The characteristic polynomial of the join of two regular graphs G1 and G2 is given by the formula P G1 (x)P G2 (x) ((x − r1 )(x − r2 ) − n1 n2 ) , (x − r1 )(x − r2 ) where n i and r i denote the order and degree of G i (1 ≤ i ≤ 2). P G1 ∇G2 (x) =

The last result can be derived from the formula for the characteristic polynomial of the join of two arbitrary graphs (obtained by Cvetković, cf. [87, Theorem 2.7]). It reads as follows: P G1 ∇G2 (x)

=

(−1)n2 P G1 (x)P G2 (−x − 1) + (−1)n1 P G2 (x)P G1 (−x − 1)

−(−1)n1 +n2 P G1 (−x − 1)P G2 (−x − 1).

A successive application of the last theorem leads to the following result. Theorem 2.2.7 (Finck and Grohmann [126]). For regular graphs G1 , G2 , . . . , G k , assume that G i have n i vertices and degree r i (1 ≤ i ≤ k). If n1 − r1 = n2 − r2 = ⋅ ⋅ ⋅ = n k − r k , then the graph G = G1 ∇G2 ∇ ⋅ ⋅ ⋅ ∇G k has n = ∑ki=1 n i vertices and is regular of degree n − n1 + r1 . It also holds k

P G (x) = (x − r)(x + n − r)k−1 ∏ i=1

P G i (x) . x − ri

Line graphs We first establish a short proof of the well known spectral property of generalized line graphs. Theorem 2.2.8 (cf. [290, Theorem 3.12]). The least eigenvalue of any generalized line graph is greater than or equal to −2. Proof. Define the vertex-edge incidence matrix R of a multigraph ̂ G described in the definition of a generalized line graph H ≅ Line(G; a1 , a2 , . . . a n ) (page 5) as a usual incidence matrix with one exception: If e and f are the edges between u and v in a petal at u, then exactly one of the entries (v, e), (v, f) is 1 and the other is −1. Then the adjacency matrix of H is R T R − 2I, which yields λ n (G) ≥ −2. Even more, we may classify all connected regular generalized line graphs. Theorem 2.2.9 (cf. [94, Proposition 1.1.9]). A connected regular generalized line graph is either a line graph or a cocktail party graph. Proof. Assume that a connected regular graph Line(G; a1 , a2 , . . . , a n ) is neither a line graph nor a cocktail party graph. Then a i > 0 holds for some vertex v i of G. But then,

20 | 2 Spectral properties the degree of an edge incident with v i as a vertex of the graph under consideration is greater than the degree of an edge in any of the a i petals at v i , and we are done. The line graph Line(G) is regular if and only if its root graph G is regular or bipartite semiregular. In both situations the degree of Line(G) is equal to the edge degree of G and the characteristic polynomial of G determines the characteristic polynomial of Line(G). Namely, the following statements hold. Theorem 2.2.10 (cf. [89, Theorem 2.4.1]). If G is an r-regular graph with n vertices and m edges, then PLine(G) (x) = (x + 2)m−n P G (x − r + 2). Proof. Considering the incidence matrix R (of G), we get the identities RR T = A(G)+ rI and R T R = A(Line(G))+2I. Now, the result follows from the fact that the matrices RR T and R T R have the same non-zero eigenvalues. The proof of the next theorem can be found in the corresponding references. Theorem 2.2.11 (cf. [87, Theorem 2.16] or [89, Theorem 2.4.2]). Let G be a bipartite (r1 , r2 )-semiregular graph with n1 vertices in the first part and n2 vertices in the second. If n1 ≥ n2 , then PLine(G) (x) = (x + 2)b P G (√a1 a2 )√(

a1 n1 −n2 , ) a2

(2.3)

where a i = x − r i + 2 (1 ≤ i ≤ 2) and b = n1 r1 − n1 − n2 . In the context of the last two theorems, one may ask the following question: Whether the spectrum of a regular line graph determines the spectrum of its root graph or not? We will consider this question much later in Subsection 4.5.1. We will see that, in general case, the answer to this question is negative. Even more, in general case, the spectrum of a regular line graph does not provide the information on whether its root graph is regular or bipartite semiregular. Certain unary operations The subdivision graph Sd(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Theorem 2.2.12 (Cvetković [84], cf. [87, Theorem 2.17]). If Sd(G) is the subdivision graph of an r-regular graph G with n vertices and m edges, then PSd(G) (x) = x m−n P G (x2 − r). Proof. Since the subdivision graph is bipartite we may assume that its adjacency matrix has the form as in (1.8), and then we have

2.2 Operations on regular graphs and resulting spectra | 21

Fig. 2.1. Petersen graph.

󵄨󵄨 󵄨 xI m PSd(G) (x) = 󵄨󵄨󵄨󵄨 T 󵄨󵄨 −N

󵄨 󵄨󵄨 󵄨󵄨 I −N 󵄨󵄨󵄨 󵄨󵄨 = x m 󵄨󵄨󵄨󵄨xI n − N T m N 󵄨󵄨󵄨󵄨 . 󵄨 󵄨󵄨 x 󵄨󵄨 xI n 󵄨󵄨

Since the last determinant is equal to x−n |x2 I n −N T N|, which can also be expressed as n − A(G) − rI n |, we get the assertion. x−n |x2 I

Here is a similar result. Exercise 2.2.13 ([84], cf. [87, Theorem 2.18]). Let G󸀠 be the graph obtained from an r-regular graph G by adding a new vertex corresponding to each edge of G and by joining each new vertex to the endvertices of the edge corresponding to it. If G has n vertices and m edges, then prove that P G󸀠 (x) = x m−n (x + 1)n P G (

x2 − r ). x+1

We give the eigenvalues of certain particular regular graphs. Example 2.2.14. The spectrum of the (regular) graph nK1 consists of n numbers all equal to 0. The spectrum of the complete graph K n consists of n − 1 and n − 1 numbers all equal to −1. The cocktail party CP(n) is the complement of nK2 , and therefore by Theorem 2.2.1, its spectrum consists of 2n − 2, n numbers all equal to 0 and n − 1 numbers all equal to −2. By [87, p. 53], the spectrum of the cycle C n consists of the numbers 2 cos (

2π i) (1 ≤ i ≤ n). n

The spectrum of the Petersen graph (see Figure 2.1) is [3, 15 , (−2)4 ].

22 | 2 Spectral properties

2.3 Other kinds of graph spectrum Here we introduce the three matrices that are frequently associated with graphs and consider their spectra (mostly in the case when a graph under consideration is regular): the Laplacian matrix, the signless Laplacian matrix and the Seidel matrix. The distance matrix is considered later in Chapter 6.

2.3.1 Laplacian and signless Laplacian spectrum Let D denote the diagonal matrix of vertex degrees in a graph G, that is D = diag(d1 , d2 , . . . , d n ). The Laplacian matrix L (= L(G)) of G is defined by L = D − A. Similarly, the signless Laplacian matrix Q (= Q(G)) is defined by Q = D + A. In other words, the matrix Q is obtained by removing the minus sign from the Laplacian matrix. Remark 2.3.1. In literature, L is also referred to as the Kirchhoff matrix or the admittance matrix. The matrix Q is also called the co-Laplacian matrix. The eigenvalues of L and Q are respectively denoted μ1 , μ2 , . . . , μ n and κ1 , κ2 , . . . , κ n , with the assumptions that μ i ≥ μ j and κ i ≥ κ j , whenever i < j. μ1 and κ1 are respectively referred to as the L-spectral radius and the Q-spectral radius of a graph. The characteristic polynomial, the spectrum and other quantities related to the eigenvalues of the adjacency matrix are defined analogously, along with the adaptation of the corresponding terminology. For example, the eigenvalues and the spectrum of the Laplacian matrix are abbreviated to the L-eigenvalues and the L-spectrum, respectively. Similarly, we deal with the Q-eigenvalues and the Q-spectrum. The matrices L and Q are positive semidefinite, and therefore the corresponding spectra are non-negative. Observe that L is a singular matrix with all-1 eigenvector j ∈ ℝn corresponding to μ n = 0. Due to Fiedler [124, 125], the L-eigenvalue μ n−1 is called the algebraic connectivity. It is not difficult to see that the L-spectrum and the Q-spectrum coincide for bipartite graphs [290, Theorem 1.19]. Next, the L-spectrum

2.3 Other kinds of graph spectrum | 23

provides information on whether a graph is connected (the number of components is equal to the multiplicity of 0) but not whether it is bipartite. If we know that a graph is connected, then the Q-spectrum gives information on whether it is bipartite and vice versa, if we know that a graph is bipartite then the Q-spectrum gives information on whether it is connected. The L-spectrum of any graph gives full information about the L-spectrum of its complement [290, Theorem 1.11]. It is also known that the L-spectrum determines the number of spanning trees τ in a connected graph. This result follows from a classical theorem of the algebraic graph theory known as the matrix-tree theorem, which states that all cofactors of L are mutually equal and their common value is just the number of spanning trees. Precisely, if μ1 ≥ μ2 ≥ ⋅ ⋅ ⋅ > μ n = 0 are the eigenvalues of L, then the number of spanning trees in G is given by τ(G) = 1n ∏n−1 i=1 μ i . There is an analogue to the interlacing theorem. Theorem 2.3.2 (cf. [91] or [145, Theorem 13.6.2]). For a graph G, let ν1 (G) ≥ ν2 (G) ≥ ⋅ ⋅ ⋅ ≥ ν n (G) be either its L-eigenvalues or Q-eigenvalues, and let H be a graph obtained by removing k (k < n) edges from G. The eigenvalues ν1 (H) ≥ ν2 (H) ≥ ⋅ ⋅ ⋅ ≥ ν n (H) of H satisfy ν i+k (G) ≤ ν i (H) ≤ ν i (G) (1 ≤ i ≤ n − k). In the case of an r-regular graph, the eigenvector j that corresponds to the eigenvalue r may be extended to an orthogonal basis of ℝn of eigenvectors that are common to the matrices A, rI ± A and J − I − 2A. This enables us to compute the eigenvalues of D ± A and obtain the following theorem. Theorem 2.3.3 (cf. [89, p. 5]). If the eigenvalues of an r-regular graph are given by r ≥ λ2 ≥ ⋅ ⋅ ⋅ ≥ λ n , then its L-eigenvalues are r − λ n ≥ r − λ n−1 ≥ ⋅ ⋅ ⋅ ≥ r − λ1 = 0 and its Q-eigenvalues are r + λ1 ≥ r + λ2 ≥ ⋅ ⋅ ⋅ ≥ r + λ n . Since there are different matrices that can be used in a study of graphs by means of the corresponding eigenvalues, we may speak about different theories of graph spectra where each of them is based on the corresponding matrix. For example, this matrix can be any of the matrices A, L or Q. For each of these three matrices the corresponding theory is highly developed. But, by the last theorem, in the case of regular graphs the spectrum of any of these matrices contains full information about the other two spectra, and therefore for regular graphs all results concerning the eigenvalues of one of these matrices can be expressed in terms of the eigenvalues of any of the other two. (For example, for an r-regular graph G, the above result counting the number of spanning trees in terms of the L-eigenvalues can be rephrased as τ(G) = 1n ∏ni=2 (r − λ i ).) According to these observations, in the largest part of the book our considerations are mostly focused on the adjacency matrix and the corresponding spectrum of a regular graph.

24 | 2 Spectral properties More on the signless Laplacian spectrum Since the signless Laplacian matrix Q of a connected graph is irreducible [93], Theorem 1.3.4 and Corollary 2.1.5 may be applied to its spectrum. Next, there are connections between the characteristic polynomial Q G of the matrix Q of a graph and the characteristic polynomial of its line graph or subdivision graph. Namely, the identity RR T = Q where R stands for the incidence matrix, can be verified directly. This identity, together with the one for R T R mentioned in the proof of Theorem 2.2.10, gives PLine(G) (x) = (x + 2)m−n Q G (x + 2).

(2.4)

Next, following the proof of Theorem 2.2.12 we get PSd(G) (x) = x m−n Q G (x2 ). Since the signless Laplacian matrix of any graph can be considered as the adjacency matrix of the corresponding multigraph with d i loops at vertex i (1 ≤ i ≤ n), by Remark 1.3.11, we conclude that Theorems 1.3.9 and 1.3.10 hold for the Q-spectrum. In Theorem 2.2.6 we provided a formula for the characteristic polynomial of the join of two regular graphs. As we said, this result is a consequence of a more general formula considering the join of any pair of graphs. It seems that there is no similar general result for the signless Laplacian, but we can offer the following result. Theorem 2.3.4 (Anđelić et al. [14]). The Q-polynomial of the join of two regular graphs G1 and G2 is given by the formula Q G1 ∇G2 (x) =

Q G1 (x − n2 )Q G2 (x − n1 ) ((x − 2r1 − n2 )(x − 2r2 − n1 ) − n1 n2 ), (x − 2r1 − n2 )(x − 2r2 − n1 )

where n i and r i denote the order and degree of G i (1 ≤ i ≤ 2). Proof. Using the equality P G (x) = Q G (x + r) (cf. Theorem 2.3.3) for regular graphs of degree r, we compute the signless Laplacian matrix of G1 ∇G2 as follows: Q(G1 ∇G2 ) = (

Q(G1 ) + n2 I n1 JT

J ). Q(G2 ) + n1 I n2

Since the divisor of this matrix has the form (

2r1 + n2 n1

n2 ), 2r2 + n1

we get that the Q-polynomial of G1 ∇G2 contains ((x − 2r1 − n2 )(x − 2r2 − n1 ) − n1 n2 ) as a factor. Assume first that both G1 and G2 are connected. Let κ2 (G1 ), . . . , κ n1 (G1 ) and κ2 (G2 ), . . . , κ n2 (G2 ) be their non-main Q-eigenvalues, and let e11 , e12 , . . . , e1n1

2.3 Other kinds of graph spectrum | 25

and e21 , e22 , . . . , e2n2 be the corresponding Q-eigenvectors. Since each of these Q-eigenvectors is orthogonal to all-1 vector, by direct computation we get that κ2 (G1 ) + n2 , . . . , κ n1 (G1 ) + n2 and κ2 (G2 ) + n1 , . . . , κ n2 (G2 ) + n1 are the Q-eigenvalues of G1 ∇G2 where κ i (G1 )+ n2 (resp. κ i (G2 )+ n1 ) corresponds to the Q-eigenvector whose first n1 (resp. last n2 ) coordinates coincide with the coordinates of e1i (resp. e2i ), while the remaining coordinates are 0’s. Therefore, we get our formula. Let now G1 and G2 be regular graphs having k and l components (with k + l ≥ 3), respectively. Then each non-main Q-eigenvalue of G1 and G2 gives the corresponding Q-eigenvalue of G1 ∇G2 in the same way as in the above. Since the sum of all Q-eigenvalues of G1 ∇G2 is equal to the trace of Q(G1 ∇G2 ), it is a mater of routine to check that k − 1 (resp. l − 1) of the remaining Q-eigenvalues are equal to 2r1 − n2 (resp. 2r2 − n1 ). If a graph G is bipartite (r, s)-semiregular, then its line graph is regular and the Q-spectrum of G and the spectrum of Line(G) are connected by the relation (2.4). The Q-eigenvalues of G are located by the following theorem. Theorem 2.3.5 ([280]). Let G be a bipartite (r, s)-semiregular graph with n1 (resp. n2 ) vertices of degree r (resp. s) (n1 ≥ n2 ). Then its Q-spectrum lies in [0, r] ∪ [s, r + s]. Proof. First, n1 ≥ n2 implies r ≤ s. Furthermore, using the relations (2.3) and (2.4) we get n2

Q G (x) = (x − r)n1 −n2 ∏ ((x − r)(x − s) − λ2i ) ,

(2.5)

i=1

where λ i (1 ≤ i ≤ n2 ) are the n2 largest eigenvalues of G. Since the Q-eigenvalues are real, we have (x − r)(x − s) ≥ 0, and the result follows. Setting λ1 = √rs in (2.5), we get the largest and the least Q-eigenvalue: r + s and 0, respectively. It is easy to see that the spectrum of G contains at least n1 −n2 eigenvalues equal to 0. The Q-eigenvalue r appears whenever G is not a regular graph, while the Q-eigenvalue s appears whenever the multiplicity of 0 is greater than n1 − n2 . The following exercise can be solved by using Theorem 1.3.12 and the relation (2.5). Exercise 2.3.6 ([280]). The Q-spectrum of a bipartite (r, s)-semiregular graph with the Q-eigenvalues r and s excluded is symmetric with respect to r+s 2 . More details on the signless Laplacian spectrum can be found in the three consecutive papers [96, 97, 98].

26 | 2 Spectral properties 2.3.2 Seidel spectrum The Seidel matrix of a graph is the matrix S = J − I − 2A, while its eigenvalues are denoted σ1 , σ2 , . . . , σ n , with the assumption that σ i ≥ σ j , whenever i < j. This matrix is mostly encountered in the context of regular graphs. We continue with a natural adaptation of the terminology introduced for the adjacency matrix. So, the above eigenvalues are referred to as the S-eigenvalues forming the S-spectrum, and so on. The following result can easily be deduced from the definition of the Seidel matrix. Theorem 2.3.7. If r, λ2 , . . . , λ n are the eigenvalues of an r-regular graph, then its S-eigenvalues are n − 1 − 2r, −1 − 2λ2 , . . . , −1 − 2λ n . Here is a direct corollary. Corollary 2.3.8. The number of distinct S-eigenvalues of a regular graph does not exceed the number of its distinct eigenvalues. The Seidel switching in a graph is described as follows. For any U ⊂ V(G), the graph H obtained from G by the Seidel switching (with respect to U) has the same set of vertices and two vertices are adjacent in H if and only if either (i) (ii)

they are adjacent in G and both belong to U or both do not belong to U or they are non-adjacent in G and exactly one of them belongs to U.

It is straightforward to verify that if G is r-regular with n vertices, then H is also r-regular if and only if U induces a k-regular subgraph of G, where |U| = n − 2(r − k). If S is the Seidel matrix of G, then the Seidel matrix of H is given by E−1 SE, where E is the diagonal matrix whose diagonal is a modified characteristic vector of U ⊂ V(G): An entry is 1 if the corresponding vertex belongs to U, and −1 otherwise. If so, then we may conclude that the Seidel switching is an equivalence relation on graphs that preserves eigenvalues of Seidel matrices (since they are similar). According to the last theorem, if G and H are non-isomorphic and regular of the same degree, then they are cospectral. At the first sight, the Seidel switching may be imagined as a powerful tool for constructing cospectral regular graphs. In fact, the following considerations give certain restrictions. Namely, if an r-regular graph G can be switched into an s-regular graph with respect to a subset U, then U and its complement (in G) determine a divisor of G with the adjacency matrix − n−r−s−t 2

(

r−s+t 2

n+r−s−t 2 ), r+s−t 2

2.4 Deducing regularity from the spectrum | 27

where t is the size of U. Since s − 2n is the eigenvalue of this matrix, by Theorem 1.3.9 it appears in the spectrum of G. In this way, we get the following result. Theorem 2.3.9 ([85]). If an r-regular graph with n vertices can be switched into an s-regular graph, then s − 2n is an eigenvalue of G. Moreover, the eigenvalue s − 2n must be integral, and therefore we have that a regular graph of an odd order cannot be switched into another regular graph. Moreover, if an r-regular graph with least eigenvalue λ n can be switched into another r-regular graph, then we have r − 2n ≥ λ n ≥ −r, which gives r ≥ 4n . The following observation is an immediate consequence. Corollary 2.3.10. If two r-regular graphs, with r < 4n , are cospectral then there is no Seidel switching from one of them into another. In particular, this apply for any pair of cospectral cubic graphs. The particular case follows from the computational result stating that there is no pair of cospectral cubic graphs with fewer than 14 vertices [89, p. 109]. If a graph under consideration is the line graph, then we can say more. Namely, if the root graph G is r-regular and has n vertices, then Line(G) is 2(r − 1)-regular and has nr 2 vertices. By the last theorem, if Line(G) can be switched into another 2(r − 1)-regular graph, then we have 2(r − 1) − nr 4 ≥ −2, giving n ≤ 8 (which is a strong restriction). If the root graph G is bipartite (r, s)-semiregular with n1 (resp. n2 ) vertices of degree r (resp. s), then Line(G) is (r + s − 2)-regular and has n1 r vertices. Using the same reasoning, we get r + s − 2 − n21 r ≥ −2, which yields n1 + n2 ≤ 9. In addition, G can have at most 18 edges. Strong restrictions, again.

2.4 Deducing regularity from the spectrum Obviously, Corollary 2.1.3 is sufficient to deduce, from the spectrum of the adjacency matrix, whether a graph is regular or not. Similarly, it is well known (cf. [66, 106]) that a graph is regular if and only if its L-eigenvalues satisfy n

n

i=1

i=1

2

n ∑ μ i (μ i − 1) = ( ∑ μ i ) . The following generalization gives an affirmative answer for the Q-spectrum, as well. Theorem 2.4.1 (Van Dam and Haemers [106]). A regular graph cannot be cospectral with a non-regular one with respect to the matrix M = a1 A + a2 J + a3 D + a4 I (a1 ≠ 0)

28 | 2 Spectral properties except possibly when a3 = 0 and −1
λ i+2 (C n ) = λ i+3 (C n ) (i even, 2 ≤ i ≤ n − 3) and λ i (C n ) > λ i (P n ) > λ i+1 (P n ) > λ i+1 (C n ) (i odd, 1 ≤ i ≤ n − 2), λ n (C n ) > λ n (P n ). Using the above (in)equalities, we obtain

σ(P n , C n )

=

n

λ1 (C n ) − λ1 (P n ) + ∑ |λ i (P n ) − λ i (C n )| i=2

=

λ1 (C n ) − λ1 (P n ) +

n−1

(λ i (P n ) − λ i (C n ) + λ i (C n ) − λ i+1 (P n ))

∑

i even, i=2

=

n

2 + ∑ (−1)n λ i (P n ) = 2, i=1

where the last equality follows from the facts that P n is bipartite and n is odd. Assuming that n is even, we compute λ1 (C n )

>

λ i (C n ) = λ i+1 (C n ) > λ i+2 (C n )

=

λ i+3 (C n ) > λ n (C n ) (i even, 2 ≤ i ≤ n − 4)

and λ i (C n ) > λ i (P n ) > λ i+1 (P n ) > λ i+1 (C n ) (i odd, 1 ≤ i ≤ n − 1). Thus, we have

2.5 Spectral distances of regular graphs | 31

n

n

i=1

i=1

σ(P n , C n ) = 2λ1 (C n ) + ∑ (−1)i λ i (P n ) = 4 − 2 ∑ (−1)i cos (

π i) = 2, n+1

where the last equality follows from the bipartiteness of P n and the trigonometrical n/2 π i) = 21 . identity ∑i=1 (−1)i cos ( n+1 We next consider the spectral distance between a bipartite r-regular graph and its complement. Theorem 2.5.2 ([189]). Given a bipartite r-regular graph G with n vertices such that its degree is not greater than degree of G, if |λ i (G) − λ i+1 (G)| ≤ 1 where 1 ≤ i ≤ ⌊ 2n ⌋, then we have σ(G, G) = 2(n − 2r − 1).

(2.7)

Proof. We compute

σ(G, G)

=

n

n

∑ |λ i (G) − λ i (G)| = (n − 2r − 1) + ∑ |λ i (G) + λ n−i+2 (G) + 1| i=1

=

i=2 ⌊n/2⌋

(n − 2r − 1) + 2 ∑ |λ i (G) − λ i+1 (G) − 1|. i=1

= 0 we get For n odd, since λ n+1 2 ⌊n/2⌋

2 ∑ |λ i (G) − λ i+1 (G) − 1| = (n − 1) − 2 (λ1 (G) − λ n+1 ) = n − 2r − 1. 2 i=1

In the same way, for n even we get ⌊n/2⌋

2 ∑ |λ i (G) − λ i+1 (G) − 1| = n − 2r − 1, i=1

and so in both cases we arrive at the equality (2.7). Spectral distances with respect to the Laplacian or the signless Laplacian matrix are defined analogously. We just add an appropriate subscript in the notation. Using Theorem 2.3.2, we get that σ L (G, G󸀠 ) = σ Q (G, G󸀠 ) = 2k holds for any graph G and its subgraph G󸀠 obtained by removing exactly k edges. Return now to regular graphs. Clearly, if G1 and G2 are such graphs of the same degree, then we have

32 | 2 Spectral properties

σ(G1 , G2 ) = σ L (G1 , G2 ) = σ Q (G1 , G2 ). The next slight modification is an easy exercise for the reader. Exercise 2.5.3 ([189]). Let O(G) stand for taking the line or the complement of a graph applied arbitrary number of times and in arbitrary order. If G1 and G2 are r-regular graphs with n vertices, then prove that σ(G1 , G2 ) = σ(O(G1 ), O(G2 )) = σ L (O(G1 ), O(G2 )) = σ Q (O(G1 ), O(G2 )). The next result is proved by direct computation. Theorem 2.5.4. Let G1 and G2 be regular graphs with n vertices and of degree r1 and r2 , respectively. If r1 ≤ r2 , then the following inequalities hold: (i) (ii)

σ Q (G1 , G2 ) ≤ σ(G1 , G2 ) + n(r2 − r1 ); σ L (G1 , G2 ) ≤ σ(G1 , G2 ) + n(r2 − r1 ).

The next inequality compares the spectral distance and the L-spectral distance between a regular graph and its complement. Theorem 2.5.5 ([189]). For an r-regular graph G with n vertices such that its degree is not greater than degree of G, we have σ L (G, G) ≤ σ(G, G) + (n − 2)(n − 2r − 1).

(2.8)

Proof. It holds μ n (G) = μ n (G) = 0 and, by the assumption, n − 2r − 1 ≥ 0. We compute

σ L (G, G)

=

n−1

∑ |μ i (G) − μ i (G)|

i=1

= = ≤ = =

n−1

∑ |(r − λ n−i+1 (G)) − (n − r + λ i+1 (G))|

i=1 n

∑ |λ i (G) + λ n−i+2 (G) + n − 2r| i=2 n

n

∑ |λ i (G) + λ n−i+2 (G) + 1| + ∑ |n − 2r − 1|

i=2 n

i=2

∑ |λ i (G) − (−1 − λ n−i+2 (G))| + |λ1 (G) − (n − 1 − r)| + (n − 2)(n − 2r − 1)

i=2 n

∑ |λ i (G) − λ i (G))| + |λ1 (G) − λ1 (G)| + (n − 2)(n − 2r − 1)

i=2

=

σ(G, G) + (n − 2)(n − 2r − 1),

and the assertion follows.

2.6 Inequalities for eigenvalues | 33

Observe that equality in (2.8) is attained for the Petersen graph. In Subsection 3.4.5 we consider more results regarding spectral distances and establish an infinite family of regular graphs attaining this equality.

2.6 Inequalities for eigenvalues Here we consider inequalities that involve the second largest and the least eigenvalue of regular graphs as well as the spectral radius of some graphs that are close to regular ones. We also focus on the role of these eigenvalues in the study of the structure of regular graphs.

2.6.1 Second largest eigenvalue Since the spectral radius of a regular graph is equal to its degree, it does not provide any additional information, and so it seems natural to consider the second largest eigenvalue. For example, a classical result is the ordering of cubic graphs with respect to their second largest eigenvalues given in [55]. The next result reveals a connection between this eigenvalue and the structure of a regular graph. Theorem 2.6.1 (Cvetković [83], Koledin and Stanić [199]). For a non-complete r-regular graph G with n vertices, let d be the average vertex degree of a subgraph induced by the vertices non-adjacent to an arbitrary vertex of G. Then d≤r

λ22 + λ2 (n − r) . λ2 (n − 1) + r

(2.9)

If, in addition, G is triangle-free and λ22 < r, then n≤

r2 (λ2 + 2) − rλ2 (λ2 + 1) − λ22 r − λ22

.

(2.10)

Proof. We partition the vertices of G into the three parts (an arbitrary vertex v, the vertices adjacent to v and the remaining vertices) and consider the corresponding blocking A = (A ij ), 1 ≤ i, j ≤ 3, of its adjacency matrix. The matrix whose entries are the average row sums in A ij has the form 0 B = (1 0

r r − ev − 1 r−d

0 ev ) , d

where e v is the average number of edges between a vertex adjacent to v and vertices . Now, non-adjacent to v. Counting the number of such edges, we get e v = (n−r−1)(r−d) r

34 | 2 Spectral properties the characteristic polynomial of B is (x − r) (x2 − (d − e v − 1)x − d). By Theorem 1.3.8, the eigenvalues of B interlace those of A, which yields the inequality λ22 − (d − e v − 1)λ2 − d ≥ 0. Substituting the above expression for e v in the last inequality, we get (2.9). If G is triangle free, we have (1, 0, r − 1) for the second row of B and, in a similar way, we get λ22 − (d − r)λ2 − d ≥ 0. Furthermore, there are r2 edges between the set of vertices adjacent to v and the set of remaining vertices, so we have d = r(n−2r) n−r−1 which, together with the previous inequality, gives (2.10). The first inequality of the last theorem gives a close relation between the second largest eigenvalue and the structure of a regular graph. There, the upper bound for d decreases as λ2 decreases. In other words, a decrease in λ2 reduces the number of edges in the subgraph induced by the vertices non-adjacent to v, and simultaneously gives more edges between this subgraph and the subgraph induced by the neighbours of v. This leads to the conclusion that, in an informal sense, regular graphs with small λ2 have more ‘round’ shape (smaller diameter and higher connectivity), while those with large values of λ2 have larger diameter and lower connectivity [57, 89]. The second inequality gives an estimate for the order of a triangle-free regular graph in terms of its degree and second largest eigenvalue. Here is another result concerning diameter. Theorem 2.6.2 ([198]). Let G be a connected r-regular (r ≥ 3) graph satisfying λ2 < √r. Then its diameter is at most 3. Proof. Assume that the diameter is greater than 3 and consider two vertices u and v at distance 4. Then there are no edges between closed neighbourhoods N[u] and N[v], and so λ2 (G) ≥ λ2 (G[N[u] ∪ N[v]]) ≥ min (λ1 (G[N[u]]), λ1 (G[N[v]])) ≥ λ1 (K1,r ) = √r. A contradiction. If we consider bipartite graphs instead of those that are triangle-free, then we can obtain a simple expression for (2.10). Namely, the inequality n≤2

r2 − λ22 r − λ22

(2.11)

holds for any connected bipartite r-regular graph with diameter 3 (with r > λ22 ) [199].

2.6 Inequalities for eigenvalues | 35

More results on triangle-free regular graphs are given in the form of exercises (for all of them, see [199]). Exercise 2.6.3. If a connected non-bipartite triangle-free r-regular graph has the property λ2 ≤ c where c > 0, then prove that r ≤ c4 + 2c3 + 4c2 + 2c + 3. Using the previous exercise, one may prove the following two. In both cases, the second largest eigenvalue is used to check whether a given regular graph is bipartite. Exercise 2.6.4. Prove that a connected triangle-free r-regular graph with the property r > λ42 + 2λ32 + 4λ22 + 2λ2 + 3 is bipartite. Exercise 2.6.5. Prove that a connected {C3 , C5 }-free r-regular graph with the property r > 2λ22 + λ2 + 1 is bipartite. Apart from Theorem 2.6.1, inspecting connectivity of a graph by means of its eigenvalues is considered from many aspects in many sources. Here we give more results on this topic along with a remark that Chapter 5 is devoted to results concerning the same problems. There is a classical result of Fiedler [124] which states that c v ≥ μ n−1 holds for any non-complete graph. Here, c v stands for the vertex connectivity and μ n−1 for the algebraic connectivity. Recall that for r-regular graphs we have μ n−1 = r − λ2 . Considering the edge connectivity of regular graphs, Cioabă [73] proved that c e ≥ k holds for any connected r-regular graph with the property λ2 ≤ r − 2(k−1) r+1 , where r ≥ k ≥ 2. For k ∈ {2, 3}, the last result is improved to the following: c e ≥ 2 (resp. c e ≥ 3) holds for any connected r-regular graph (with r ≥ 3) whose second largest eigenvalue is less than the largest root of x3 − (r − 3)x2 − (3r − 2)x − 2 = 0 (resp. less 2 −16 √ than r−3+ (r+3) ). These results extend a result of Krivelevich and Sudakov [206] 2 who proved that λ2 ≤ r − 2 implies c e = r. We mention without proof a result of Nilli [243]. If G is a connected r-regular graph which contains two edges whose distance apart is at least 2k + 2, then λ2 (G) ≥ 2√r − 1 (1 −

1 1 . )+ k+1 k+1

Continue now with an interplay of λ2 and the girth g of a regular graph. Theorem 2.6.6 ([194]). If G is a connected r-regular graph with girth at least 5, then r ≤ λ22 + λ2 + 1. If, in addition, G is bipartite then r ≤ λ22 + 1. Proof. Consider two adjacent vertices, u and v. Since G is triangle-free, the subgraphs induced by the neighbourhoods N(u) and N(v) consists of isolated vertices, and since G is also quadrangle-free, there are no edges between the sets N(u)\v and N(v)\u. By direct computation, we get that the second largest eigenvalue of G[N[u] ∪ N[v]] is −1+√4r−3 , and thus λ2 (G) ≥ −1+√24r−3 . After simplifying, we obtain the result. 2

36 | 2 Spectral properties If G is bipartite, consider two vertices, u and v, such that d(u, v) = 2. Since the girth of G is at least 6, there is a unique common neighbour of u and v, say w. We see at once that the subgraph G[(N[u] ∪ N[v])\w] is isomorphic to 2K1,r−1 . Hence, λ2 (G) ≥ λ2 (2K1,r−1 ) = λ1 (K1,r−1 ) = √r − 1, and we are done. To meet the graphs attaining the bounds of the last theorem, we need to develop more theory. Required non-bipartite graphs are given on page 41. The bipartite graphs attaining the corresponding bound are given much later in Corollary 3.8.26 (page 118).

2.6.2 Toughness of regular graphs The toughness of a connected graph G is a structural invariant defined by t(G) = min {

|U| : U ⊆ V(G), G[V(G)\U] is disconnected} , comp(G[V(G)\U])

where comp(G[V(G)\U]) denotes the number of components of a graph obtained by removing the vertices of U. This invariant was introduced by Chvátal in 1973 [72] who investigated the cyclic structure of a graph. The toughness of a graph is related to many other important properties such as Hamiltonicity or the existence of spanning trees [27]. For example, it is not difficult to see that t(G) ≥ 1 holds for any Hamiltonian graph G [71]. The same author asked does there exist a constant c such that any graph with toughness greater than c is Hamiltonian. Bauer et al. [28] established that if such a constant exists, then it must be at least 49 . For regular graphs, there is the following result. Theorem 2.6.7 (Brouwer [47]). For a connected r-regular graph G, t(G) >

r − 2, Λ

(2.12)

where Λ = max{|λ2 |, |λ n |}. Brouwer conjectured that the lower bound of the previous theorem can be improved to t(G) > Λr − 1. This bound would be the best possible as there exists bipartite regular graphs with toughness arbitrarily close to 0 (cf. [319]). Here we give two results taken from the doctoral thesis of Wong [319] who improved the inequality (2.12) in certain cases. Theorem 2.6.8 ([319]). Let G be an r-regular graph (r ≥ 3) with n vertices and edge cr connectivity c e . If c ≤ cre is a positive number such that λ2 < r − r+1 , then t(G) ≥ c.

2.6 Inequalities for eigenvalues | 37

Proof. Following the original reference, we prove the contrapositive. So, assume that there is a constant c satisfying c ≤ cre ≤ 1 for which we have t(G) < c. Then there is a set U ∈ V(G) of size s such that k = comp(G[V(G)\U]) > cs ≥ s. Let G1 , G2 , . . . , G k be the components of G[V(G)\U] and let, in addition, n i and m i (G i , U) denote the order of G i and the number of edges between G i and U (1 ≤ i ≤ k), respectively. We compute ∑ki=1 m(G i , U) ≤ rs < rck. Observe that there exist at least two m(G i , U)’s that are not greater than rc (otherwise, we would have ∑ki=1 m(G i , U) ≥ c e + (k − 1)rc = rck + c e − rc ≥ rck). Hence, we may set the following: max {m(G1 , U), m(G2 , U)} ≤ rc < r. A simple combinatorial observation provides min{n1 , n2 } ≥ rc , we have r + 1. Then, since the average vertex degree in G1 is r − m(Gn11,U) ≥ r − r+1 rc rc λ1 (G1 ) ≥ r − r+1 and, by the same argument, λ1 (G2 ) ≥ r − r+1 . Application of the inrc terlacing theorem gives λ2 (G) ≥ λ2 (G1 ∪ G2 ) ≥ r − r+1 , and the proof is complete. The proof of the next result can be found in the original reference. Theorem 2.6.9 ([319]). If G is a connected r-regular graph and { λ2 < { {

r−2+√r2 +8 , 2 2 √ r−2+ r +12 , 2

when r is odd, when r is even,

then t(G) ≥ 1. It is proved in [319], by constructing examples of r-regular graphs whose second largest eigenvalue equals the right hand side of the corresponding inequality, that the condition of the last theorem is the best possible.

2.6.3 More inequalities We start with an inequality that involves an arbitrary eigenvalue distinct from the degree, and then give inequalities that involve the least eigenvalue of a regular graph. In the sequel, we consider estimates for the spectral radius of nearly regular graphs, that is graphs obtained from regular ones by removing or adding an edge. Theorem 2.6.10 (Rowlinson and Tayfeh-Rezaie [262]). Let G be a connected r-regular graph and λ its eigenvalue distinct from r. If m λ stands for the multiplicity of λ and |V(G)| = r + m λ + 1, then r−(

mλ + 1) λ2 − 2λ ≥ 0. r

(2.13)

Proof. There are r eigenvalues of G distinct from r or λ. We denote them λ󸀠1 , λ󸀠2 , . . . , λ󸀠r . Computing the spectral moments, we obtain the equalities r + m λ λ + ∑ri=1 λ󸀠i = 0 and 2 r2 + m λ λ2 + ∑ri=1 λ󸀠i = (r + m λ + 1)r. By setting ̃λ = 1r ∑ri=1 λ󸀠i , we get

38 | 2 Spectral properties

r

r

2 mλ 2 + 1) λ2 − 2λ) , ∑ (λ󸀠i − ̃λ) = ∑ λ󸀠i − r ̃λ2 = m λ (r − ( r i=1 i=1

and (2.13) follows. Remark 2.6.11. Observe that equality holds in (2.13) if and only if all λ󸀠i ’s are equal to ̃λ. In other words, if and only if G has 3 distinct eigenvalues. In Section 3.4 (in particular, Theorem 3.4.7) we will see that a connected regular graph has 3 distinct eigenvalues if and only if it belongs to a special class of regular graphs called strongly regular. In this spirit, we can say that equality holds in (2.13) if and only if G is a strongly regular graph. The following result is a sort of generalization of Theorem 2.6.1. It may be proved in a similar way, that is by considering blocking of the adjacency matrix. Theorem 2.6.12 (cf. [89, Theorem 3.5.7]). Given an r-regular graph with eigenvalues r, λ2 , . . . , λ n , let G󸀠 be its induced subgraph with n󸀠 vertices and average vertex degree d󸀠 . Then n󸀠 (r − λ n ) n󸀠 (r − λ2 ) + λ n ≤ d󸀠 ≤ + λ2 . n n We mention a classical result of Delsarte and Hoffman concerning the upper bound for the independence number (cf. [144]) and stating that α≤

−λ n n r − λn

(2.14)

holds for any r-regular graph. The last inequality may also be proved by blocking the adjacency matrix. In addition, it can be derived from a result of Bolobás and Nikiforov (concerning the upper bound for λ n ) [39] or from a result of Godsil and Newman (concerning the upper bound for the size of an independent set) [144]. Spectral radius of nearly regular graphs Here are the effects of removing or adding an edge to an r-regular graph on a spectral radius. Theorem 2.6.13 (Cioabă [75]). Let G be an r-regular graph and e be an edge of G such that H = G − e is connected. Then 1 2 < r − λ1 (H) < , nD n where D is the diameter of H. The proof is a direct consequence of the inequality λ1 < ∆ −

1 , nD

(2.15)

2.7 Extremal regular graphs | 39

that holds for any connected non-regular graph (see the same reference or [290, Theorem 2.11]). Later on, Nikiforov [242] generalized the left inequality of (2.15) by showing that it holds for any proper induced subgraph of H and when D stands for the diameter of G. The next theorem can be found in [290] where we gave a slight modification of a result of Cioabă [75]. Theorem 2.6.14 (cf. [290, Theorem 2.44]). Let G be a connected r-regular graph and H be obtained by adding an edge e to G. In addition, assume that r − λ2 (G) > 1. Then 2 1 2 1 + < λ1 (H) − r < ( + 1) . n n(r + 2) + 2 n r − λ2 (G) − 1 The left inequality follows from the inequality for non-regular graphs

λ1 −

2m { >{ n {

1 n(∆+2) , 1 2(m+n) ,

if ∆ − δ = 1 and all but one vertices have equal degree, otherwise

(cf. [239]). Namely, using this result we get that 2m 1 + , n 2(m + n) where m is the number of edges of H. Since G is r-regular, we have r = 2(m−1) , equivan nr+2 lently m = 2 . Setting this in the last inequality, we get the assertion. The right inequality is obtained in [75] by a straightforward computation that includes the technique of Maas based on intermediate eigenvalue problems of the second type (for more details, see [220]). λ1 (H) >

2.7 Extremal regular graphs Given an inequality that involves some graph eigenvalues, it is of great importance to determine which graphs the equality is attained for. It occurs that, in many situations, these graphs are regular ones. We give a brief proofless review of such results. Eigenvalues of the adjacency matrix The first is an upper bound for the spectral radius obtained independently by several authors [176, 241, 322], δ − 1 + √(δ + 1)2 + 4(2m − δn) . (2.16) 2 Equality holds if and only if a corresponding graph is regular or has a component of order ∆ + 1 in which every vertex is of degree δ or ∆, and all other components are δ-regular. λ1 ≤

40 | 2 Spectral properties We note that the well known Stanley inequality λ1 ≤

1 √ ( 8m + 1 − 1) 2

and the Schwenk inequality λ1 ≤ √2m − n + 1 follow from the previous one [290, p. 45]. The result of Shu and Wu [278] considers connected graphs with vertex degrees d1 ≥ d2 ≥ ⋅ ⋅ ⋅ ≥ d n for which we have d i − 1 + √(d i + 1)2 + 4(i − 1)(d1 − d i ) , 2 where 1 ≤ i ≤ n. If i = 1, then equality holds if and only if our graph is regular. If 2 ≤ i ≤ n, then equality holds if and only if it is either a regular graph or a bidegreed graph with ∆ = d1 = d2 = ⋅ ⋅ ⋅ = d i−1 = n − 1 and d i = d i+1 = ⋅ ⋅ ⋅ = d n = δ. Feng et al. [123] considered upper bounds for λ1 of connected graphs in terms of vertex degrees and average 2-degrees (the average 2-degree f v of a vertex v is the average degree in its neighbourhood). The three inequalities read as follows: λ1 ≤

{ d2 + d v f v } λ1 ≤ max {√ v : v ∈ V} , 2 { } { d u (d u + f u ) + d v (d v + f v ) } λ1 ≤ max {√ : u ∼ v} 2 { } and λ1 ≤ max {

∆ + √∆2 + 8(d v f v ) : v ∈ V} . 4

In all cases, equality holds if and only if a graph is regular. The next inequality obtained by Cioabă et al. [76] gives a lower bound for the spectral radius expressed in terms of order and diameter 1

λ1 ≥ (n − 1) D . Equality can only hold for D ≤ 2, and it indeed holds for D = 1, while for D = 2 it holds if and only if a graph under consideration is the star K1,n−1 or a Moore graph [109]. Recall from [290, p. 42] that a Moore graph is a graph with diameter D and girth 2D + 1, for some D > 1. It is well known that any Moore graph must be regular [89], and then we have the following bound for n in terms of diameter D and degree r:

2.7 Extremal regular graphs |

41

D

−1 { r (r−1) + 1, if D > 2, r−2 n≤{ 2D + 1, if D = 2. { Hence, a Moore graph can also be defined as a connected regular graph that attains the above bound for n. Next, the Moore graph can only exist if r = 2 (when it is the cycle C2D+1 ) or if D = 2 and r ∈ {3, 7, 57}. Here, for r = 3 and r = 7 there exists a unique Moore graph: the Petersen graph and the Hoffman–Singleton graph. The latter is a 7-regular graph with 50 vertices named after its discoverers [175]. More details on this graph are given on page 143. For r = 57, the answer is not known [34, 174]. Moore graphs belong to a wider class of graphs called cages. A cage is an r-regular graph of girth g having the minimal possible number of vertices. Examples of non-bipartite graphs attaining the bound of Theorem 2.6.6 are the Moore graphs: the pentagon, the Petersen graph and the Hoffman–Singleton graph. According to Koledin [194], it can be verified that this bound can only be attained by graphs with diameter 2 (thus making the above mentioned graphs and the 57-regular Moore graph, if it exists, the only examples). The detailed proof requires considerable computation. Nikiforov [240] controlled the clique number by the order, average vertex degree and the least eigenvalue of an arbitrary graph:

ω ≥1+

dn (n − d)(d − λ n )

.

Equality holds if and only if a graph under consideration is complete ω-partite and regular. Concerning the second largest eigenvalue, it is noteworthy to say that cubic graphs for which λ2 is maximal are identified in [46]. For n ≥ 10 these graphs are illustrated in Figure 2.2. There, for the first (resp. second) family we have n ≡ 2 (mod 4) (resp. n ≡ 0 (mod 4)). In this context the reader may tray to solve the following, pure combinatorial, exercise. Exercise 2.7.1 ([51]). Prove that a cubic graph with diameter 4 cannot have more than 38 vertices. Laplacian eigenvalues Concerning the L-spectral radius, we first give the result of Guo [153]: { } } { d v + √d2v + 8d v f v󸀠 μ1 ≤ max { : v ∈ V} , } { 2 { } ∑u∼v (d v −|N(u)∩ N(v)|) 󸀠 where f v = . Equality holds if and only if a graph is bipartite regular. dv

42 | 2 Spectral properties

Fig. 2.2. Cubic graphs with at least 10 vertices for which λ2 is maximal.

The next result is attributed to Zhu [323]: μ1 ≤ max {

du fu dv fv + : u ∼ v} . dv du

Again, we have the equality in the case of regular graphs. This result improves the two classical results. The first belongs to Anderson and Morley [12] who proved that μ1 ≤ max{d u + d v : u ∼ v}, while the second we due to Merris [229] who proved that μ1 ≤ max{d v + f v : v ∈ V}. Shu et al. [279] used the concept of line graphs to prove the following upper bound for connected graphs μ1 ≤ δ +

1 2 n 1 + √(δ − ) + ∑ d i (d i − δ), 2 2 i=1

where equality holds if and only if a graph is bipartite regular. By [214], G is an r-regular graph with the property μ1 = n if and only if G ≅ G1 ∇G2 , where G i is an r i -regular graph of order n i (1 ≤ i ≤ 2) and n1 +r2 = n2 +r1 = r. Thus, we have r ≥ 2n . Now, is it true that there exists an r-regular graph with μ1 = n whenever n 2 ≤ r ≤ n − 1? The answer is given in the following exercise. Exercise 2.7.2 ([214]). Show that there exists a connected r-regular graph with μ1 = n if and only if nr is even and 2n ≤ r ≤ n − 1. In a recent paper [212] Lin and Weng demonstrated an approach based on eigenvector technique for obtaining bounds for Laplacian eigenvalues of connected graphs. In particular, they gave an upper bound for the Laplacian spectral radius and a lower bound for the algebraic connectivity that read as follows: μ1 ≤

2∆ + c + √(2∆ + c)2 − 4c2 n 2

2.8 Eigenvalue estimation |

43

and 2δ + c − √(2δ + c)2 − 4(c2 n + δ2 − ∆2 ) , 2 where c1 and c2 are constants satisfying μ n−1 ≥

c1 ≤ min |N(u) ∩ N(v)|, c2 ≤ min |N(u) ∩ N(v)| and c = c2 − c1 . u∼v

u≁v

Combining these bounds one may obtain an upper bound for the difference μ1 − μ n−1 (also known as the Laplacian spectral spread), which in the case of r-regular graphs reduces to μ1 − μ n−1 ≤ √(2r − c1 + c2 )2 − 4c2 n. According to [212], equality holds for strongly regular graphs (as we already said, these graphs are defined in Section 3.4). Signless Laplacian eigenvalues The next result for the Q-spectral radius is a direct consequence of the inequality (2.16):

κ1 ≤ min {2∆,

∆ + 2δ − 1 + √(∆ + 2δ − 1)2 + 16m − 8(n − 1 + ∆)δ }, 2

with equality if and only if a corresponding graph is regular or has a component of order ∆ + 1 in which every vertex is of degree δ or ∆, and all other components are δ-regular. The three consequences of this inequality are given in [290, pp. 208–209]. We conclude the section by the result of De Lima et al. [211] bounding the least Q-eigenvalue in terms of the order, size and chromatic number: κ n ≤ (1 −

2m 1 , ) χ−1 n

where equality is attained for χ-partite regular graphs.

2.8 Eigenvalue estimation We provide Cioabă’s proof of the Serre theorem that gives estimates for largest eigenvalues of regular graphs. We also give an analogue of this theorem regarding smallest eigenvalues and supply a discussion on related results. Theorem 2.8.1 (Serre Theorem, cf. [74, 113]). For every ε > 0, there exists a positive constant c = c(ε, r) such that for any r-regular graph G with n vertices the number of eigenvalues λ of G with the property

44 | 2 Spectral properties

λ ≥ (2 − ε)√r − 1, is at least cn. Proof. Following [74], let n0 denote the number of eigenvalues λ with λ ≥ (2−ε)√r − 1. Since n − n0 eigenvalues are less than (2 − ε)√r − 1, for s ≥ 1 we have

tr(rI + A)2s

=

n

∑ (r + λ i )2s i=1


(2√r − 1) s−j+1 s−j (s − j + 1)2

(where the first inequality follows from a simple combinatorial result stating that the number of closed walks of length 2(s − j) (≠ 0) starting at a fixed vertex of an 1 s−j−1 , while the second follows from r-regular graph is at least s−j+1 (2(s−j) s−j )r(r − 1) s−j 2(s−j) 4 ( s−j ) ≥ s−j+1 ), we get s

∑( j=0

2s 2j )r tr (A2(s−j) ) 2j

>

s 2s n ∑ ( )r2j (2√r − 1)2(s−j) 2 (s + 1) j=0 2j

>

n (r + 2√r − 1)2s . 2(s + 1)2

Collecting the upper and lower estimates for tr(rI + A)2s , we get n0 > n

1 (r 2(s+1)2

+ 2√r − 1)2s − (r + (2 − ε)√r − 1)2s

(2r)2s − (r + (2 − ε)√r − 1)2s

.

Now, it is a matter of routine to see that there exists a constant s0 (ε, r) such that the inequality 1 (r + 2√r − 1)2s − (r + (2 − ε)√r − 1)2s > (r + (2 − ε)√r − 1)2s 2(s + 1)2

2.8 Eigenvalue estimation | 45

holds for all s ≥ s0 , and so if we choose c(ε, r) =

(r + (2 − ε)√r − 1)2s0 , − (r + (2 − ε)√r − 1)2s0

(2r)2s0

we would have c(ε, r) > 0 and n0 > c(ε, r)n, and the result follows. The last theorem can be derived from the results of Friedman [132] or Nilli [244]. For example, the second result considers r-regular graphs containing s vertices so that the distance between any pair is at least 4k. Then at least s eigenvalues of such graphs are π greater than or equal to 2√r − 1 cos ( 2k ). The reader may use Theorem 2.8.1 to solve the following exercise. Exercise 2.8.2 ([74]). Let {G i }, for i ∈ ℕ, be a sequence of r-regular graphs such that limi→∞ n(G i ) = ∞ (where n(G i ) is the order of G i ). Prove that the inequality lim inf i→∞ λ j (G i ) ≥ 2√r − 1 holds for every j ≥ 1. For this exercise, the reader can also consult [113, 132]. Observe that, for j = 2, it gives lim inf λ2 (G i ) ≥ 2√r − 1, i→∞

which is an asymptotic version of the Alon–Boppana theorem [8, 218] stating that λ2 ≥ 2√r − 1 − o(1)

(2.17)

holds for connected regular graphs with fixed r and, in a precise sense, large n. The bound for j = 2 is sharp at least when r − 1 is a prime and r − 1 ≡ 1 (mod 4) [218]. Moreover, Alon conjectured [8] and Friedman proved [131] that the second largest eigenvalue of almost all r-regular graphs does not exceed 2√r − 1 + o(1) (as n tends to infinity). There is an analogous theorem for the smallest eigenvalues. We use oddg(G) to denote the oddgirth of a graph G, that is the length of a shortest odd cycle in G. If G does not contain odd cycles, then we take oddg(G) = ∞. Theorem 2.8.3 ([74]). For every ε > 0, there exist positive constants c = c(ε, r) and k = k(ε, r) such that for any r-regular graph G with n vertices and oddg(G) > k the number of eigenvalues λ of G with the property λ ≤ −(2 − ε)√r − 1, is at least cn. We conclude by an exercise analogous to the previous one. Exercise 2.8.4 ([74]). Let {G i }, for i ∈ ℕ, be a sequence of r-regular graphs such that limi→∞ oddg(G i ) = ∞. If the eigenvalues of G i are λ󸀠1 ≤ λ󸀠2 ≤ ⋅ ⋅ ⋅ , then prove that the inequality lim supi→∞ λ󸀠j (G i ) ≤ −2√r − 1 holds for every j ≥ 1.

46 | 2 Spectral properties The case j = 1 is proved in [210]. In the same reference we can find a stronger result proved by Serre.

2.9 Eigenvalue distribution The question on how the eigenvalues of a random regular graph are distributed along the corresponding segment of a real line is one of the oldest in the entire theory of graph spectra. Computational results show that, for r ≥ 3, r-regular graphs should often have simple spectra (that is, without repeated eigenvalues). The following data (provided by Royle) confirms this estimate for connected cubic graphs. n 10 12 14 16 18

# connected cubic graphs # those with simple spectrum 19 85 509 4060 41 301

6 (31 %) 18 (21 %) 316 (62 %) 2181 (54 %) 26 446 (64 %)

Therefore, it could be expected that the eigenvalues of a random regular graph of large order and fixed degree are densely distributed along the segment [λ n , r]. Semicircle distribution for random graphs The Erdős–Rényi random graph G(n, p) is formed by taking the graph nK1 and adding each edge independently with probability p [121]. There is a theorem. Theorem 2.9.1 (Semicircle Distribution, cf. [68, 321]). Assume that p = ω ( 1n ) and p ≤ 21 . If A is the adjacency matrix of a random graph G(n, p) then, as n → ∞, the spectral distribution of the scaled matrix 1 A √np(1 − p) converges in distribution to the semicircle distribution which has a density ρ G(n,p) (x) =

1 √ 4 − x2 2π

with support on [−2, 2]. Here, ω stands for the asymptotic notation ‘little ω’. The graph of the function ρ G(n,p) is a semicircle of radius 2 centred at the origin and then suitably normalized (so, it is in fact a semiellipse). According to Wigner [317], the same distribution is encountered for many random symmetric matrices as the size of the matrix approaches infinity. In the context of these matrices, an analogue to the last theorem is known as the

2.9 Eigenvalue distribution |

47

Wigner distribution. According to [321], the condition p = ω ( 1n ) is essential as confirmed by many examples. Note that the restriction p ≤ 12 is taken in order to simplify the consideration as the other case can be analyzed by taking the graph complement. Finally, the result can be derived by considering the spectral moments of a graph. McKay distribution for random regular graphs Let G n,r be a random r-regular graph with n vertices. Observe that, contrary to the Erdős–Rényi random graphs, the entries of the corresponding adjacency matrix are not independent. There are several algorithms for constructing random regular graphs [117, 185], and one of them reads as follows. Take a graph nK1 and attach r lines at each vertex. Each line is an edge with one end determined and the other not. At each step identify a pair of uniformly random lines (i.e., make their ends neighbours). Over the procedure, disallow identifying the lines if this would create a loop or a multiple edge. If we arrive at the situation in which all the remaining lines are attached at the same vertex (which can occur with very small probability), we may restart the procedure. At the end of the procedure, we obtain a random r-regular graph G n,r . As claimed in the following theorem, the spectral distribution of random regular graphs G n,r where r is fixed and n → ∞ in general case is not of semicircle distribution. Theorem 2.9.2 (McKay Distribution, cf. [225]). Let r ≥ 2 be a fixed integer. As n → ∞, the spectral distribution of a random r-regular graph converges in distribution to the distribution which has a density { r√4(r−1)−x , 2 2 ρ G n,r (x) = { 2π(r −x ) { 0, 2

if |x| ≤ 2√r − 1, otherwise.

This distribution was previously discovered by Kesten [192] in the context of random walks on groups, and so it is also known as the Kesten–McKay distribution. It was recently proved that if r approaches infinity but slowly with n, then the corresponding distribution is the semicircle distribution. Theorem 2.9.3 (Tran et al. [312]). Let r → ∞ and r ≤ 2n . If A is the adjacency matrix of a random regular graph G n,r then, as n → ∞, the empirical spectral distribution of the scaled matrix 1 √r (1 − nr )

A

converges in distribution to the semicircle distribution of Theorem 2.9.1.

3 Particular types of regular graph Here we focus on the most extensively investigated types of regular graph. Regular graphs obtained by an operation called a non-complete extended p-sum are examined in Section 3.1. In particular, we meet two operations commonly known as the tensor product and the Cartesian product of graphs. In a short Section 3.2 we give the main characterizations of walk-regular graphs. As we will see, many regular graphs considered in the subsequent sections are in fact walk-regular. Distance-regular graphs are considered in Section 3.3. We show that the structure of these graphs is determined by a specified sequence of parameters and derive a connection between these parameters and the spectrum. We meet many examples such as halved graphs, antipodal covers, distance-transitive graphs, Johnson graphs, Hamming graphs or Grassman graphs. We also present a set of inequalities that include certain eigenvalues of these graphs, and we examine whether the distance-regularity of a graph can be deduced from its spectrum. In Section 3.4 we play with strongly regular graphs which, under certain restrictions, can be considered as distance-regular. Acting in the same manner, we mostly balance between their parameters and spectra. In particular, we emphasize a beautiful relation between the spectrum and the structure of a connected regular graph stating that such a graph is strongly regular if and only if it has 3 distinct eigenvalues. We also meet some specified strongly regular graphs known as conference graphs (in particular, Paley graphs), generalized Latin squares or projective planes (in particular, generalized quadrangles). We conclude by inspecting spectral distances between strongly regular graphs where we observe that the spectral distance between some generalized quadrangles and their complements is extremal in sense that it exceeds the maximal graph energy in the set of graphs with fixed order. A short Section 3.5 is devoted to the eigenvalues of Kneser graphs and their bipartite counterparts. The spectra of regular graphs that correspond to line systems are considered in Section 3.6. In particular, we deal with the two types of line system: In the first the angles between lines taken pairwise are mutually equal, while in the second the lines are at angle 3π or 2π . We explain how these line systems produce regular graphs with interesting spectral properties. In particular, the maximal number of lines in the first system is closely related to particular strongly regular graphs, while the second system is used in characterizing (regular) graphs whose least eigenvalue is greater than or equal to −2. Induced subgraphs called star complements are the subject of Section 3.7. We use the main tool for studying star complements, so-called reconstruction theorem, to derive a background for a spectral technique that builds large graphs from their smaller

DOI: 10.1515/9783110351347-003

3.1 Non-complete extended p-sums |

49

parts. We prove that some well known regular graphs are uniquely determined by fixed star complements. Various block designs are closely related to their incidence graphs and corresponding spectra. In Section 3.8 we compute the spectra of incidence graphs of specified partially balanced incomplete block designs (in particular, balanced incomplete block designs). We also establish that every symmetric 2-class partially balanced incomplete block design is based on a fixed strongly regular graph whose parameters give full information about the spectrum of the corresponding incidence graph. We meet regular incidence graphs of finite projective planes, finite affine planes, group divisible designs or triangular designs.

3.1 Non-complete extended p-sums Let B denote a set of binary k-tuples (b1 , b2 , . . . , b k ) with all-0 k-tuple excluded. The non-complete extended p-sum (for short, NEPS) of the graphs G1 , G2 , . . . , G k is the graph whose vertex set is the Cartesian product V(G1 ) × V(G2 ) × ⋅ ⋅ ⋅ × V(G k ) and in which two vertices (u1 , u2 , . . . , u k ) and (v1 , v2 , . . . , v k ) are adjacent if and only if there is a k-tuple such that u i = v i holds exactly when b i = 0 and u i is adjacent to v i in G i exactly when b i = 1. If all k-tuples of B contain exactly p 1’s, the operation is called the incomplete p-sum; If B contains all k-tuples with p 1’s, we deal with the complete p-sum. By choosing B = {(1, 1)} and B = {(0, 1), (1, 0)} we get two operations known as the tensor product and the Cartesian product of two graphs, respectively. We denote these products by × and ◻, respectively. More precisely, two vertices (u1 , u2 ) and (v1 , v2 ) are adjacent in G1 × G2 if and only if u i is adjacent to v i in G i , for 1 ≤ i ≤ 2. Similarly, they are adjacent in G1 ◻G2 if and only if either u1 = v1 and u2 is adjacent to v2 in G2 or u2 = v2 and u1 is adjacent to v1 in G1 . Remark 3.1.1. The tensor product and the Cartesian product can be found in numerous sources under different names. For example, the tensor product is also known as the product, the Kronecker product or the direct product. In [87], the latter product is called the β-product or the sum. Anyone who needs a practice with these products may tray to solve the following exercise. Exercise 3.1.2. Show that the graphs (G◻K2 ) × K2 and (G × K2 )◻K2 are isomorphic for any graph G. The Kronecker product A ⊗ B of a p × q matrix A = (a ij ) and an r × s matrix B = (b ij ) is the pr × qs matrix obtained by replacing each element a ij of A by the block a ij B. Using

50 | 3 Particular types of regular graph associativity of this product in conjunction with distributivity of the matrix product over the Kronecker product, we get the following result. Theorem 3.1.3 (Cvetković, cf. [87, Theorem 2.21]). The NEPS with the basis B of graphs G1 , G2 , . . . , G k with adjacency matrices A1 , A2 , . . . , A k has the adjacency matrix A=

b

∑ (b1 ,b2 ,...,b k )∈B

b

b

A11 ⊗ A22 ⊗ ⋅ ⋅ ⋅ ⊗ A k k .

(3.1)

Considering the spectra of all summands in (3.1), we arrive at the next result giving the spectrum of a NEPS. Theorem 3.1.4 (Cvetković, cf. [87, Theorem 2.23]). If a graph G i has n i vertices and the spectrum λ i1 , λ i2 , . . . , λ in i , then the spectrum of the NEPS with the basis B of graphs G1 , G2 , . . . , G k consists of all possible numbers λ i1 ,i2 ,...,i k of the form λ i1 ,i2 ,...,i k =

∑ (b1 ,b2 ,...,b k )∈B

b

b

b

λ1i11 λ2i22 ⋅ ⋅ ⋅ λ kikk ,

where 1 ≤ i j ≤ n j and 1 ≤ j ≤ k. In particular, the spectrum of the tensor product of two graphs consists of all possible products of the eigenvalues of these graphs. Similarly, all possible sums of the same eigenvalues form the spectrum of their Cartesian product. Graphs defined by specified operations For a reason that will be revealed on page 188, the tensor product of any graph G with K2 is called a bipartite double of G. Obviously, the adjacency matrix of G × K2 is given by A ⊗ (J2 − I2 ), where A is the adjacency matrix of G. If the eigenvalues of G are λ1 , λ2 , . . . , λ n , then the eigenvalues of G × K2 are ±λ1 , ±λ2 , . . . , ±λ n . If the vertices of K2 are v1 and v2 , then the extended bipartite double of G is obtained from the bipartite double by inserting an edge between the vertices (u, v1 ) and (u, v2 ), for all u ∈ V(G). Observe that its adjacency matrix is given by (A + I) ⊗ (J2 − I2 ), and then its eigenvalues are ±(λ1 + 1), ±(λ2 + 1), . . . , ±(λ n + 1). Both, bipartite and extended bipartite double, are bipartite. Bipartite double is connected if and only if G is connected and non-bipartite. Extended bipartite double is connected if and only if G is connected. An example is illustrated in Figure 3.1. We may use the tensor product to compute the spectrum of an arbitrary r-dimensional cube. Example 3.1.5. Observe that the r-dimensional cube Q r (r ≥ 2) is the tensor product of Q r−1 and K2 . Using this, we recursively compute that the spectrum of Q r consists of the numbers r − 2i each with multiplicity (ri), where 0 ≤ i ≤ r. A lattice graph is the Cartesian product of the complete graphs K n1 and K n2 . Equivalently, it is the line graph of K n1 ,n2 . Therefore, it has n1 n2 vertices, n1 n2 (n1 +n22 −2) edges,

3.2 Walk-regular graphs | 51

Fig. 3.1. C6 (top), its bipartite double (left) and extended bipartite double (right).

and consequently its degree is n1 + n2 − 2. It is not difficult to see that, if we exclude some special cases of small values of n1 and n2 , the diameter of a lattice graph is 3. Using Theorem 3.1.4, we deduce that the spectrum of Line(K n1 ,n2 ), for n1 ≥ n2 ≥ 2, is given by [n1 + n2 − 2, (n1 − 2)n2 −1 , (n2 − 2)n1 −1 , (−2)n1 n2 −n1 −n2 +1 ] .

3.2 Walk-regular graphs A graph is called walk-regular if for each its vertex i, the number of closed k-walks that start at i is constant for each non-negative integer k. Setting k = 3, we obtain that every walk-regular graph is regular. In the forthcoming sections we will meet various types of regular graph that are all walk-regular. See page 54 for vertex-transitive graphs, page 55 for distance-regular graphs or Exercise 3.4.1 for strongly regular graphs. The graph of Figure 3.2 provided by Godsil and McKay [142] is an illustrating example of a walk-regular graph that is

52 | 3 Particular types of regular graph

Fig. 3.2. A walk-regular graph with 13 vertices.

neither of these types. Following the same authors [142, 143], we give basic properties of walk-regular graphs. Lemma 3.2.1 ([142, 143]). The following statements are equivalent: (i) G is walk-regular; (ii) The diagonal entries of A k (k ≥ 1) are mutually equal; (iii) The characteristic polynomials P G−i are identical for all i ∈ V(G). Proof. The equivalence of (i) and (ii) follows from the fact that the number of closed k-walks starting at vertex i, w k (i), is equal to the ith entry of A k−1 . Let ∞

W i (x) = ∑ w k+1 (i)x k

(3.2)

k=0

be the generating function for the number of closed walks starting at i. Observing that W i (x) is the ith diagonal entry of (I − xA)−1 =

−1 1 1 ( I − A) x x

and noting that the corresponding cofactor in 1x I−A is just P G−i ( 1x ), we get the identity

3.2 Walk-regular graphs | 53

W i (x) =

P G−i ( 1x ) xP G ( 1x )

,

which gives the equivalence of (i) and (iii). If G is an arbitrary graph, then using the expression (1.4) (page 11) we get w k (i) = ∑ λ k−1 (X λT ei )T X λT ei ,

(3.3)

where X λ is the matrix whose columns make the complete system of orthonormal eigenvectors corresponding to an eigenvalue λ, while the summation goes over all distinct eigenvalues of G. Using the expression (3.3) and Corollary 2.2.2, we immediately get the following result. Theorem 3.2.2. A graph is walk-regular if and only if its complement is walk-regular. Here is an exercise. Exercise 3.2.3 ([142]). Show that a disconnected graph is walk-regular if and only if its components are walk-regular and share the same characteristic polynomial. The expression (3.3) is useful if we consider simple eigenvalues of a walk-regular graph (since then X λ reduces to a single column). If λ is a simple eigenvalue of a walk-regular graph and x denotes a corresponding unit eigenvector, then we immediately deduce that the entries of x are equal in absolute value. Considering the first row of the identity Ax = λx, we get the following theorem. Theorem 3.2.4 ([142]). Every simple eigenvalue λ of a walk-regular graph of degree r is integral and satisfies (r − λ) ≡ 0 (mod 2). We say more about the number of simple eigenvalues. Theorem 3.2.5 ([142]). If p stands for the number of simple eigenvalues of a walk- regular graph G with n vertices, then the following statements hold: (i) If p ≥ 2, then n is even; (ii) If p ≥ 3, then n ≡ 0 (mod 4); (iii) If n ≥ 3, then p ≤ 2n . Proof. If there exists a simple eigenvalue other than the spectral radius, then the entries of a corresponding unit eigenvector x have equal absolute value. Hence, the orthogonality of x and j gives (i). If there is another simple eigenvalue, then the orthogonality of a corresponding eigenvector, the previous vector x and j gives (ii). We may assume that n is even, since otherwise (iii) follows from (i). By Theorem 2.2.1, the complementary graph G has p simple eigenvalues, as well. By the result

54 | 3 Particular types of regular graph

Fig. 3.3. 8-crossed prism.

of Exercise 3.2.3, both G and G must be connected, but then by applying Theorem 3.2.4 to both of them we get the assertion. Vertex-transitive graphs A vertex-transitive graph is a graph such that for every pair i, j of its vertices, there is an automorphism that maps i to j. In other words, a graph is vertex-transitive if its automorphism group acts transitively upon the vertices. It follows directly that, for vertex-transitive graphs, the generating function (3.2) is independent of the vertex i, and therefore every vertex-transitive graph is walk-regular. Example 3.2.6. An n-crossed prism for an even n (n ≥ 4) is a graph obtained by taking two copies of the cycle C n (whose vertices are labelled 0, 1, . . . , n − 1) and inserting additional edges between the vertices i (of the first copy) and i − 1 (of the second), and i + 1 (of the first copy) and i + 2 (of the second), for i ∈ {2j : 0 ≤ j < 2n } where all operations are mod n. An illustration is given in Figure 3.3. Observe that all vertex-deleted subgraphs of an n-crossed prism are isomorphic, and so their characteristic polynomials are identical which (by Lemma 3.2.1) leads to the conclusion that every n-crossed prism is walk-regular. In particular, it is vertex-transitive.

3.3 Distance-regular graphs | 55

3.3 Distance-regular graphs A distance-regular graph is a connected non-trivial¹ regular graph such that for every two vertices u and v at distance i = d(u, v), the number of vertices at distance j from u and at distance k from v depends only upon i, j and k. In other words, a connected graph G with diameter D is distance-regular if there exist integers a i , b i , c i (called the intersection numbers) such that for all i (0 ≤ i ≤ D) and all vertices u and v at distance i, there are precisely a i neighbours of v at distance i from u, b i neighbours of v at distance i + 1 from u and c i neighbours of v at distance i − 1 from u. Obviously, the graph G is regular of degree r = b0 . It also follows that a i + b i + c i = r holds for all values of i. Consequently, the intersection numbers a i can be expressed in terms of b i and c i . Observe that, by definition, the intersection numbers bD and c0 are fixed and equal to 0. If so, then the following array (b0 , b1 , . . . , bD−1 ; c1 , c2 , . . . , cD )

(3.4)

is sufficient to determine all intersection numbers. This array is called the intersection array of a distance-regular graph G. One may notice that the intersection array includes 2D intersection numbers and also that, for convenience, we included the number c1 = 1. It follows from definition that any power of the adjacency matrix of a distance-regular graph has a constant main diagonal, and thus by Lemma 3.2.1, every distance-regular graph is walk-regular. It also follows that every vertex of a distance-regular graph has a constant number k i of vertices that are at distance i from it. Namely, we have k0 = 1, and then for given k i by computing the number of edges between the sets of vertices at distance k i and that at distance k i+1 from a fixed vertex, we get the recurrence relation {

k0 = 1, k i+1 =

bi c i+1 k i

(0 ≤ i ≤ D − 1).

(3.5)

Thus, for the number of vertices we have n = ∑D i=0 k i , and so n is determined by the intersection array. Since we have 2m = nb0 , the same holds for the number of edges. There is a simple exercise. Exercise 3.3.1. Show that a distance-regular graph with diameter D is bipartite whenever a i = 0 (0 ≤ i ≤ D).

1 Throughout this section and later on, whenever deal with distance-regular graphs we always assume (without noting!) that the trivial graph is excluded from our consideration. For example, according to this convention it may occur that we say that every graph of a specified family is distance-regular even if the trivial graph is also recognized as a member of that family.

56 | 3 Particular types of regular graph In other words, bipartiteness of a distance-regular graph can be deduced from its intersection array. For a graph G, we simultaneously deal with the distance-i graph G i (0 ≤ i ≤ D) having the same vertex set and in which two vertices are adjacent if and only if they are at distance i in G. The adjacency matrix of G i is denoted A i . In particular, we say that G is an antipodal graph if GD is a disjoint union of complete graphs. There is another exercise. Exercise 3.3.2 ([111]). Show that a distance-regular graph with diameter D is antipodal whenever b i = cD−i (0 ≤ i ≤ D) holds except possibly for i = ⌊ D 2 ⌋. So, besides bipartiteness, antipodality of a distance-regular graph is also deducible from the intersection array. We denote by G i (v) the ith subconstituent of G, that is the graph induced by the vertices at distance i from a fixed vertex v in G. The first subconstituent is also called a local graph. Following [111, 308], we say that G is of order (s, t) if all local graphs are disjoint unions of t + 1 cliques with s vertices.

3.3.1 Basic properties The study of distance-regular graphs often deals with obtaining a list of possible intersection arrays, and then trying to construct the corresponding graphs. In this context, the following restrictions on intersection numbers are very useful. Theorem 3.3.3 (cf. [48, Chapter 4] or [111]). For a distance-regular graph G with diameter D and intersection array (3.4), we have (i) (ii) (iii) (iv) (v) (vi) (vii)

b0 ≥ b1 ≥ ⋅ ⋅ ⋅ ≥ bD−1 , c1 ≤ c2 ≤ ⋅ ⋅ ⋅ ≤ cD , b i ≥ c j for i + j ≤ D, 1 ⋅⋅⋅b i k i+1 = cb10cb2 ⋅⋅⋅c is an integer for 0 ≤ i ≤ D − 1, i+1 nk i and k i a i are ≡ 0 (mod 2) for 1 ≤ i ≤ D, nra1 ≡ 0 (mod 6) and there is an i such that k0 ≤ k1 ≤ ⋅ ⋅ ⋅ ≤ k i and k i+1 ≥ k i+2 ≥ ⋅ ⋅ ⋅ ≥ kD ,

where k i ’s are given by (3.5). Proof. (i) Let u and v be two vertices at distance i and w a neighbour of u at distance i − 1 from v. Since b i neighbours of v that are at distance i + 1 from u are at distance i from w, we have b i−1 ≥ b i . Part (ii) is considered in the same way. (iii) Considering the vertices u and v at distance i + j and a vertex w such that d(u, w) = i and d(w, v) = j (clearly, there is such a triplet of vertices), we conclude

3.3 Distance-regular graphs | 57

that c i neighbours of w that are at distance i − 1 from u are at distance j + 1 from v, and we are done. (iv) The equality follows from (3.5). All k i ’s are integers since they stand for the numbers of vertices that are at distance i from a vertex of G. (v) The first number is twice the number of edges in G i , while the second is twice the number of edges in a local graph G(v) (for arbitrary v). (vi) Computing the spectral moment M3 of G, we get that G contains 61 nra1 triangles, and we are done. (vii) It follows from (i), (ii) and (iv) that k2i ≥ k i−1 k i+1 (1 ≤ i ≤ D − 1) which gives the assertion. An intersection array is called feasible if the corresponding intersection numbers satisfy some set of feasibility conditions. For example, these could be the conditions given in the last theorem. Of course, the feasibility of an intersection array does not necessary implies the existence of a corresponding distance-regular graph. Another problem that naturally arises is as follows: Given a distance-regular graph, does there exist another graph sharing the same intersection array? There is a simple exercise. Exercise 3.3.4 ([111]). Show that the Petersen graph is a unique distance-regular graph with intersection array (3, 2; 1, 1). The last exercise is singled out from a wider result concerning so-called odd graphs. For an integer k ≥ 2, the vertices of an odd graph are identified with the subsets of size k − 1 of a set of size 2k − 1, and two vertices are adjacent if and only if the corresponding subsets have trivial intersection. Any odd graph is distance-regular with diameter k − 1. Now, by Koolen [200], any non-bipartite distance-regular graph with diameter at least 4 and intersection numbers a1 = a2 = a3 = 0, c1 = c2 = 1 and c3 = c4 = 2 is an odd graph. Spectrum In what follows we show that the spectrum of a distance-regular graph (with diameter D) can easily be computed from its intersection array (3.4). For this purpose, we need the following (D + 1) × (D + 1) tridiagonal matrix: 0 c1 ( T = ( ... 0 0 (

b0 a1 ⋅⋅⋅ ⋅⋅⋅

0 b1

0 cD−1 0

⋅⋅⋅ ⋅⋅⋅ .. . aD−1 cD

0 0 .. ) . . )

(3.6)

bD−1 aD )

Theorem 3.3.5 (cf. [48, pp. 128–129] or [111]). A distance-regular graph G with diameter D has D + 1 distinct eigenvalues. These eigenvalues are precisely the eigenvalues of the matrix T from (3.6).

58 | 3 Particular types of regular graph Proof. The proof is separated into two parts. We first show that G has D + 1 distinct eigenvalues. Assume that the number of distinct eigenvalues is k. By Theorem 1.3.3, we have D ≤ k − 1. To conclude this part it is sufficient to show that k ≤ D + 1. Using distance-regularity of G, it is not difficult to verify the following recurrence relation AA i = b i−1 A i−1 a i A i + c i+1 A i+1 (0 ≤ i ≤ D),

(3.7)

with convention that b−1 A−1 = cD+1 AD+1 = O. This relation implies the existence of polynomials P i (of degree i) such that A i = P i (A), where these polynomials satisfy similar recurrence relation, and so form the system of orthogonal polynomials. Using D ∑D i=1 A i = J and AJ = rJ (where r = b 0 ), we get (A − rI) ∑i=1 P i (A) = O, and consequently {I, A, . . . , AD } is a basis of the matrix algebra spanned by {I, A, A2 , . . . , A k−1 }. Thus, the dimension of this algebra does not exceed D + 1. Since the minimal polynomial of A has degree k, the dimension is precisely k. In other words, k ≤ D + 1. Observing that the intersection numbers c i (1 ≤ i ≤ D) are all non-zero, we get that the rank of xI − T is at least D for all x ∈ ℝ, which means that T has D + 1 distinct eigenvalues. In this part we prove that all eigenvalues of T are the eigenvalues of G. If λ is an eigenvalue of T, then we have Ty = λy for some y = (y0 , y1 , . . . , yD )T . Considering the matrix T, we get y0 = 1, { { { y1 = λr , { { { { c i y i−1 + a i y i + b i y i+1 = λy i (1 ≤ i ≤ D).

(3.8)

Let u be a vertex of G. We claim that λ is an eigenvalue of G with the corresponding eigenvector x = (x1 , x2 , . . . , x n )T defined by x v = y d(u,v) . Namely, denoting by ui the column of A i that corresponds to u, we get x = ∑D i=0 y i ui . Using (3.7), D we get Ax = ∑i=0 y i (b i−1 ui−1 + a i ui + c i+1 ui+1 ). The last sum may be rewritten as D ∑D i=0 (c i y i−1 + a i y i + b i y i+1 ) ui . Now, by (3.8), the last is equal to ∑i=1 λy i ui = λx, and we are done. Remark 3.3.6. In the previous proof we mentioned the existence of orthogonal polynomials P i (0 ≤ i ≤ D) satisfying P i (A) = A i . In fact, Fiol and Gariga [127] established a similar system of orthogonal polynomials that can be constructed from distinct eigenvalues of an arbitrary regular graph. These polynomials are called the predistance polynomials. In [48, p. 130], Brouwer et al. gave another form of a tridiagonal matrix together with a slightly different formulation of the corresponding result. We formulate it in the form of an exercise which can be solved either by using the result of Theorem 3.3.5 or without using it but by following its proof.

3.3 Distance-regular graphs | 59

Exercise 3.3.7. Apart from its degree, all distinct eigenvalues of a distance-regular graph with diameter D are the eigenvalues of the following D × D matrix −c1 c1 ( .. ( . 0 ( 0

b1 b0 − b1 − c2

0 b2

⋅⋅⋅ ⋅⋅⋅

0

⋅⋅⋅ ⋅⋅⋅ .. .

cD−2 0

b0 − bD−2 − cD−1 cD−1

0 0 .. .

) ).

bD−1 b0 − bD−1 − cD )

Using the connection between intersection numbers and coordinates of y given in (3.8), we can obtain the exact multiplicities of eigenvalues of G. Theorem 3.3.8 (Biggs Formula, cf. [48, 111]). Given a distance-regular graph G with n vertices and diameter D, let λ be its eigenvalue and T be the matrix determined by the intersection array as in (3.6). If y = (y0 , y1 , . . . , yD )T is an eigenvector satisfying Ty = λy, then the multiplicity of λ (in the spectrum of G) is given by mλ =

n 2 ∑D i=0 k i y i

,

where k i ’s are given by (3.5). Note that the fact that multiplicities of the eigenvalues are positive integers together with the Biggs formula can be added to the set of conditions for the intersection numbers listed in Theorem 3.3.3. According to the last two theorems, if an intersection array is given then we can compute the spectrum of the corresponding distance-regular graph. The opposite implication is considered in the next theorem. Theorem 3.3.9. If G is distance-regular, then its intersection array is determined by its spectrum. It is sufficient to show that the spectrum determines the orthogonal predistance polynomials P i (0 ≤ i ≤ D) satisfying P i (A) = A i , since then the intersection numbers can be derived by using relations (3.7). An elegant proof based on the Gram–Schmidt orthogonalization can be found in the Appendix of [105]. We may say more about the multiplicities by considering their bounds. Another notion is needed: The head of a distance-regular graph G with intersection numbers a i , b i , c i (0 ≤ i ≤ D) is a positive integer defined by h = |{i : 1 ≤ i ≤ D−1, (a i , b i , c i ) = (a1 , b1 , c1 )}|. The following result gives a lower bound for the multiplicity of an arbitrary eigenvalue. Theorem 3.3.10 (Hiraki and Koolen [165]). Given a distance-regular graph G of order (s, t) and with head h, let λ be an eigenvalue distinct from its degree and set p = ⌊ h+1 2 ⌋.

60 | 3 Particular types of regular graph We have: (i) (ii) (iii) (iv)

If h is odd and λ ≠ −t − 1, then m λ ≥ (t + 1)s p t p−1 ; If h is even and λ ≠ −t − 1, then m λ ≥ (s + 1)(st)p ; p −1 + 1; If h is odd and λ = −t − 1, then m λ ≥ (s − 1)(t + 1) (st) st−1 p+1 (st) 1 If h is odd and λ = −t − 1, then m λ ≥ s ((s − 1)(s + 1) st−1−1 + 1).

From the last theorem one may derive a practical result called the Terwilliger tree bound [313] which reads as follows. If a distance-regular graph G contains a spanning tree such that the distance between every two vertices in G is equal to the distance between them in that tree, then the multiplicity of any eigenvalue distinct from the degree of G is at least the number of pendant vertices in the tree. Two more consequences are the lower bound obtained by Zhu [324] and that of Bannai and Ito [22]. The former p states that m λ ≥ (s − 1)(t + 1) (for p = 1), while the later says that m λ ≥ ( 2r ) (where r is the degree of G) holds whenever a1 ≠ 0. In the same paper, Hiraki and Koolen obtained another useful result bounding the diameter in terms of the multiplicity of an eigenvalue. Namely, if G has an eigenvalue λ such that m λ ≥ 3, then D{

< m λ + log5 m λ + 2,

if h = c2 = 1,

≤ m λ + 6,

otherwise.

The last upper bound for diameter generalizes the upper bounds obtained by Godλ +2) sil [138] stating that D ≤ 3m λ − 4 and r ≤ (m λ −1)(m hold whenever m λ ≥ 3. Finally, 2 note that Terwilliger (see [111]) proved that the inequalities m λ < r and −r < λ < r imply that λ is either the second largest or the least eigenvalue of G. We conclude by the following result giving a condition for a connected regular graph to be distance-regular. We use the orthogonal predistance polynomials encountered in Remark 3.3.6. Theorem 3.3.11 (Spectral Excess Theorem, cf. [48, 111, 127]). For a connected r-regular graph G with n vertices, D + 1 distinct eigenvalues and corresponding orthogonal predistance polynomials P i (0 ≤ i ≤ D), let kD (v) denote the number of vertices at distance D from a vertex v. Then G is distance-regular if and only if kD (v) = PD (r) holds for all v ∈ V(G). The quantity PD (r) is called the spectral excess of G. The name ‘excess’ is taken from [35], where Biggs defined the ith vertex excess as the number of vertices at distance greater than i from it. According to the last theorem, a connected regular graph is distance-regular if and only if the (D − 1)th excess of every vertex coincide with the spectral excess. By [107], the spectral excess can be computed from the spectrum by −1

n D+1 1 PD (r) = 2 ( ∑ ) p1 i=1 m i p2i

,

3.3 Distance-regular graphs |

61

where p i = ∏j=i̸ |θ i − θ j | (1 ≤ i ≤ D + 1). Non-inclusivity with vertex-transitivity Considering the (vertex-transitive) n-crossed prism of Example 3.2.6, we easily deduce that not every vertex-transitive graph is distance-regular. Conversely, here we mention an infinite family of distance-regular graphs with unbounded diameter that are not vertex-transitive. These graphs are know as the twisted Grassmann graphs, and construction is due to Van Dam and Koolen [110]. Given a prime power q and an integer d ≥ 2, let W be a (2d + 1)-dimensional vector space over the finite field GF(q), and let H be a hyperplane in W. The vertices of a twisted Grassmann graph G q (d) are identified with the (d +1)-dimensional subspaces that are not contained in H and the (d − 1)-dimensional subspaces of H. Two vertices of the first kind are adjacent precisely if they intersect in a d-dimensional subspace, two vertices of the second kind are adjacent precisely if they intersect in a (d −2)-dimensional subspace, while a vertex of the first kind is adjacent to a vertex of the second precisely if the subspace of the first contains the subspace of the second. This graph is obviously distance-regular. The twisted Grassmann graph is not vertex-transitive since it has two orbits of vertices. In the next subsection we will meet a better known family of Grassmann graphs (that are vertex-transitive).

3.3.2 Classical examples Here we give a list of well known examples (that in some cases consist of entire families) of distance-regular graphs. A distance-regular graph is called imprimitive if at least one of graphs G i , for 1 ≤ i ≤ D, is disconnected. Otherwise, it is primitive. If G is bipartite, then G2 is disconnected, so bipartite distance-regular graphs are imprimitive. The same holds for antipodal distance-regular graphs (this follows directly from their definition on page 56). Similar examples of imprimitive distance-regular graphs can be encountered by examining odd cycles. In fact, the following result says that there are no other examples of such graphs. Theorem 3.3.12 (Smith Theorem, cf. [48, Theorem 4.2.1]). Every imprimitive distance-regular graph whose degree is at least 3 is bipartite, antipodal or both. If G is bipartite, then its halved graph is defined in the following way. The vertices belong to one part of G, and two vertices are adjacent precisely when they are at distance 2 in G. It is a matter of routine to verify that both halved graphs of a bipartite distance-regular graph are distance-regular. Example 3.3.13. A halved r-dimensional cube has 2r−1 vertices and, according to definition, two vertices are adjacent when they are at distance 2 in Q r . In some literature

62 | 3 Particular types of regular graph

Fig. 3.4. Halved 4-dimensional cube.

it is denoted 12 Q r . A halved 4-dimensional cube is illustrated in Figure 3.4 (for the 4-dimensional cube, see Figure 1.1). In fact, the graph 21 Q4 is isomorphic to the cocktail party CP(4). We already said that if a graph G is antipodal, then GD is a disjoint union of cliques. If so, then we may define the folded graph of G by identifying its vertices with these cliques and saying that two vertices are adjacent precisely when there is an edge (in G) between the corresponding cliques. Clearly, the order of a folded graph does not exceed the order of G. We also say that G is an antipodal k-cover of its folded graph, where k stands for the order of the cliques in GD . The antipodal cover is a special case of a graph cover defined later on page 188. As before, we have that a folded graph of an antipodal distance-regular graph is distance-regular. Example 3.3.14. Recall from the introductory chapter that K −n,n denotes the graph obtained by removing a perfect matching from K n,n . For n ≥ 3, K −n,n is connected and distance-regular with diameter 3. It is also antipodal, since its distance-3 graph consists of n copies of K2 . Its folded graph is K n , and so K −n,n is the antipodal 2-cover of K n (for n ≥ 3). A graph with diameter D is called distance-transitive if for all i (0 ≤ i ≤ D) and all pairs of vertices (u1 , v1 ) and (u2 , v2 ) such that d(u1 , v1 ) = d(u2 , v2 ) = i, there is an automorphism that maps u1 to u2 and v1 to v2 . The property of distance-regularity can be derived from the previous condition, and thus each distance-transitive graph is simultaneously distance-regular. Considering the opposite direction, we may say that many families of distance-regular graphs are also distance-transitive (for example,

3.3 Distance-regular graphs |

63

Fig. 3.5. Shrikhande graph.

this holds for the forthcoming Johnson and Hamming graphs), but in general case, a distance-regularity does not imply distance-transitivity. The smallest example that confirms this is the Shrikhande graph illustrated in Figure 3.5. This graph shares the spectrum [6, 26 , (−2)9 ] with Line(K4,4 ). A Taylor graph is a distance-regular graph whose intersection array has the form (r, s, 1; 1, s, r), for feasible integers r and s. Obviously, its degree is r, it has 2(r + 1) vertices and the diameter is 3. The Taylor graph with (r, s) = (2, 1) is just the hexagon, that with (r, s) = (3, 2) is the 3-dimensional cube (or simply, cube) and that with (r, s) = (5, 2) is the icosahedron (graph of a Platonic solid) illustrated in Figure 3.6. These are the smallest examples of Taylor graphs. Using Theorem 3.3.5, we compute that the distinct eigenvalues of a Taylor graph are r, −1 and 1 2 2 √ 2 (r − 2s − 1 ± (r + 1) − 4(r − 1)s + 4s ). Using the Biggs formula, we compute that the multiplicity of −1 is r, while the multiplicities of the remaining two eigenvalues depend on whether they are integral or not. For example, the spectrum of the icosahedron is [5, (±√5)3 , (−1)5 ]. The vertices of a Johnson graph J(k, d) are identified with the subsets of size d of a set of size k. Two vertices are adjacent if and only if the size of intersection of the corresponding subsets is d − 1 (in other words, the subsets differ in exactly one element). Clearly, there must hold k ≥ d, but since J(k, d) is isomorphic to J(k, k − d), we may restrict ourselves to k ≥ 2d. For example, J(k, 1) is the complete graph K k , J(4, 2) is the octahedron and J(5, 2) is the complement of the Petersen graph. In general case, the k graph J(k, d) has (dk ) vertices, (k−d)d 2 (d) edges, and consequently its degree is (k − d)d.

64 | 3 Particular types of regular graph

Fig. 3.6. Taylor graph with (r, s) = (5, 2) (icosahedron), Johnson graph J(4, 2) (octahedron) and Hamming graph H(3, 2) (lattice Line(K3,3 )).

According to [111], a Johnson graph is the unique distance-regular graph with given intersection array, unless (k, d) = (8, 2). The vertices of a Hamming graph H(k, d) are identified with the ordered d-tuples of elements of a set of size k. (Here, we do not prerequisite the previous condition k ≥ d.) Two vertices are adjacent if and only if the corresponding d-tuples differ in precisely one coordinate. Again, H(k, 1) is the complete graph K k , H(k, 2) is the lattice Line(K k,k ), while H(2, d) is the d-dimensional cube. In general case, the d graph H(k, d) has k d vertices, (k−1)d 2 k edges, and consequently its degree is (k − 1)d. A Hamming graph is the unique distance-regular graph with given intersection array, unless k = 4, d > 1 [111]. Hamming graphs are alternatively defined as the Cartesian products of the d complete graphs K k . If so, then according to Theorem 3.1.4, the spectrum of H(k, d) consists of the numbers k(d − i) − d each with multiplicity (di)(k − 1)i , where 0 ≤ i ≤ d. Bang et al. [20] proved that a Hamming graph H(k, 3) is determined by the spectrum whenever k ≥ 36. The vertices of a Grassmann graph G q (k, d) are identified with the d-dimensional subspaces of a vector space of dimension k over a finite field GF(q), where q is a prime power. Two vertices are adjacent if and only if the corresponding subspaces intersect in a (d − 1)-dimensional subspace. Since G q (k, d) and G q (k, k − d) are isomorphic, we may restrict ourselves to k ≥ 2d. A Grassmann graph is the unique distance-regular graph with given intersection array whenever d ∉ {2, 2k , k−1 2 }, for any q, k−2 k−3 and also whenever (d, q) ∉ {( k−2 , 2) , , 3) , , 2)} [111]. Observe that the ( ( 2 2 2 twisted Grassmann graph G q (d) defined in the previous subsection shares the intersection array with G q (2d + 1, d). The last three families of graphs have intersection numbers that can be expressed in terms of only four parameters d, t, α, β, in the following way:

j where Σ d = ∑d−1 j=0 t .

bi

=

(Σ d − Σ i ) (β − αΣ i ) (0 ≤ i ≤ d − 1),

ci

=

Σ i (1 + αΣ i−1 ) (1 ≤ i ≤ d),

3.3 Distance-regular graphs |

65

A distance-regular graph whose intersection numbers can be expressed in the above form is said to have the classical parameters (d, t, α, β). According to this, the classical parameters of mentioned Johnson, Hamming and Grassmann graphs are given by

(d, 1, 1, k − d), (d, 1, 0, k − 1) and (d, q, q,

q k−d+1 − 1 − 1) , q−1

respectively (where for Johnson and Grassman graphs we still assume that k ≥ 2d). The diameter of a Johnson or a Grassman graph is D = min{d, k − d}, but whenever restricted to k ≥ 2d, we may write D instead of the parameter d. For a Hamming graph, we have D = d.

3.3.3 Inequalities for eigenvalues There is a number of inequalities obtained in terms of intersection numbers, mostly expressed in the form of bounds on a fixed structural invariant (like diameter) of a distance-regular graph. Since the spectrum of a distance-regular graph can be computed from its intersection numbers (recall Theorems 3.3.5 and 3.3.8), here we mention just few of them. Bang et al. [21] obtained the following upper bound for a distance-regular graph of degree r and diameter D, D< 1

1 c r k + 1, 2

1

where c = inf{x > 0 : 4 x − 2 x ≤ 1} ≈ 1.44 and k = |{i : c i = 1}|. Hiraki [164] proved that k ≤ 2h + 2, where h is the head of G. As a consequence we have D ≤ r c (h + 1) + 1, which implies the Ivanov bound D < 2r+1 (h + 1). Concerning distance-regular graphs of order (s, t), Faradjev et al. [122] established a bound that is finer than the previous general ones. It states that D ≤ th + 1 holds whenever D ≥ 2 and c h+1 > 1. The following inequality, obtained by Jurišić et al. [191], that includes the second largest and the least eigenvalue of a distance-regular graph G of degree r and diameter D ≥ 3, is known as the fundamental bound: (θ2 +

r ra1 b1 r . ) (θD+1 + )≥− a1 + 1 a1 + 1 (a1 + 1)2

Observe that, if r, λ2 , . . . , λ n are all eigenvalues of G, then since G is connected we have λ2 = θ2 and λ n = θD+1 . If we define the (primitive idempotent) matrix D+1

Ei = ∏

j=i,j=1 ̸

A − θj I (1 ≤ i ≤ D + 1), θi − θj

66 | 3 Particular types of regular graph then it can be seen that E i is an orthogonal projection onto the eigenspace of the adjacency matrix A corresponding to θ i . If x is an eigenvector of A associated with θ j , then E i x = δ ij x. This means that {E1 , E2 , . . . , ED+1 } makes another basis of the matrix space spanned by {I, A, . . . , AD }. Using this, the authors of [191] proved the fundamental bound by taking adjacent vertices u and v and considering the determinants of the Gram matrices of vectors Eeu , Eev and E1uv , where E ∈ {E1 , ED+1 } and eu , ev belong to the canonical basis of ℝn , while 1uv is the characteristic vector of N(u)∩ N(v). Different proofs and more details can be found in [190, 247]. If K(= K k ) is a clique in a distance-regular graph G (with n vertices), then considering the primitive idempotent E corresponding to λ n we deduce that 1K T E1K ≥ 0 (where 1K is the characteristic vector of V(K)), which yields an upper bound for the clique number of G (cf. [137, p. 261]): ω≤

θD+1 − r . θD+1

(3.9)

The reader may compare this bound with the upper bound for the independence number given in (2.14). Cioabă and Koolen [77] considered an interplay between the (i + 1) × (i + 1) top-left block and the (D− i +1)×(D− i +1) bottom-right block of the tridiagonal matrix (3.6) to obtain a sequence of conditions for the second largest eigenvalue. In particular, they proved that θ2 lies between a3 and

a1 +√ a21 +4r . 2

In other words,

{ a1 + √a21 + 4r } { } θ2 ≥ min {a3 , . } } { 2 { } In particular, for D ≥ 4, we have θ2 ≥ matrix, we may conclude that θD+1
(2l − 1)(b󸀠 − 1) − 1; 󸀠 −1) . Maximal vertex degree in G is less than (l + 1)(a󸀠 + 1) − l(l+1)(b 2

If a maximal clique K with the property |V(K)| ≥ a󸀠 + 2 − (l − 1)(b󸀠 − 1) is called a line, then every vertex of G belongs to at most l lines, and every two adjacent vertices belong to a unique line. Metsch used this result to characterize Grassmann graphs. More details can be found in the original reference. Here we prove the following results giving sufficient conditions for a distance-regular graph to be geometric. Both are taken from [111], but also can be found in [201]. Theorem 3.3.16. Let G be a distance-regular graph with n vertices and diameter D satisfying the property that there is a positive integer l and a set K of cliques such that every edge of G belongs to exactly one clique and every vertex of G belongs to exactly l cliques of K. If |K| < |V(G)|, then G is geometric with least eigenvalue −l.

68 | 3 Particular types of regular graph Proof. Consider the |V(G)| × |K| incidence matrix N whose rows and columns are indexed by the vertices of G and the cliques of K in such a way that the (v, K)-entry is 1 if v ∈ K, and 0 otherwise. Since NN T = lI + A, we have θD+1 ≥ −l. If, in addition, |K| < |V(G)|, then NN T is a singular matrix, and so θD+1 = −l. It remains to show that all cliques of K are Delsarte. By (3.9), each of these cliques has at most l+r l vertices. Counting the number of vertices of l cliques containing a fixed vertex, we deduce that they must have exactly l+r l vertices, and the result follows. Note that the inequality min{|K| : K ∈ K} > l assures the condition |K| < |V(G)|. Theorem 3.3.17. Let G be a non-complete distance-regular graph of degree r satisfying 2 +1) (l−1)(a1 +1) < r < l(l+a1 ), for some integer l ≥ 2. If a1 ≥ l(l+1)(c , then G is geometric 2 with least eigenvalue −l (where a1 and c2 are the corresponding intersection numbers). Proof. The lower bound on a1 and the upper bound on r of this theorem can be 2 −1) rewritten as a1 ≥ (2l − 1)(c2 − 1) and r ≤ (l + 1)(a1 + 1) − l(l+1)(c , giving the as2 sumptions (iii) and (iv) of Theorem 3.3.15. Hence, the set K of cliques with at least a1 + 2 − (l − 1)(c2 − 1) vertices is a set of lines such that every vertex of G belongs to at most l lines and every edge of G is in exactly one line. Using the remaining (lower) bound on r, we conclude that every vertex belongs to exactly l lines. Since a1 + 2 − (l − 1)(c2 − 1) ≥ l + 1, applying the previous theorem, we get that G is geometric with least eigenvalue −l.

3.3.5 Deducing distance-regularity from the spectrum By Theorem 3.3.5, the intersection array uniquely determines the spectrum of a distance-regular graph. Conversely, if we know that a graph is distance-regular, then its spectrum uniquely determines the intersection array (by Theorem 3.3.9). Therefore, a natural question that arises is: Does a graph that shares the spectrum with a distance-regular graph must necessary be distance-regular? The answer is affirmative for complete graphs (which is an obvious fact) and all distance-regular graphs with diameter 2 (compare the forthcoming Remark 3.4.5 in conjunction with Remark 3.4.8). In 1963, Hoffman [167] constructed a non-distance-regular graph cospectral with the 4-dimensional cube (which is distance-regular). This graph (named after its discoverer) is illustrated in Figure 3.7. Since that, many similar counterexamples appeared in literature, but also there are more cases in which the answer is affirmative. For example, we mentioned on page 64 that every Hamming graph H(k, 3), for k ≥ 36, is unique with the corresponding spectrum, and consequently the answer is affirmative for these graphs. Other cases are summarized in the following theorem. Theorem 3.3.18 (cf. [3, 111]). If a distance-regular graph G with diameter D, girth g and distinct eigenvalues θ1 , θ2 , . . . , θD+1 satisfies one of the following conditions, then every graph cospectral with G is also distance-regular (with the same intersection array):

3.3 Distance-regular graphs |

69

Fig. 3.7. Hoffman graph.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

g ≥ 2D − 1; g ≥ 2D − 2 and G is bipartite; g ≥ 2D − 2 and cD−1 cD < −(cD−1 + 1) ∑D+1 i=2 θ i ; a1 = a2 = ⋅ ⋅ ⋅ = aD−1 = 0, but aD ≠ 0; c1 = c2 = ⋅ ⋅ ⋅ = cD−1 = 1; D = 3 and c2 = 1; D = 3 and G is bipartite; D = 4, G is bipartite and every pair of non-adjacent vertices have the same number of common neighbours.

All quantities mentioned in (i)–(vi) can be derived from the spectrum of G (for the girth, see page 17). Part (i) is proved in [50], for (ii), (iii) and (v) see [105], for (iv) see [180] and for (vi) see [156]. Part (vii) follows directly from (ii); It is considered in a different context in Theorems 3.8.3 and 3.8.5. The details for part (viii) can be found in [3]. Graphs having the properties of (iv) are known as generalized odd graphs. Concerning individual graphs, we remark that Haemers and Spence [160] determined all graphs cospectral with distance-regular graphs with at most 27 vertices, while Van Dam et al. [108] extended this list to those with at most 70 vertices. It further occurs that the answer to the above question is affirmative for the icosahedron or the dodecahedron [111], or any of the graphs with intersection arrays (7, 6, 4, 4; 1, 1, 1, 6), (3, 2, 2, 2, 1, 1, 1; 1, 1, 1, 1, 1, 1, 3) (Biggs–Smith graph), (21, 20, 16; 1, 2, 12) or (24, 22, 20; 1, 2, 12) [105]. Some distance-regular graphs that are cospectral with non-distance-regular graphs are Johnson graphs J(k, 3), for k ≥ 6, [156] or those with intersection ar-

70 | 3 Particular types of regular graph rays (5, 4, 1, 1; 1, 1, 4, 5) (Wells graph), (7, 6, 6, 1, 1; 1, 1, 6, 6, 7) (bipartite double of the Hoffman–Singleton graph) or (3, 2, 2, 2, 2, 1, 1, 1; 1, 1, 1, 1, 2, 2, 2, 3) (Foster graph) [105]. For more details on this topic, see [4, 105, 106, 111].

3.4 Strongly regular graphs An r-regular graph with n vertices that is neither complete nor totally disconnected is said to be strongly regular with parameters (n, r, a, b)

(3.11)

if there exist non-negative integers a and b such that every two adjacent vertices have exactly a common neighbours and every two non-adjacent vertices have exactly b common neighbours. Hence, the number of common neighbours of two vertices of a strongly regular graph depends only on whether they are adjacent or not. The graphs K n and nK1 are excluded since the parameters b and a are undefined for them. It follows directly that the diameter of a connected strongly regular graph is 2. Exercise 3.4.1. Use induction to prove that the number of closed k-walks (k ≥ 1) in a strongly regular graph does not depend on a starting vertex, and consequently that every strongly regular graph is walk-regular. Examples of strongly regular graphs appear sporadically in the previous part of the book. We recall some of them. Example 3.4.2. The quadrangle and the pentagon are strongly regular with parameters (4, 2, 0, 2) and (5, 2, 1, 0), respectively. The disjoint union kC3 (k ≥ 2) is also strongly regular with parameters (3k, 2, 1, 0). Moreover, there are no other strongly regular graphs of degree 2. The Petersen graph is strongly regular with parameters (10, 3, 0, 1), and there are no other strongly regular graphs sharing the same parameter set. The Hoffman–Singleton graph (see page 143) is also a unique strongly regular graph with parameters (50, 7, 0, 1). Line graphs Line(K n ) (n ≥ 4) and Line(K n,n ) (n ≥ 2) are strongly regular with parameters ((2n), 2(n − 2), n − 2, 4) and (n2 , 2(n − 1), n − 2, 2), respectively. For the next exercise, the reader should observe the diameter of graphs under consideration. Exercise 3.4.3. Determine which Johnson, Hamming or Grassmann graphs are strongly regular?

3.4 Strongly regular graphs | 71

3.4.1 Basic properties Throughout this subsection, we assume that G is a strongly regular graph with parameters (3.11). If G is bipartite, then a = 0. Considering the Petersen graph, we conclude that the converse is not true. (Connected bipartite strongly regular graphs are characterized in the forthcoming Exercise 3.4.10.) Next, simple combinatorial arguments show that the complement G is also strongly regular with parameters (n, r, a, b), where r = n − r − 1, a = n − 2(r + 1) + b, b = n − 2r + a.

(3.12)

Disconnected strongly regular graphs are characterized by the following lemma. Lemma 3.4.4 (cf. [145, Lemma 10.1.1]). If G is a strongly regular graph with parameters (n, r, a, b), then the following properties are equivalent: (i) (ii) (iii) (iv)

G is disconnected; b = 0; a = r − 1; G is a disjoint union of (at least two) complete graphs K r+1 .

Proof. Implications (i)⇒(ii) and (iv)⇒(i) are trivial. If b = 0, then every two neighbours of a fixed vertex must be adjacent which yields a = r − 1, and we get (ii)⇒(iii). If a = r − 1, then every two adjacent vertices share the same closed neighbourhood, and so every component of G must be the complete graph K r+1 which gives (iii)⇒(iv), and we are done. Hence, a strongly regular graph is disconnected if and only if it is a disjoint union of mutually isomorphic complete graphs. Moreover, if G is r-regular and disconnected, then G is complete (r + 1)-partite. Remark 3.4.5. If G is connected, then it is straightforward to conclude that G is distance-regular with diameter 2 and intersection array (r, r − a − 1; 1, b). In other words, apart from disconnected strongly regular graphs (which are characterized by the previous lemma), the remaining ones are distance-regular. In this spirit, this section may be considered as an extension of the previous one. In Subsection 3.3.2, we dealt with imprimitive distance-regular graphs. In that context, the imprimitivity of a distance-regular graph was assured by disconnectedness of at least one distance-i graph. Following the similar reasoning, we say that a strongly regular graph G is primitive if both G and G are connected. There is a simple feasibility condition for the parameters. Lemma 3.4.6. The parameters (n, r, a, b) of a strongly regular graph G satisfy

72 | 3 Particular types of regular graph

r(r − a − 1) = b(n − r − 1).

(3.13)

Proof. Let S and T stand for the set of neighbours of a fixed vertex v and the set of its non-neighbours (i.e., S = N(v) and T = V(G)\N[v]). We have |S| = r and |T| = n − r − 1. Each vertex of S is adjacent to a vertices in S, and so to r − a − 1 vertices in T. Thus, the total number of edges between S and T is m(S, T) = r(r − a − 1). Similarly, each vertex of T is adjacent to b vertices in S, giving m(S, T) = b(n − r − 1), and we are done. Additional feasibility conditions are given later on by the identities (3.18) and by the result of Theorem 3.4.12. Spectrum If A stands for the adjacency matrix of G, then any entry of A2 represents the number of 3-walks starting in one and terminating in the other of the corresponding vertices. Since G is strongly regular, we may consider all possible 3-walks between two vertices by distinguishing cases in which they are identical, adjacent or distinct and non-adjacent. In this way we get A2 = rI + aA + b(J − I − A), that is A2 − (a − b)A − (r − b)I = bJ.

(3.14)

We use this identity to prove the following theorem. Theorem 3.4.7 (Shrikhande and Bhagawandas [277]). A connected r-regular graph with n vertices is strongly regular if and only if it has 3 distinct eigenvalues r, θ2 and θ3 . In addition, if G is strongly regular then a = r + θ2 θ3 + θ2 + θ3 and b = r + θ2 θ3 .

(3.15)

Proof. Assume that G is connected regular graph with 3 distinct eigenvalues denoted as in the theorem. By Theorem 2.1.6, the Hoffman polynomial of G satisfies P(A) = c1 A2 + c2 A + c3 I = J,

(3.16)

where c1 ≠ 0 (otherwise, we would have less than 3 distinct eigenvalues), while θ2 and θ3 are the roots of c1 x2 + c2 x + c3 = 0. Concerning diagonal elements in (3.16), we may eliminate c3 by computing c3 = 1 − c1 r. Concerning non-diagonal elements, we 2 deduce that the number of 3-walks between adjacent vertices of G is 1−c c1 , while the 1 number of 3-walks between distinct non-adjacent vertices is c1 . Thus, G is strongly regular. By comparing the coefficients in (3.14) and (3.16), we get both equalities of (3.15). Assume now that G is strongly regular. By (3.14), we have that G has at most 3 distinct eigenvalues (since the degree of its minimal polynomial is at most 3). The eigenvalues of G cannot be mutually equal since G is not totally disconnected (by defini-

3.4 Strongly regular graphs | 73

tion). Similarly, G cannot have (exactly) 2 distinct eigenvalues since then its diameter would be equal to 1 which is impossible (again, by definition), and we are done. In general case, a graph with 3 distinct eigenvalues does not need to be regular, but nevertheless its regularity can be checked by Corollary 2.1.3. Non-regular graphs with 3 distinct eigenvalues are studied in [102]. Remark 3.4.8. The last theorem states that we may take any connected regular graph and conclude whether is strongly regular or not by counting its distinct eigenvalues. Observe that this information is not easily accessible if we use the definition of a strongly regular graph, since we would need to check the common neighbourhood of (2n) pairs of vertices. Bearing in mind that a connected graph has only one positive eigenvalue if and only if it is complete multipartite, we deduce the following result. Theorem 3.4.9. A strongly regular graph G has the eigenvalue 0 if and only if it is a complete multiparite regular graph. Proof. By Lemma 3.4.4, G cannot be disconnected. If r, 0 and θ3 are distinct eigenvalues of G, then by inserting θ2 = 0 in both expressions of (3.15) and then inserting these expressions in (3.13), we conclude that G has r − θ3 vertices. Thus, by Theorem 2.2.1, the distinct eigenvalues of the complementary graph G are −θ3 −1 and −1, which gives the first implication. The opposite implication follows directly, and we are done. Using the fact that a strongly regular graph has 3 distinct eigenvalues, we arrive at the following result which is an easy exercise. Exercise 3.4.10. Show that a connected strongly regular graph is bipartite if and only if it is complete bipartite. For G connected, the eigenvalues θ2 and θ3 can be expressed in terms of parameters (3.11) as follows. Corollary 3.4.11. The eigenvalues θ2 and θ3 of a connected strongly regular graph G with parameters (n, r, a, b) are given by θ2 = θ3 =

a−b+√(a−b)2 +4(r−b) , 2 a−b−√(a−b)2 −4(r−b) . 2

(3.17)

Proof. The result follows by solving the system (3.15) in θ2 , θ3 . Since G has at least one edge, θ3 is negative. On the contrary, inspecting the expression for θ2 we deduce that this eigenvalue must be positive, unless G is complete multipartite in which case we have θ2 = 0. The multiplicities of the eigenvalues θ2 and θ3 read as follows:

74 | 3 Particular types of regular graph

1 (n − 1 − √2r+(n−1)(a−b) ), (a−b)2 −4(r−b) 2 (3.18) 1 m3 = (n − 1 + √2r+(n−1)(a−b) . ) (a−b)2 −4(r−b) 2 A simple way to derive these equalities is to use the spectral moments to obtain m2 + m3 = n − 1 and m2 θ2 + m3 θ3 = −r, which gives m2 =

m2

=

3 − (r+n−1)θ θ2 −θ3 ,

m3

=

r+(n−1)θ3 θ2 −θ3 .

If so, then both equalities of (3.18) are obtained by substituting the values for θ2 and θ3 (given in (3.17)) in the last expressions. The equalities of (3.18) give two feasibility conditions for the parameter set of a putative connected strongly regular graph. Namely, for given parameters we may compute the corresponding multiplicities, and if at least one of them is not a positive integer, then there is no connected strongly regular graph with these parameters. Another pair of feasibility conditions is given in the next theorem. Theorem 3.4.12 (Krein Bounds, cf. [48] or [145, Theorem 10.7.1]). For a primitive strongly regular graph with parameters (n, r, a, b), we have θ2 θ23 − 2θ22 θ3 − θ22 − r(θ2 − θ23 − 2θ3 ) ≥ 0, θ22 θ3 − 2θ2 θ23 − θ23 + r(θ22 − θ3 + 2θ2 ) ≥ 0. If equality holds in the first (resp. second) case, then r ≥ m2 (resp. r ≥ m3 ). There are similar inequalities for distance-regular graphs and the last result can be derived from them (see [48]). For a long but straightforward proof which is restricted to strongly regular graphs and provides information about the graphs attaining equalities, we recommend the book of Godsil and Royle [145]. If we partition the vertices of a primitive strongly regular graph G as described in the proof of Theorem 2.6.1, that is into the three parts containing a single vertex v, its neighbours and the remaining vertices of a graph, we get the following blocking of the adjacency matrix: 0 A = (j O

jT A1 B

OT BT ) , A2

where O is an all-0 vector. Clearly, A1 is the adjacency matrix of the subgraph induced by neighbours of v, while A2 is the adjacency matrix of the subgraph induced by vertices at distance two. Resuming terminology for distance-regular graphs, we call these two subgraphs the first and the second subconstituent of G (with respect to v). According to [145], if x is an eigenvector of A1 orthogonal to j, then the corresponding eigenvalue, say λ, is located by the value of Bx. Namely,

3.4 Strongly regular graphs | 75

{

λ ∈ {θ3 , θ2 },

if Bx = O,

λ ∈ (θ3 , θ2 ),

if Bx ≠ O.

By considering G, we get the similar conclusion for the second subconstituent. We say that an eigenvalue of a subconstituent is local if it is not an eigenvalue of G and has an eigenvector x orthogonal to j. In other words, if Bx ≠ O (for A1 ) or B T x ≠ O (for A2 ). We conclude by another result. Theorem 3.4.13 (cf. [145, Theorem 10.6.3]). If G is a primitive strongly regular graph with parameters (n, r, a, b), then λ is a local eigenvalue of one subconstituent of G if and only if a − b − λ is a local eigenvalue of the other (with equal multiplicity).

3.4.2 Small examples Constructions of strongly regular graphs usually start with determining parameters that satisfy the set of feasibility conditions obtained throughout the previous section. Note that passing these conditions does not assure the existence of any graph corresponding to them. Small strongly regular graphs can be constructed by hand, and for example there are exactly 13 primitive strongly regular graphs with at most 16 vertices. All of them can be identified from the following list (the asterisk indicates the existence of the non-isomorphic complement). Graph Pentagon Line(K3,3 ) Petersen graph∗ 13-Paley graph

Parameters (5,2,0,1) (9,4,1,2) (10,3,0,1) (13,6,2,3)

Graph

Parameters

Line(K6 Clebsch graph∗ Shrikhande graph∗ Line(K4,4 )∗ )∗

(15,8,4,4) (16,5,0,2) (16,6,2,2) (16,6,2,2)

The 13-Paley graph and the Clebsch graph are illustrated in Figure 3.8, while the remaining graphs from the list are already encountered in this book. Paley graphs are special strongly regular graphs; The definition is given later on page 78. Once we have determined a strongly regular graph with fixed set of parameters, the next natural problem is inspecting whether there exist other strongly regular graphs sharing the same parameter set. One method is based on the Seidel switching described on page 26. Although all switchings are with respect to sets that induce a regular subgraph whose degree is determined by its order, in practice this method is oriented to a computer search and even then constrained by the value of n. Example 3.4.14. According to the previous list, there are two non-isomorphic strongly regular graphs with parameters (16, 6, 2, 2): the Shrikhande graph and the lattice Line(K4,4 ). Their cospectrality was also mentioned on page 63. Since they are

76 | 3 Particular types of regular graph

Fig. 3.8. 13-Paley graph and Clebsch graph.

cospectral and satisfy the conclusion of Theorem 2.3.9, we may look for a Seidel switching that transforms one of them into another. Indeed, if we apply the Seidel switching with respect to a co-clique with 4 vertices in Line(K4,4 ), we get the Shrikhande graph. Strongly regular graphs of larger order are considered in Subsection 4.4.1, while in the following two subsections we focus on certain particular families.

3.4.3 Conference graphs A connected strongly regular graph with m2 = m3 is called a conference graph. These graphs are named after symmetric conference matrices. A conference matrix C is an (n + 1) × (n + 1) matrix with 0’s on the main diagonal and 1’s and −1’s outside, such that CC T = nI. The last identity requires n to be odd. The name arises from the problem of constructing telephone conference networks (see a work of Belevitch [29]). If, in addition, C is symmetric then there must hold (n + 1) ≡ 2 (mod 4), which is a simple combinatorial fact. Less elementary is the result reported by Van Lint and Seidel [213] (see also [29]) stating that n must be a sum of two (perfect) squares. Considering candidates for the order of a conference matrix, we recall that, by the Fermat two squares theorem, an odd prime n is a sum of two squares if and only if n ≡ 1 (mod 4). Furthermore, if two integers n1 and n2 are expressible as sums of two squares, say n1 = s211 + s212 and n2 = s221 + s222 , then the same holds for their product since n1 n2 = (s11 s22 − s12 s21 )2 + (s11 s21 + s12 s22 )2 . Consequently, any prime power n satisfying n ≡ 1 (mod 4) is always expressible as a sum of two squares. Considering odd sums of two squares, we conclude that there is no conference matrix of size 22 nor 34. They exist for all other sizes (congruent to 2 mod 4) with up to 46.

3.4 Strongly regular graphs | 77

By permuting rows and columns of a symmetric conference matrix, we can obtain a similar conference matrix in which, apart from the diagonal one, all entries of the (n + 1)th row and the (n + 1)th column are 1’s. It is not difficult to see that the n × n submatrix obtained by deleting mentioned row and column is a Seidel matrix of some conference graph (if necessary, see [213, 271]). Conversely, a conference matrix can be reconstructed from a conference graph by extending its Seidel matrix in the described way (cf. [271]). Therefore, the order n of a conference graph satisfies n ≡ 1 (mod 4) and it is a sum of two squares. We may say this in a more concise way. Namely, since any square is either ≡ 0 (mod 4) or ≡ 1 (mod 4), we have that a sum of two squares cannot be ≡ 3 (mod 4). Hence, if we already know that the order n of a conference graph must be an odd integer expressible as a sum of two squares, then it is needless to add that there must be n ≡ 1 (mod 4). To get more properties, we need the following lemma. Lemma 3.4.15. For a connected strongly regular graph G with parameters (n, r, a, b), we have m2 m3 =

nr(n − r − 1) . (θ2 − θ3 )2

The proof may be obtained by using the expressions for multiplicities given by (3.18). Now, we may compute the parameters of a conference graph. Theorem 3.4.16 (cf. [145, Lemma 10.3.1]). The parameters of a conference graph G with n vertices are (n,

n−1 n−5 n−1 , , ). 2 4 4

Proof. Since G is a conference graph, we have m2 = m3 = n−1 2 . By the previous lemma, 2 n−1 divides nr(n − r − 1), but since we get that ( n−1 and n are coprime this implies ) 2 2 n−1 2 that ( 2 ) divides r(n − r − 1). The last is possible only if r = n−1 2 , giving the second parameter. Using the spectral moment m2 (θ2 + θ3 ) = −r we get θ2 + θ3 = a − b = −1, while using the feasibility condition (3.13) we get b = 2r , giving the remaining parameters. If G is a connected strongly regular graph, then by using (3.18) we get m3 − m2 =

2r + (n − 1)(a − b) . √(a − b)2 − 4(r − b)

(3.19)

If G is not a conference graph then (a − b)2 −4(r − b) is a square, and then by (3.17), θ2 and θ3 are rational. Moreover, since both of them are obtained as roots of a monic quadratic polynomial with integral coefficients (that arises from (3.15)), they are integers. Thus, we get the following result.

78 | 3 Particular types of regular graph Theorem 3.4.17 (cf. [145, Lemma 10.3.3]). A strongly regular graph with parameters (n, r, a, b) having non-integral eigenvalues is a conference graph. Observe that Line(K3,3 ) and the 13-Paley graph are conference graphs with all eigenvalues integral, and so the converse is not true. We characterize strongly regular graphs of a prime order. Corollary 3.4.18. A strongly regular graph G of a prime order n is a conference graph. Proof. Since n is a prime, we have that G cannot be disconnected (by Lemma 3.4.4). By Lemma 3.4.15, it holds (θ2 − θ3 )2 = nr(n−r−1) m2 m3 . If G is not a conference graph, then θ 2 and θ3 are integers. Thus, the right hand side of the last equality is a square containing a prime n as a factor, and consequently n must divide r(n−r−1) m2 m3 , which is impossible since the last value is less than n. Using the similar reasoning, the reader may try to solve the following two exercises giving the parameters of specified strongly regular graphs. Exercise 3.4.19 ([145]). Prove that if a primitive strongly regular graph of degree r < has an eigenvalue λ with multiplicity 2n , then its parameters are given by

n 2

((2λ + 1)2 + 1, λ(2λ + 1), λ2 − 1, λ2 ) . (Hint: Since the multiplicity is 2n , the graph is not conference, and thus its spectrum is integral. Show first that λ must be positive, and then use the spectral moments to get the parameters.) Observe a consequence that there is no primitive strongly regular graph such that its degree and degree of its complement are coprime. Exercise 3.4.20 ([145]). Prove that if a primitive strongly regular graph has n vertices where 2n is prime, then its or the parameters of its complement are the same as in the previous exercise. (Hint: The graph is not conference since it has an even number of vertices, and thus its spectrum is integral. Use the previous exercise to obtain the parameters.) Paley graphs We proceed with an abstract definition. In the sequel, we show that Paley graphs make a particular family of conference graphs. For a prime power n such that n ≡ 1 (mod 4), the vertices of an n-Paley graph G are identified with the elements of the finite field GF(n), and two vertices are adjacent if and only if the difference between the corresponding elements is a non-zero square in GF(n). Formally, V(G) = GF(n) and E(G) = {{u, v} : u, v ∈ GF(n), u − v ∈ (GF(n)∗ )2 } ,

3.4 Strongly regular graphs | 79

where GF(n)∗ = GF(n)\{0} and (GF(n)∗ )2 = {u2 : u ∈ GF(n)∗ }. Recall from the field theory that there is exactly one finite field of order n (where n is as in the above) and that GF(n)∗ form a cyclic group under multiplication. To show that the edge set is well defined (i.e., that G is undirected), let u denote a generator of GF(n)∗ . Then n − 1 is the least positive integer such that u n−1 = 1, which yields (u Since u

n−1 2

n−1 2

− 1) (u

n−1 2

+ 1) = 0.

≠ 1, we have u

n−1 2

= (u

n−1 4

2

) = −1,

n−1

that is u 4 is the square root of −1, and hence −1 ∈ (GF(n)∗ )2 . If so, then the equality u − v = −1(v − u) implies that u − v ∈ (GF(n)∗ )2 is equivalent to v − u ∈ (GF(n)∗ )2 , which means that G is undirected, and thus well defined. Example 3.4.21. The GF(5) is just the finite field of residue classes mod 5, denoted ℤ5 (actually, similar holds for any GF(n), where n is a prime). The vertex set of the 5-Paley graph is identified with the set of class representatives {0, 1, 2, 3, 4} (recall that in ℤ5 , we have −1 = 4). The edge set is {{0, 1}, {1, 2}, {2, 3}, {3, 4}, {4, 0}}, giving the pentagon. The GF(9) is the finite field ℤ3 [x]/P(x), where P is an irreducible monic polynomial of degree 2 over ℤ3 [x] (similar holds for any GF(n), where n is prime powered by k ≥ 2). Taking P(x) = x2 + 1, we get that the vertices of the 9-Paley graph are identified with the elements of the field GF(9) = {0, 1, 2, t, 2t, t + 1, 2t + 1, t + 2, 2t + 2}, where t is a root of x2 + 1. We next obtain (GF(9)∗ )2 = {1, 2, t, 2t} (here we have −1 = 2), and then the edge set is determined by computing E = {{u, v} : u, v ∈ GF(9), u − v ∈ (GF(9)∗ )2 }. This gives Line(K3,3 ). Both graphs are listed on page 75, together with the 13-Paley graph. An edge-transitivity of a graph is defined in the same way as a vertex-transitivity (page 54) with edges instead of vertices. In addition, it follows directly that a graph is edge-transitive if and only if its line graph is vertex-transitive. Clearly, edge-transitive graphs are regular. We say that a graph is symmetric if it is simultaneously vertex-transitive and edge-transitive. We believe that the proof of the following result is an adequate exercise for the reader. Theorem 3.4.22. A Paley graph is symmetric. There is another interesting property. Theorem 3.4.23 (cf. [48, 120]). A Paley graph is self-complementary.

80 | 3 Particular types of regular graph Proof. Consider an n-Paley graph G. Let r be a quadratic non-residue of n (i.e., r is a positive integer coprime with n such that x2 ≡ r (mod n) has no solution in the set of positive integers less than n). Consider the function g : V(G) 󳨀→ V(G), g(u) = ru, where the vertex set is defined by the corresponding finite field and all operations are in that field. Observe that if uv is an edge of G, then u − v ∈ (GF(n)∗ )2 but r(u − v) ∉ (GF(n)∗ )2 , and so g(u)g(v) is an edge in G. To show that g is an isomorphism between G and G, it remains to prove its bijectivity. The injectivity follows by u − v = 0 ⇔ 0 = r(u − v) = g(u) − g(v). Since r and n are coprime, there exists an integer z such that rz ≡ 1 (mod n), and thus g(zu) = rzu = u gives the surjectivity, and we are done. We conclude by the announced result. Theorem 3.4.24 (cf. [48, 120]). An n-Paley graph is strongly regular with parameters (n,

n−1 n−5 n−1 , , ). 2 4 4

Proof. For a vertex u ∈ GF(n), we have N(u) = {v ∈ GF(n), u − v ∈ (GF(n)∗ )2 }. If u − v1 = u − v2 = w then v1 = u − w = v2 , and so since the surjectivity is clear, there is a bijection between N(u) and (GF(n)∗ )2 . In other words, the degree of u is |(GF(n)∗ )2 |. Since |GF(n)∗ | = n − 1, we conclude that for distinct vertices u, v ∈ GF(n)∗ , the equality u2 = v2 yields u = −v, and so |(GF(n)∗ )2 | = n−1 2 , which gives the second parameter. For the remaining parameters, let u be a fixed vertex, and let N(u) be as in the above. Denote U = GF(n)\N[u]. We need to prove that |N(u) ∩ N(v)| = n−5 4 holds for all v ∈ N(u), and that |N(u) ∩ N(w)| = n−1 holds for all w ∈ U. Since our graph is 4 symmetric, every vertex v ∈ N(u) is adjacent exactly to n−1 vertices in U, and since 4 it is self-complementary, every vertex w ∈ U is non-adjacent to the same number of n−1 vertices in N(u). Thus, we have |N(v) ∩ U| = n−1 4 and |(GF(n)\N(w)) ∩ N(u)| = 4 . n−1 Now, since 2 = N(v) = |N(u) ∩ N(v)| + |U ∩ N(v)| + |{u} ∩ N(v)|, we conclude that |N(u) ∩ N(v)| = n−5 4 . Similarly, since n−1 2 = |N(u)| = |(GF(n)\N(w)) ∩ N(u)| + |N(w) ∩ N(u)|, we get |N(u) ∩ N(w)| = n−1 , and the proof is complete. 4 Since for n ≥ 5 we have n−1 4 ≥ 1, every Paley graph is connected. According to definition, a Paley graph exists for any prime power n ≡ 1 (mod 4), and also there is a conference graph of the same order. Observe next that conference graphs and Paley graphs of the same order share the same set of parameters (by Theorems 3.4.16 and 3.4.24). Since the parameters determine the multiplicities of eigenvalues (by (3.18)), we conclude that Paley graphs satisfy the property that defines conference graphs, and therefore all Paley graphs are conference graphs.

3.4 Strongly regular graphs | 81

3.4.4 Generalized Latin squares and partial geometries We start with Latin squares, orthogonal arrays and generalized Latin squares, and then proceed with partial geometries and generalized quadrangles. We show that all graphs that arise from these combinatorial structures are strongly regular. We also consider the spectra of these graphs. Generalized Latin squares An n×n Latin square consists of n n-tuples of numbers 1, 2, . . . , n arranged in a square matrix in such a way that no row or column contains the same number repeated. Here is an example of a Latin square 1 2 (3 ( (4 5 (6

2 3 4 5 6 1

3 4 5 6 1 2

4 5 6 1 2 3

5 6 1 2 3 4

6 1 2) ). 3) 4 5)

(3.20)

Now, the vertices of a Latin square graph are identified with the n2 elements of the corresponding Latin square, and two vertices are adjacent precisely when the corresponding elements are identical or belong to the same row or column. We may use the elements of an n × n Latin square to form a 3 × n2 matrix M whose columns are indexed by the row number i, column number j and the (i, j)-entry of the Latin square. If so, then another way to define a Latin square graph is to take the columns of M to be its vertices and say that two vertices are adjacent if and only if the corresponding columns agree in exactly one position. We say that two n × n Latin squares are orthogonal if the n2 ordered pairs formed by the entries in the same positions are all distinct. If so, then we may generalize the above 3 × n2 matrix in the following way. Given a set of (r − 2) (where r ≥ 3) orthogonal n × n Latin squares. Then the first two rows (in general case, any pair of rows can play this role) of the r × n2 matrix M are indexed by the row numbers and column numbers of Latin squares, while there is a bijection between the remaining rows and the set of given Latin squares, and each row is indexed by the entries of one of them. Of course, we may form such a matrix independently of Latin squares in the following way. For r ≥ 2 and n as in the above, an orthogonal array OA(r, n) is an r×n2 matrix whose entries belong to {1, 2, . . . , n} such that the n2 ordered pairs determined by any two rows of the matrix are all distinct. Clearly, the matrix M (formed on the basis of a set of orthogonal Latin squares) is an orthogonal array, but also vice versa, an orthogonal array OA(r, n) gives rise to r − 2 Latin squares which are mutually orthogonal. The last follows by definition.

82 | 3 Particular types of regular graph Exercise 3.4.25. Determine the orthogonal array that corresponds to the Latin square of (3.20) and confirm that the same Latin square may be reconstructed from it. Now, we are in position to generalize the above definition of a Latin square graph. Given an orthogonal array OA(r, n), the vertices of a generalized Latin square graph are identified with the n2 columns of OA(r, n), and two vertices are adjacent if and only if the corresponding columns agree in exactly one position. Remark 3.4.26. Observe that if we use a set of Latin squares to form an orthogonal array OA(r, n), then the first parameter is at least 3. On the contrary, in our definition of an orthogonal array this parameter can be equal to 2. The generalized Latin square graph that arises for r = 2 is Line(K n,n ). We show that a generalized Latin square graph is strongly regular. Theorem 3.4.27. For n ≥ 2, a generalized Latin square graph determined by OA(r, n) is strongly regular with parameters (n2 , (n − 1)r, n − 2 + (r − 1)(r − 2), r(r − 1)) . Proof. Let u = (u1 , u2 , . . . , u r )T and v = (v1 , v2 , . . . , v r )T be the vertices identified with two arbitrary columns of OA(r, n). The vertex u is adjacent to n − 1 vertices that have the same element in the ith position for 1 ≤ i ≤ r, and thus the degree of u is (n − 1)r, which gives the second parameter. Assume that u and v are adjacent and let the equality u i = v i holds for some fixed i ∈ {1, 2, . . . , n}. A vertex w = (w1 , w2 , . . . , w r )T is adjacent to both u and v if either w i = u i or w j = u j and w k = v k for some distinct indices j, k ∈ {1, 2, . . . , r}\{i}. Since we have n − 2 possibilities for the first choice and (r − 1)(r − 2) for the second, the third parameter follows. Assume that u and v are non-adjacent. A vertex w is their common neighbour if w i = u i and w j = v j for some distinct i, j ∈ {1, 2, . . . , r}. We may choose such i and j in r(r − 1) ways, which gives the last parameter. Partial geometries For positive integers s, t and k, a partial geometry is a finite linear space satisfying the following conditions: (i) (ii) (iii) (iv)

Any two points lie on at most one line; Each line contains exactly s + 1 points; Each point lies on exactly t + 1 lines; Given a point P not on line l, there are exactly k lines through P and also through some point lying on l.

3.4 Strongly regular graphs | 83

Fig. 3.9. Generalized quadrangle GQ(2, 4).

A partial geometry graph PG(s, t, k) is a graph whose vertices are identified with the points of a partial geometry, and two vertices are adjacent precisely when the corresponding points are collinear (this graph is also called the point graph or the collinearity graph of the corresponding partial geometry). Observe that PG(s, t, k) does not necessary stand for a unique graph. From definition, each point is collinear with exactly s(t + 1) other points, which means that PG(s, t, k) is regular. Fixing a point and counting the number of points that lie on lines passing through the fixed one, we get that there are exactly (s+1)(st+k) k points and (t+1)(st+k) lines. Moreover, following the proof of the previous theorem, we k get the next result. Theorem 3.4.28. A partial geometry graph PG(s, t, k) is strongly regular with parameters (

(s + 1)(st + k) , s(t + 1), s − 1 + t(k − 1), (t + 1)k) . k

Setting k = 1, we get a special case of partial geometry called a generalized quadrangle. We use GQ(s, t) to denote a corresponding generalized quadrangle graph.

84 | 3 Particular types of regular graph If either s = 1 or t = 1, the corresponding generalized quadrangle is called trivial. Thus, the smallest non-trivial generalized quadrangle is obtained for (s, t) = (2, 2). There is a unique corresponding graph GQ(2, 2) (the complement of Line(K6 )). Here is a corollary. Corollary 3.4.29. A generalized quadrangle graph GQ(s, t) is strongly regular with parameters ((s + 1)(st + 1), s(t + 1), s − 1, t + 1). Considering the (multiplicities of) eigenvalues which can be derived from the parameter set, one may conclude that if a generalized quadrangle graph of order (2, t) exists, then there must hold t ∈ {1, 2, 4}. Furthermore, there is a unique graph for each possibility for t. These graphs are Line(K3,3 ), Line(K6 ) and a unique strongly regular graph with parameters (27, 10, 1, 5) illustrated in Figure 3.9. Its complement is known as the Schläfli graph. Exercise 3.4.30. Show that the spectrum of GQ(2, 4) is [10, 120 , (−5)6 ] and that its second subconstituent is the Clebsch graph (see Figure 3.8).

3.4.5 Spectral distances We consider spectral distances defined in Section 2.5 (in particular, see the expression (2.6)) in the context of strongly regular graphs. Since the spectrum of a strongly regular graph and that of its complement are determined by the parameters (n, r, a, b), the following result is straightforward. Theorem 3.4.31 ([187]). For a primitive strongly regular graph G with parameters (n, r, a, b), σ(G, G)

=

|n − 2r − 1| + (n − 1) |a + 1 − b| 2 + (1 − |a − b + 1|) |2r + (n − 1)(a − b)| . √(a − b)2 + 4(r − b)

We give more graphs attaining equality in (2.8) (page 32). Theorem 3.4.32 (cf. [187]). For a primitive strongly regular graph G with parameters (n, r, a, a + 1), σ L (G, G) = σ(G, G) + (n − 2)(n − 2r − 1).

(3.21)

Proof. We may assume that 2r ≤ n − 1 (otherwise, we consider G instead of G). Using the expressions (3.17) and Theorem 2.3.3, we deduce that the L-spectrum of G and that of G are given by [(r − q)l , (r − p)k , 0] and [(n − r + p)k , (n − r + q)l , 0] respectively,

3.4 Strongly regular graphs | 85

where p = 12 (√4(r − a − 1) + 1 − 1) and q = 12 (−√4(r − a − 1) + 1 − 1), while k and l stand for the multiplicities. According to our assumption, we have k ≥ l. If k = l, then G and G share the same set of parameters, and thus the equality (3.21) holds trivially (since we have σ L (G, G) = σ(G, G) = 0). If k > l, using the above expressions for p and q, we make the following algebraic computation:

σ L (G, G)

=

l

k

∑ |r − q − n + r − p| + ∑ |r − p − n + r − p| i=1

i=l+1

n−1

+ ∑ |r − p − n + r − q| i=k+1

=

2l(n − 2r − 1) + (k − l)(n − 2r − 1 + √4(r − a − 1) + 1)

=

n(n − 2r − 1).

Applying the previous theorem, we obtain σ(G, G) = 2(n − 2r − 1), which gives the assertion. In our paper [188], we conjectured that the spectral distance between two graphs with n vertices does not exceed the maximal graph energy (the graph energy is equal to the sum of the absolute values of the eigenvalues) in the set of all graphs with n vertices. Although this conjecture is supported by computational results reported on page 29, in general case it is not true, and the counterexamples are found by Jovanović [187] in a set of generalized quadrangles. Namely, it is well known that the energy of any graph with n vertices does not exceed 2n (1 + √n) (see [202]). On the contrary, the spectral distance between the generalized quadrangle graph GQ(2, 4) illustrated in Figure 3.9 and its complement is equal to 84, while since we are dealing with graphs having 27 vertices, the above upper bound is close to 83.65. The situation is similar in the case of GQ(3, 9) which is a strongly regular graph with parameters (112, 30, 2, 10). Since the spectrum of this graph is [30, 290 , (−10)21 ], we find σ(GQ(3, 9), GQ(3, 9)) = 690 > 648.65 ≈

n (1 + √n) . 2

At the end, recall from [202] that the above upper bound for the graph energy is attained exactly for strongly regular graphs with parameters (n,

n + √n n + √n n + √n , , ), 2 4 4

and so an extremal case is encountered in the set of (strongly) regular graphs.

86 | 3 Particular types of regular graph

3.5 Kneser graphs The vertices of a Kneser graph K(n, k) are identified with the subsets of size k of a set of size n. Two vertices are adjacent ifn the corresponding subsets have trivial inter( )(n−k) section. Therefore, it has (nk) vertices, k 2 k edges and it is (n−k k )-regular. A Kneser graph K(n, k) is connected if n > 2k. For n < 2k it consists entirely of isolated vertices, while for n = 2k it is a disjoint union of K2 ’s. The Kneser graph K(n, 1) is the complete graph K n . Kneser graphs with small parameter n can easily be identified. For example, K(5, 2) is the frequently mentioned Petersen graph, while K(6, 2) is the recently mentioned generalized quadrangle graph GQ(2, 2) ≅ Line(K6 ). The definition of a symmetric graph is given on page 79. Theorem 3.5.1. Every Kneser graph is symmetric. Proof. Let S, T ⊂ {1, 2, . . . , n}, with |S| = |T| = k, be a pair of vertices of the Kneser graph K(n, k). If Perm stands for a permutation of {1, 2, . . . , n}, then the set SPerm = {Perm(i) : i ∈ S} has k elements, and hence represents a vertex of K(n, k). If S and T are adjacent then S ∩ T = 0, which yields SPerm ∩ T Perm = 0. Thus, Perm is an automorphism of K(n, k). Let, without loss of generality, S = {1, 2, . . . , k} and T = {1󸀠 , 2󸀠 , . . . , k󸀠 }, where 󸀠 1 , 2󸀠 , . . . , n󸀠 is a rearrangement of 1, 2, . . . , n. Then the permutation i 󳨃→ i󸀠 maps S to T, and therefore K(n, k) is vertex-transitive. Let S, T and S󸀠 , T 󸀠 be pairs of adjacent vertices. If S = {1, 2, . . . , k}, T = {k + 1, k + 2, . . . , 2k} and S󸀠 = {1󸀠 , 2󸀠 , . . . , k󸀠 }, T 󸀠 = {(k + 1)󸀠 , (k + 2)󸀠 , . . . , (2k)󸀠 }, where we rearrange the elements in a similar way, then the same permutation maps the edge ST to S󸀠 T 󸀠 . This assures the edge-transitivity, and we are done. Godsil and Royle [145, Theorem 9.4.3] used a specified vertex partitioning and a corresponding quotient matrix to prove that the spectrum of the Kneser graph K(n, k) consists of the numbers (−1)i (

1, n−k−i ), with multiplicity { n n k−i ( i ) − (i−1 ),

for i = 0, for i > 0,

where 0 ≤ i ≤ k. Another proof can be found in [219]. The reader may consider the subsets of size two to solve the following exercise. Exercise 3.5.2. Show that the Kneser graph K(n, 2) (where n ≥ 5) is distance-regular with intersection array ((

n−2 n−3 ), 2(n − 4); 1, ( )) , 2 2

and consequently strongly regular.

3.5 Kneser graphs | 87

Fig. 3.10. Desargues graph and its cospectral mate.

Consider now the Kneser graph K(8, 3). We compute d(S, T) = d(S, R) = 2, where S = {1, 2, 3}, T = {1, 7, 8} and R = {1, 2, 7}. Now, the number of neighbours of T that are at distance 1 from S is 1 (it is {4, 5, 6}), while the number neighbours of R that are at the same distance from S is 4 (they are {4, 5, 6}, {4, 5, 8}, {4, 6, 8} and {5, 6, 8}). Therefore, K(8, 3) is not distance-regular. By virtue of Theorem 3.5.1, this is another example of a vertex-transitive, but not distance-regular graph (compare the paragraph starting at page 61). Other structural properties of Kneser graphs can be found in [145]. For example, k−1 if K(n, k) is connected then its diameter is ⌈ n−2k ⌉ + 1 (see also [315]), the indepenn−1 dence number is ( k−1) (this is known as the Erdős–Ko–Rado theorem) and the chromatic number is n − 2k + 2. Finally, a bipartite Kneser graph BK(n, k) is a bipartite graph whose vertices in different parts are identified with the subsets of size k and those of size n − k of a set of size n, respectively. Two vertices from different parts are adjacent if and only if one of the corresponding sets is a subset of another. Next, BK(n, k) has 2(nk) vertices and is regular of degree (n−k k ). Moreover, BK(n, k) is the bipartite double of K(n, k), that is the tensor product of K(n, k) and K2 . According to Theorem 3.1.4, its spectrum is determined by the spectrum of K(n, k). Example 3.5.3. The bipartite Kneser graph BK(5, 2) is the bipartite double of the Petersen graph K(5, 2). It is known as the Desargues graph. Although the Petersen graph is determined by its spectrum, the Desargues graph is not (see Figure 3.10). Its spectrum is [3, 24 , 15 , (−1)5 , (−2)4 , −3].

88 | 3 Particular types of regular graph

3.6 Regular graphs of line systems A line system refers to a set of 1-dimensional subspaces of ℝk . So, its elements are lines passing through the origin of ℝk . The distance between a pair of lines is equal to the (smaller) angle between them. Every line l i may be represented by a unit vector yi (or −yi ) spanning it, and so the finite line system {l1 , l2 , . . . , l n } is uniquely determined by a set of corresponding unit vectors {y1 , y2 , . . . , yn }. If the angle between distinct lines l i and l j is ϕ ij , then it holds yi T yj = ± cos ϕ ij . Given a set {y1 , y2 , . . . , yn }, let M denote the corresponding Gram matrix, that is the matrix of all possible inner products yi ⋅ yj (1 ≤ i, j ≤ n). In other words, if N = (y1 | y2 | ⋅ ⋅ ⋅ | yn ), then M = N T N. Clearly, M is symmetric positive semidefinite. Conversely, if M is an n × n positive semidefinite matrix of rank k, then it is not difficult to see that M can be expressed as M = N T N, for some k × n matrix N (in this case, its columns does not need to be unit).

3.6.1 Equidistant lines If the distance between lines taken pairwise is constant, the lines are said to be equidistant (in many sources, equiangular). Determining the maximal number of equidistant lines in Euclidean space ℝk is a difficult task. The reader may recognize it as a part of a more general problem concerning determining the maximal number of elements of a simplex (i.e., a set of points that are at equal distance) in a metric space. Example 3.6.1. The 6 lines determined by pairs of antipodal vertices of an icosahedron in ℝ3 are equidistant. The cosine of the angle between any two of them is √15 . Exercise 3.6.2. Prove that there are no 7 equidistant lines in ℝ3 nor ℝ4 . To avoid trivial cases, we assume that the common distance ϕ is less than 2π . By setting σ = cos ϕ, we get that the corresponding Gram matrix has the form M = I + σS,

(3.22)

where S is the Seidel matrix of some graph, with the (i, j)th entry equal to 1 if the angle between yi and yj is acute, and −1 otherwise. Clearly, replacing yi with −yi does not affect the line system, but changes the corresponding graph. In fact, graphs obtained in this way are equivalent with respect to the Seidel switching (introduced on page 26). If the vectors y1 , y2 , . . . , yn are linearly dependent, the matrix M is singular, and thus − 1σ is the least eigenvalue of S. If n > k, then its multiplicity satisfies m− 1σ ≥ n − k. Conversely, if − 1σ is the least eigenvalue of an n × n Seidel matrix S, then the matrix 1σ I + S is symmetric positive semidefinite of rank n − m− 1σ , and so it can be represented as 1σ I + S = N T N. Therefore, 1σ I + S is the Gram matrix of n vectors in ℝ

n−m− 1 σ

.

3.6 Regular graphs of line systems | 89

1 and the cosine of the angle between any All these vectors have common length √σ pair of distinct is ±σ. So, in this context there is a bijective correspondence between the Seidel switching classes of graphs with n vertices and the sets of equidistant lines represented by linearly dependent vectors (cf. [213]). According to the above discussion, determining the minimal integer k such that there exist n equidistant lines in ℝk can be considered by determining a graph with n vertices such that the multiplicity of the least eigenvalue of the corresponding Seidel matrix is maximal.

Relative bound Assume in further that n > k and let σ1 , σ2 , . . . , σ n denote the eigenvalues of the Seidel matrix S from (3.22). We have σ k+1 = σ k+2 = ⋅ ⋅ ⋅ = σ n = − 1σ . Considering the spectral moments of S arising from tr(S) = 0 and tr(S2 ) = n(n − 1), we compute k

0 ≤ k ∑ (σ i − i=1

n(n − k) (n − k) 2 , ) = n(n − 1)k − σk σ2

and thus the inequality n − k ≤ σ2 (n − 1)k

(3.23)

holds. This bound is known as the relative bound (for the number of equidistant lines in a k-dimensional Euclidean space). The bound is ‘relative’ because the expression includes σ (the cosine of the common angle). The relative bound can be rewritten in the form of an upper bound for n whenever σ2 < 1k . Equality in (3.23) is attained if and only if σ1 = σ2 = ⋅ ⋅ ⋅ = σ k . In other words, if and only if S has 2 distinct eigenvalues: n−k 1 σk and − σ (with multiplicity k and n − k, respectively). In some literature (say, [89]) equidistant lines for which equality in (3.23) is attained are called extremal. This case is of particular interest and by the next theorem, it is closely related to specified strongly regular graphs. In what follows we denote distinct eigenvalues of S by σ̇ 1 =

1 n−k and σ̇ 2 = − . σk σ

(3.24)

Theorem 3.6.3 (cf. [89, Theorem 6.6.4]). Equality in (3.23) holds if and only if the corresponding graph is equivalent (with respect to the Seidel switching) to K1 ∪ G, where G is a strongly regular graph with parameters (n − 1, 2b, a, b).

(3.25)

Proof of the first implication. If H is the corresponding graph, then we may apply a particular Seidel switching to obtain a graph containing an isolated vertex, that is the graph of the form K1 ∪ G (if necessary, see the details on page 76). If so, then the Seidel matrix of K1 ∪ G has the form

90 | 3 Particular types of regular graph

S=(

0 j

jT ), S(G)

where S(G) is the Seidel matrix of G. If σ̇ 1 > σ̇ 2 are distinct eigenvalues of S, since it holds S2 = (

n−1 S(G)j

jT S(G) ) J + S(G)2

and x2 − (σ̇ 1 + σ̇ 2 )x + σ̇ 1 σ̇ 2 represents the minimal polynomial of S, we arrive at the identities S(G)j = (σ̇ 1 + σ̇ 2 )j and J + S(G)2 − (σ̇ 1 + σ̇ 2 )S(G) + σ̇ 1 σ̇ 2 I = O.

(3.26)

Since S(G) = J − I − 2A (where A = A(G)), the former identity implies that G is regular ̇ ̇ of degree − (σ1 +1)(2 σ2 +1) , while the latter yields 4A2 + 2(σ̇ 1 + σ̇ 2 + 2)A + (σ̇ 1 + 1)(σ̇ 2 + 1)I = (σ̇ 1 − σ̇ 2 − σ̇ 1 σ̇ 2 )J,

(3.27)

and so G is strongly regular (cf. (3.14)). By computing the last parameter in terms of the degree obtained in the above, we get the assertion. Straightforward proofs of the second implication can be found in [89, 145]. The eigenvalues of G other than its degree are easily deduced from (3.27). They are − σ̇ 22+1 and − σ̇ 12+1 or by (3.24), 1−σ n − k + σk and − . 2σ 2σk

(3.28)

Remark 3.6.4. Given a system of n equidistant lines, there are 2n possibilities for unit vectors along these lines (obtained by changing their signs). As we pointed out, they determine a collection of graphs with similar Seidel matrices. Moreover, if one of these graphs is given, then any other is obtained by some Seidel switching. In some literature this collection of graphs is called a two-graph. Two-graphs that consists of graphs with 2 distinct S-eigenvalues are called regular two-graphs. As we have seen, equality in (3.23) yields regularity of the corresponding two-graph. Observe that the relative bound can be attained for different values of σ, giving various possibilities for n and k. In addition, there may exist a maximal set of equidistant lines in ℝk for which this bound is not attained. However, in the case of equality in (3.23) we have k σ̇ 1 + (n − k)σ̇ 2 = 0, which can be rewritten as (2k − n)σ̇ 2 = k(σ̇ 1 + σ̇ 2 ). This equality holds either if n = 2k and the S-eigenvalues differ only in sign or n ≠ 2k and both S-eigenvalues are integral.

3.6 Regular graphs of line systems | 91

In the former case, using (3.26) we get SS T = S2 = (n − 1)I, which means that S is an n × n conference matrix. We recall from page 76 that such matrices exist for all sizes n ≤ 46, n ≡ 2 (mod 4), except for n = 22 and n = 34. In addition, for each feasible n there exists a strongly regular graph mentioned in Theorem 3.6.3 providing the existence of the corresponding set of equidistant lines. In the latter case, both integral S-eigenvalues must be odd (since the eigenvalues of the corresponding strongly regular graph must also be integral). Using the spectral moments of S, one may impose additional conditions for them, and subsequently obtain feasible values for n. Some of them, for which the existence of a corresponding strongly regular graph is confirmed, are 16, 26, 28, 36, 126, 176 and 276. Determining the maximal number of equidistant lines in an Euclidean space consists on further examinations of putative Seidel matrices and their eigenvalues. Here is an example. Example 3.6.5. Setting σ = 31 in the relative bound (3.23), we deduce that equality is attained, say in the case (n, k) = (28, 7). To determine the set of 28 lines at distance arccos 31 , we need to find a regular graph with 28 vertices whose Seidel matrix has 2 distinct eigenvalues such that −3 is one of them. By Theorem 3.6.3, its existence is equivalent to the existence of a strongly regular graph with parameters (3.25). Now, the parameters of the generalized quadrangle graph GQ(2, 4) from Exercise 3.4.30 have the required form. Observe that its eigenvalues distinct from the degree match the values of (3.28). Following the given part of the proof of Theorem 3.6.3, we reconstruct from the Seidel matrix of our generalized quadrangle graph the Seidel matrix of a graph we are looking for. It appears that this graph is Line(K8 ) (with S-spectrum [97 , (−3)21 ]). We next reconstruct from the identity 1 I + S(Line(K8 )) = N T N 3 that N can be taken to be the 8×28 matrix whose columns are all possible vectors of the form √124 y, where y contains two 3’s and six −1’s. Since all these vectors are orthogonal to all-1 vector, they can be embedded into ℝ7 , giving a system of 28 equidistant lines in the 7-dimensional Euclidean space. In a recent paper [150], the authors gave current bounds for the maximal number of equidistant lines in ℝk where k ≤ 41. Here we repeat the part of the corresponding table: k Maximal number

2 3

3-4 6

5 10

6 16

7-13 28

14 28-29

15 36

16 40-41

92 | 3 Particular types of regular graph For k ∉ {14, 16} and (n, k) ∈ {(14, 28), (16, 40)}, the existence of the given number of equidistant lines is confirmed by reasonings similar to these of the previous example (see [209] and for k = 15, see also the forthcoming Remark 4.4.1). For k = 14 and k = 16, the authors of [150] emphasized a possibility of the existence of 29 and 41 equidistant lines, respectively. At this moment, their existence is not confirmed nor disproved. Note that the relative bound would not be attained for these systems. In the same reference the reader can find a slight generalization of this bound. Absolute bound The following proof for the upper bound on the number of equidistant lines appeared in the work of Seidel [272]. Theorem 3.6.6 ([272], cf. [89, Theorem 6.6.3]). If there are n equidistant lines in ℝk , then n ≤ (k+1 2 ). Proof. If ϕ is the common distance between the lines and y1 , y2 , . . . , yn are unit vectors along these lines, then we define the functions f i (1 ≤ i ≤ n) on the unit sphere by f i (x) = (yi T x)2 − cos2 ϕ (so, here we have x = (x1 , x2 , . . . , x k )T and ∑ki=1 x2i = 1). Since f i (yj ) = δ ij sin2 ϕ, our functions are independent. In addition, all of them lie in the space of the functions having the form ∑ki=1 a i x2i + ∑i 1 in an r-regular graph, then r = 2k and a graph under consideration is strongly regular with parameters ((λ2 + 3λ − 1)2 , λ2 (λ + 3), 1, λ(λ + 1)) . We will meet star complements once again in Section 4.1.

3.8 Partially balanced incomplete block designs Suppose that we have a finite set of some elements and wish to arrange these elements into a collection of subsets (called blocks) by following fixed rules reflecting a sort of balance between blocks. Any such arrangement results in something that we call a block design. Recall that we already met some of them, say when we defined Johnson, Hamming and Grassmann graphs in Subsection 3.3.2 or when we defined combinatorial structures considered in Subsection 3.4.4. Here is the formal definition. Given a finite set P, let B be a finite collection (i.e., a multiset) of subsets of equal size of P. A pair (P, B) is referred to as a block design. The elements of P and B are called the points and the blocks, respectively. If we denote P = {p1 , p2 , . . . , p p } and B = {B1 , B2 , . . . , B b }, then the p × b incidence matrix N = (n ij ) of (P, B) is defined by n ij = 1 if p i ∈ B j , and n ij = 0 otherwise. The number of occurrences of a point in different blocks is called a replication of that point. When p = b, a design is said to be symmetric. The dual of a design (P, B) with the incidence matrix N is the design (P, B)∗ with the incidence matrix N T . The complement of the same design is the design with the incidence matrix J − N. There is a natural way to associate a bipartite graph with a block design. Its vertices are indexed by the points and the blocks, and two vertices are adjacent if and only if one is indexed by a point and other by a block containing that point. This graph is called the incidence graph³ of a given block design. According to the introduced notation, its adjacency matrix is given by (

0 NT

N ) 0

(this is the same form as in (1.8), page 13).

3 In general case, the incidence graph can be disconnected, but since the parts of a block design that correspond to the components of its disconnected incidence graph can be considered as separate designs, we will always assume (without noting!) that incidence graphs are connected.

106 | 3 Particular types of regular graph By specifying the rules for arranging points into blocks, we may obtain different types of block design. Here we consider one of them. A partially balanced incomplete block design with h (h ≥ 1) associate classes (for short, an h-class PBIBD) is an arrangement of p points into b blocks of size k, such that: (a) Each of p points occurs in exactly r blocks, and no point appears more than once in a block. (b) There exists a relationship of association between every pair of points, satisfying the following conditions: – Any two points are either first, second, ... or hth associates, and any pair of points which are sth associates occur together in exactly l s blocks (1 ≤ s ≤ h). – Each point has n s sth associates. – For any pair of points which are sth associates, the number of points that are simultaneously ith associates of the first and jth associates of the second point is q sij , and this number is independent of the pair of points we started with. Furthermore, it holds q sij = q sji (i ≠ j; 1 ≤ i, j, s ≤ h). We usually say that a PBIBD has a constant replication r. Property (b) is the ‘partial balance’ property. It is usually assumed that 2 ≤ k < p holds. The first inequality eliminates a trivial case, while the second preserves that every block is a proper subset of P. In other words, every block is ‘incomplete’ with respect to P. The parameters mentioned in the above definition satisfy the following conditions (all of them follow directly): pr = bk, ∑hs=1 n s ∑hs=1 n s l s ∑hj=1 n i q ijs

= p − 1, = r(k − 1),

if i ≠ s, { ni , = { n s − 1, otherwise, { j = n j q si = n s q sij (1 ≤ i, j, s ≤ h).

(3.33)

q sij

Example 3.8.1. Arrange the 6 points enumerated 1, 2, . . . , 6 into the 3 blocks identified with the columns of the matrix (

1 4

2 5

3 ). 6

This blocking gives rise to a 3-class PBIBD in which each point has 2 first associates (given in the same row), a single second associate (given in the same column) and 2 third associates (given in the remainder of the matrix). So, (n1 , n2 , n3 ) = (2, 1, 2) holds. The reader may compute the parameters l s , q sij (1 ≤ i, j, s ≤ 3) and check the

3.8 Partially balanced incomplete block designs |

107

conditions of (b). The equalities of (3.33) are verified by considering the remaining parameters (p, r, b, k) = (6, 1, 3, 2). Probably, the most extensively studied problem in the entire theory of block designs is that on whether an arrangement of points (into blocks) that satisfies given constraints is possible or not. Continuing with certain particular PBIBDs, we will see that in many cases their existence is conditioned by the existence of a related regular graph. We will also meet the role of the spectrum of a regular graph in the study of the corresponding design.

3.8.1 Balanced incomplete block designs An 1-class PBIBD is called a balanced incomplete block design (for short, a BIBD). There, each pair of points is simultaneously contained in l blocks (we write l instead of l1 ). Using the first three equalities of (3.33), we express the parameters b and r in terms of p, k and l by b=

(p − 1)l p(p − 1)l and r = . k(k − 1) k−1

(3.34)

Therefore, the integers (p, k, l)

(3.35)

are usually taken as the basic parameters of a BIBD.⁴ When these parameters are given, we will always bear in mind the above equalities determining b and r. A (necessary) condition for the existence of a BIBD with parameters (p, k, l) is that both fractions of (3.34) are positive integers. In addition, there may exist different arrangements of points into blocks with the same parameters, and so in general case a BIBD is not determined by its parameters. Every BIBD satisfies the Fisher inequality p ≤ b [306, Theorem 1.33]. (i.e., the number of points does not exceed the number of blocks). This inequality can be interpreted as another existence condition. If a BIBD is symmetric, then by definition it holds p = b, and therefore the corresponding incidence graph is bipartite regular. Observe that in this case we also have k = r. It follows directly from definition that the incidence matrix of a BIBD with parameters (p, k, l) satisfies the identities NN T = lJ + (r − l)I and JN = kJ.

(3.36)

4 The reader may observe that usual symbols for the first and the third parameter in many sources are v and λ, respectively. In this book, these symbols are reserved for vertices and eigenvalues.

108 | 3 Particular types of regular graph If A is the adjacency matrix of the corresponding incidence graph G, then its square can be expressed as A2 = (

NN T 0

0 ). NT N

(3.37)

Since NN T and N T N share the same non-zero eigenvalues, using the above expression for NN T and the fact that the matrices J and I can be simultaneously diagonalized, we get that A2 has the spectrum [(rk)2 , (r − l)2(p−1) , 0b−p ], and so the spectrum of G has the following form [±√rk, (±√r − l)

p−1

, 0b−p ] .

(3.38)

Hence, the spectrum of G is determined by the parameters of a BIBD, and consequently the incidence graphs of different BIBDs with the same parameters are mutually cospectral. We have seen that the incidence matrix of a BIBD satisfies the identities (3.36). Interesting, the converse is also true. Lemma 3.8.2. If a p × b (0, 1)-matrix N satisfies both identities of (3.36), then it is the incidence matrix of a BIBD with parameters (p, k, l). Proof. Consider the set of p points arranged into b blocks such that the point p i (1 ≤ i ≤ p) is contained in the block B j (1 ≤ j ≤ b) if and only if the entry n ij of N is 1. We need to show that this arrangement is a BIBD. Indeed, the second identity of (3.36) implies that each block contains k points, while the first identity implies that each pair of points occurs in exactly l blocks and each point occurs in exactly r blocks, giving the assertion. According to the last lemma, a natural way to decide whether a given connected bipartite graph is the incidence graph of a BIBD is to write its adjacency matrix in the form (1.8), and then check whether the corresponding submatrix N satisfies the identities (3.36) (for some parameters r, k and l). Note that, if a graph is semiregular (in particular, regular), then the second identity of (3.36) is satisfied automatically. We will meet such considerations, say in the forthcoming Theorems 3.8.5 and 3.8.8. Incidence graphs of symmetric BIBDs Since in the case of symmetric BIBDs we have k = r, we may write r for the second parameter in (3.35). Here is a connection with distance-regularity. Theorem 3.8.3 (cf. [48, Theorem 1.6.1]). A graph G is the incidence graph of a symmetric BIBD with parameters (p, r, l) if and only if it is bipartite distance-regular with intersection array (r, r − 1, r − l; 1, l, r).

3.8 Partially balanced incomplete block designs |

109

Proof. The incidence graph of a symmetric BIBD with parameters (p, r, l) is bipartite regular of degree r. Since each pair of points is contained in l blocks, the first implication follows by the definition of a distance-regular graph (page 55). Conversely, if G is bipartite distance-regular with intersection array (r, r − 1, r − l; 1, l, r), then by indexing the vertices in one part by the points and those in the other by the blocks, we get that each block contains r vertices, while two vertices occur in exactly l blocks, giving the last two parameters. The first parameter is uniquely determined by these two (it is the half of the number of vertices of G). Once the intersection array of a bipartite distance-regular incidence graph has been determined, we may consider its bipartite complement. Theorem 3.8.4 ([7]). Let G be a bipartite distance-regular graph with intersection array (r, r − 1, r − l; 1, l, r). If l < r − 1, then the bipartite complement G is distance-regular + 1. with intersection array (p − r, p − r − 1, r − l; 1, p − 2r + l, p − r), where p = r(r−1) l Proof. Computing the parameter p that appers in the previous theorem (see the end of the proof), we get that G is the incidence graph of a symmetric BIBD with parameters (p, r, l), where p is specified in this theorem. Now, if N is the incidence matrix of a corresponding design, then the adjacency matrix of G has the form (1.8), and then G has the adjacency matrix A(G) = (

0 J − NT

J−N ), 0

where J − N is the incidence matrix of the complementary design. Since the complement of a symmetric BIBD with parameters (p, r, l) is itself a symmetric BIBD with parameters (p, p−r, p−2r+l) (which is a clear fact), we have that G is either disconnected or distance-regular with intersection array (p − r, p − r − 1, r − l; 1, p − 2r + l, p − r). By the result of Exercise 2.2.5 we have that G is disconnected if and only if λ1 (G) + λ2 (G) = p, which is equivalent to r + √p − r = r(r−1) + 1, that is (r − 1)(r − l) = l√r − l. l Using the fact that r − l must be a square, after a direct computation, we easily arrive at a contradiction to l < r − 1. By excluding the assumption l < r − 1, we can derive a simple example of bipartite distance-regular graphs whose bipartite complements are not connected. Namely, by setting l = r − 1, we may consider a complete bipartite graph with a perfect matching removed (denoted K −r+1,r+1 ) in the role of G, and the disconnected graph (r + 1)K2 in the role of G. Setting p = b in (3.38), we get that the incidence graph of a symmetric BIBD has 4 distinct eigenvalues. The converse is considered in the next theorem that in fact gives an interesting characterization. Theorem 3.8.5 ([87, pp. 166–167]). Every connected bipartite regular graph with 4 distinct eigenvalues is the incidence graph of a symmetric BIBD.

110 | 3 Particular types of regular graph Proof. Without loss of generality we may assume that a connected bipartite regular graph G has the eigenvalues ±r with multiplicity 1 and ±√r − l with multiplicity p − 1 each. The adjacency matrix A satisfies the identity (3.37) and, since G is regular, NN T and N T N are of the same size and share the common spectrum consisting of r2 with multiplicity 1 and r − l with multiplicity p − 1. Since NN T and J commute, we deduce that the identity (NN T − lJ)x = (r − l)x holds for all their (common) eigenvectors. Therefore, it holds NN T = lJ + (r − l)I. In addition, the r-regularity of G yields JN = rJ, and thus by Lemma 3.8.2, N is the incidence matrix of a BIBD (with parameters (p, r, l)), while G is its incidence graph. Finite planes A symmetric BIBD with parameters (n2 + n + 1, n + 1, 1), where n ≥ 2, is called a finite projective plane. It is well known that a finite projective plane exists whenever n is a prime power [78, 306]. Observe that finite projective planes form an infinite family of symmetric BIBDs with a constant value for the third parameter (in this case, it is equal to 1). A classical definition reads as follows. A finite projective plane is a linear space satisfying the following conditions: (i) (ii) (iii)

Any two points lie on at most one line; Any two lines have exactly one intersection point; In any finite projective plane there exist four points, of which no three lie on the same line.

The last condition is added to avoid certain trivial cases. It follows directly from the last definition that the number of lines that contain the same point is equal to the number of points on each line. In particular, a finite projective plane with n + 1 (n ≥ 2) points on each line has n + 1 lines containing each point. By fixing any point, we get n + 1 lines containing it and that each of these lines contains n points other than fixed one. Counting the total number of points, we deduce that a finite projective plane with n +1 points on each line has exactly n2 + n + 1 points and the same number of lines. Now, considering each line as a block containing points that lie on it, we conclude that our and classical definition of a finite projective plane are equivalent. Example 3.8.6. The finite projective plane with parameters (7, 3, 1) is known as the Fano plane. It has 7 points (enumerated 1, 2, . . . , 7) arranged into the following 7 blocks: {1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 4, 6}, {2, 5, 7}, {3, 4, 7}, {3, 5, 6}.

3.8 Partially balanced incomplete block designs | 111

Exercise 3.8.7. A (non-symmetric) BIBD with parameters (n2 , n, 1), with n ≥ 2, is called a finite affine plane. Show that such a BIBD exists for any prime power n. Recall from literature the classical definition of a finite affine plane and show that these definitions are equivalent. Hadamard matrices An n×n Hadamard matrix H has entries ±1 and satisfies HH T = nI. Note that H T is also a Hadamard matrix, because H T = nH −1 giving H T H = nI. A simple combinatorial existence condition states that, except for n ∈ {1, 2}, there must be n ≡ 0 (mod 4). There is an obvious similarity between the definitions of a Hadamard matrix and a conference matrix (page 76). Moreover, if C is a symmetric 2n × 2n conference matrix (where we know that 2n − 1 is expressible as a sum of two squares), then by direct computation we get that (

C+I C−I

C−I ) −C − I

(3.39)

is a Hadamard matrix. Therefore, if there exists a conference matrix C, then there also exists a Hadamard matrix of order 4n formed as in (3.39). The most important open question in the theory of Hadamard matrices is that of their existence. It was conjectured by Hadamard in 1893 (cf. [306, p. 74]) that a Hadamard matrix of order 4n exists for every positive integer n. In the above we presented a method for constructing Hadamard matrices that is based on existing conference matrices. Several other methods can be found in literature (say in [62, 306]). According to the recent results of Kharaghani and Tayfeh-Rezaie [193], at present moment the smallest multiple of 4 for which no Hadamard matrix is known is 668. Here is a connection with symmetric BIBDs. Theorem 3.8.8. For n ≥ 2, there exists a Hadamard matrix of order 4n if and only if there exists a symmetric BIBD with parameters (4n − 1, 2n − 1, n − 1). Proof. A Hadamard matrix of order 4n is similar to (also Hadamard) matrix H whose last row and last column consist only of 1’s. If so, then the identity HH T = 4nI implies that any other row of H contains exactly 2n 1’s, and since H T is Hadamard, the same conclusion holds for any other column of H. Consider now the matrix N obtained by deleting the last row and the last column of H and replacing every −1 with 0. Since the number of 1’s in each row and each column of N is 2n − 1, we get NN T = (n − 1)J + nI and JN = (2n − 1)J.

(3.40)

By Lemma 3.8.2, N is the incidence matrix of a BIBD with parameters (4n − 1, 2n − 1, n − 1). Since N is a square matrix, this BIBD is symmetric.

112 | 3 Particular types of regular graph Since the incidence matrix of a symmetric BIBD with parameters (4n − 1, 2n − 1, n − 1) satisfies the identities of (3.40), going backward through the previous part of the proof (i.e., by replacing all 0’s with −1’s and extending the matrix with the all-1 row and the all-1 column), we get the corresponding Hadamard matrix. A symmetric BIBD with parameters (4n − 1, 2n − 1, n − 1) is called a Hadamard design. A Hadamard matrix in which every row and every column have equal number of 1’s is called a regular Hadamard matrix. We note in passing that regular Hadamard matrices have maximal number of 1’s in the set of all Hadamard matrices of fixed order [306]. Example 3.8.9. The 4 × 4 (−1, 1)-matrix with −1’s precisely on the main diagonal is a regular Hadamard matrix. If H is an n×n regular Hadamard matrix with s 1’s in every row and every column, then following the proof of Theorem 3.8.8, we get that the matrix N obtained by replacing every −1 with 0 satisfies the identities NN T = (s − 4n ) J + 4n I and JN = sJ, and thus N is the incidence matrix of a symmetric BIBD with parameters (n, s, s − 4n ). Considering the existence condition given in the form of an equality for r in (3.34), we arrive at s(s − 1) = (n − 1) (s − 4n ). Solving this quadratic equation in s, we get s = n±√n 2 . Hence, n must be a square. Since we already have n ≡ 0 (mod 4) whenever n ∉ {1, 2}, we may assume that n = 4t2 , for a non-zero integer t. If so, then our symmetric BIBD has the parameters (4t2 , t(2t + 1), t(t + 1)). Conversely, the incidence matrix of a symmetric BIBD with these parameters yields the existence of a regular Hadamard matrix of order 4t2 . Thus, we have the following conclusion. Theorem 3.8.10. For n ≥ 3, there exists a regular Hadamard matrix of order n if and only if n = 4t2 (where t ∈ ℤ\{0}) and there exists a symmetric BIBD with parameters (4t2 , t(2t + 1), t(t + 1)). A symmetric BIBD mentioned in the last theorem is called a Menon design. Note that by replacing t with −t, we obtain a pair of Menon designs whose incidence graphs are mutual bipartite complements. It is also easy to confirm that, for t ≥ 2, such designs are dual. Setting t = −1 and t = 1, we get the self-dual Menon designs whose incidence − graphs are 4K2 and K4,4 , respectively. According to [82], for t positive, a Menon design exists whenever t and 2t − 1 are prime powers and t ≡ 3 (mod 4). Example 3.8.11. Given the subset Z = {(0, 1), (0, 2), (0, 3), (1, 0), (2, 0), (3, 0)} of the additive group P = ℤ4 × ℤ4 , let us for each p ∈ P consider the following block B p = {p + z : z ∈ Z} (where the operation is mod 4). Clearly, |P| = 16, the collection B of blocks B p also counts 16 elements and the size of each block is 6. In addition, it is straightforward to verify that each element of P occurs in exactly 6 blocks, while each pair of elements occurs in exactly 2 blocks. Altogether, (P, B) is a Menon design with parameters (16, 6, 2).

3.8 Partially balanced incomplete block designs | 113

So far we have seen that Hadamard matrices are related to symmetric BIBDs, and consequently they are related to the incidence graphs of these designs. It is worth mentioning in this context that Hadamard matrices produce another family of so-called n-Hadamard graphs. Such a graph has 4n vertices, while its edges are defined in terms of an n × n Hadamard matrix in the following way. If the vertices are denoted u+i , u−i , v+i and v−i (1 ≤ i ≤ n), insert an edge between u±i and v±j whenever the corresponding element of the Hadamard matrix is 1, and also between u∓i and v±j whenever the corresponding element of the Hadamard matrix is −1. The existence of an n-Hadamard graph is equivalent to the existence of the corresponding Hadamard matrix. Example 3.8.12. The 1-Hadamard graph is just 2K2 , the 2-Hadamard graph is the octagon C8 . For n = 4, using the Hadamard matrix of Example 3.8.9, we arrive at the 4-dimensional cube. Exercise 3.8.13. Prove that an n-Hadamard graph (n ≥ 2) is distance regular with intersection array (n, n − 1,

n n , 1; 1, , n − 1, n) . 2 2

Remark 3.8.14. BIBDs can be generalized to so-called t-designs. In a t-design with parameters t-(p, k, l), the parameters p and k play the same role as before, while now each subset of t points is contained in exactly l blocks. Clearly, for BIBDs we had t = 2. More details on t-designs including Steiner systems and the Witt design can be found in [62, 145, 306].

3.8.2 Partially balanced incomplete block designs with 2 associate classes Identifying the points of a 2-class PBIBD with the vertices of a graph H in which two vertices are adjacent precisely when the corresponding points are the first associates, we get that H is strongly regular with parameters (p, n1 , q111 , q211 ). Therefore, every 2-class PBIBD is associated with the two graphs: its incidence graph (previously denoted G) and the mentioned strongly regular graph H. We say that a 2-class PBIBD is based on H. Moreover, the incidence matrix N satisfies the identity NN T = rI + l1 A + l2 (J − I − A),

(3.41)

where A is the adjacency matrix of H. A simple analysis shows that a matrix N with constant row sums and constant column sums that satisfies the above identity must be the incidence matrix of some 2-class PBIBD. Since all the matrices on the right hand side of (3.41) commute, they can be simultaneously diagonalized, and so if θ1 > θ2 > θ3 are the distinct eigenvalues of H, then the distinct eigenvalues of NN T are given by

114 | 3 Particular types of regular graph

r + (l1 − l2 )θ1 + (p − 1)l2 , r + (l1 − l2 )θ2 − l2 and r + (l1 − l2 )θ3 − l2 . The distinct eigenvalues of H and their multiplicities are determined by its parameters, and so are the eigenvalues of NN T . Consequently, the spectrum of the incidence graph G is determined by the spectrum of H. Exercise 3.8.15. Prove that a partial geometry (defined on page 82) is a 2-class PBIBD (based on a strongly regular graph, say H), where clearly the blocks are identified with the lines of a linear space. Determine the parameters of H in terms of the parameters of the corresponding incidence graph (previously denoted PG(s, t, k)). When the converse is true, that is under which conditions a 2-class PBIBD based on a strongly regular graph whose parameters are determined in the previous part of the exercise is a partial geometry graph? We mention the two special types of 2-class PBIBD. The points of a group divisible design can be divided into m groups of size n, such that two points are the first associates if they belong to the same group (and then they occur together in l1 blocks), while two points are the second associates if they belong to different groups (and then they occur together in l2 blocks). Clearly, we have p = mn, n1 = n − 1 and (n − 1)l1 + n(m − 1)l2 = r(k − 1).

(3.42)

Example 3.8.16 (cf. [162]). Let the 6 points be divided into 3 groups containing 2 of them each. This dividing produces various group divisible designs. For example, we get the two such designs by choosing blocks determined by the columns of the (incidence) matrices 1 1 (1 ( (1 0 (0

0 0 1 1 1 1

1 1 1 1 ) ( 1 0) and ( (0 0) 1 1 1) (0

1 1 1 0 0 1

1 1 0 1 1 0

1 1 0 1 0 1

1 0 1 1 1 0

1 0 1 1 0 1

0 1 1 1 1 0

0 1 1 1 0 1

1 0 1 0 1 1

1 0 0 1 1 1

0 1 1 0 1 1

0 1 0) ). 1) 1 1)

The parameters (p, b, r, k, l1 , l2 , n1 , n2 ) of these designs are respectively given by (6, 3, 2, 4, 2, 1, 1, 4) and (6, 12, 8, 4, 4, 5, 1, 4). Both designs are based on the strongly regular graph 3K2 . If a group divisible design is symmetric, then we usually deal with the five parameters arranged as (m, n, r, l1 , l2 )

(3.43)

3.8 Partially balanced incomplete block designs | 115

(observe that they are not independent). The 2(2n) points of a triangular design are arranged into an n × n symmetric matrix-like scheme with the main diagonal neglected, in such a way that the first associates occur in the same row and column, while the second associates do not. Again, we have n p = 2( ) and n1 = 2(n − 2). 2 More properties of these designs and considerations related to their existence can be found in [40, 41, 43, 44, 80]. We now continue with symmetric designs. Recall that the incidence graph of a symmetric 2-class PBIBD is regular. Exercise 3.8.17. Compute the spectrum of both regular graphs (G and H) associated with a symmetric group divisible design and a symmetric triangular design in terms of parameters (of a design). Distance-regularity As we have seen, the incidence graph G of a symmetric 2-class PBIBD (which is not a BIBD) has 3 distinct non-negative eigenvalues, that is 5 or 6 distinct eigenvalues in total depending on whether NN T has eigenvalue 0 or not. The next natural question is on whether a bipartite regular graph with 3 distinct non-negative eigenvalues is an incidence graph of some symmetric 2-class PBIBD or not. Koledin and Stanić [196] gave the affirmative answer in a particular case. Recall from page 61 the definition of a halved graph. Theorem 3.8.18. Every bipartite distance-regular graph G with 3 distinct non-negative eigenvalues is the incidence graph of a symmetric 2-class PBIBD based on a (strongly regular) halved graph of G. Proof. Let (1.8) be the adjacency matrix of G. Since G is distance-regular and has 3 distinct non-negative eigenvalues, its diameter is 5 or 4 and two vertices from the same part are either at distance 4 (in which case they do not have common neighbours) or 2. Note that the number of common neighbours of two vertices at distance 2 in G is a constant, say l. Using these facts, we easily conclude that N has constant row and column sums both equal to r, and that NN T = rI + lA, where A is the adjacency matrix of a halved graph of G. Now, the result follows by the identity (3.41) and the subsequent text. Now, when the converse of this assertion is also true, that is under which conditions we may say that the incidence graph of a symmetric 2-class PBIBD is distance-regular? If G is the incidence graph of such a design and has diameter 4 and 5 distinct eigenvalues, then the equality (3.41) is obviously satisfied and one of the parameters l1 or l2 is

116 | 3 Particular types of regular graph equal to 0. Therefore, two vertices at distance 2 in G have a constant number of common neighbours, and then according to Theorem 3.3.18, G must be distance-regular. In this way we have proved the following result. Theorem 3.8.19. Every incidence graph of a symmetric 2-class PBIBD with diameter 4 and 5 distinct eigenvalues is distance-regular. By Koledin [194], neither of the two assumptions (on diameter and on the number of distinct eigenvalues) can be omitted. For example, the Möbius–Kantor graph (the second graph of Figure 4.9 on page 162) has diameter 4 and 6 distinct eigenvalues. This graph is also the incidence graph of a symmetric 2-class PBIBD, but by Theorem 3.3.5, it is not distance-regular. Similarly, the bipartite double of the complete multipartite graph mK n , for m ≥ 3 and n ≥ 2, is bipartite (m − 1)n-regular graph with diameter 3 and 5 distinct eigenvalues (its spectrum is [±(m − 1)n, (±n)m−1 , 02m(n−1) ]). It is easily seen that such a graph is also the incidence graph of a symmetric group divisible design with parameters (m, n, (m − 1)n, (m − 2)n, (m − 1)n), but by Theorem 3.3.5, it is not distance-regular. A symmetric group divisible design with parameters (nl2 , n, nl2 , 0, l2 ) is called a symmetric transversal design. In such a design, every two points occur together either in exactly one group or in exactly l2 blocks, but not both. The spectrum of the corresponding incidence graph has the form [±nl2 , (±√nl2 )

n(n−1)l2

, 02(nl2 −1) ] .

Thus, the incidence graph has 5 distinct eigenvalues and its diameter is at most 4. Also, NN T = nl2 I + l2 A holds, where A is the adjacency matrix of nl2 K n . Since this graph is non-complete for n ≥ 2, the diameter of the incidence graph is exactly 4. Hence, by virtue of the last theorem we have the following result. Corollary 3.8.20. The incidence graph of a symmetric transversal design with parameters (nl2 , n, nl2 , 0, l2 ), where n ≥ 2, is distance-regular. Example 3.8.21 ([194]). It is known that the symmetric transversal design with l2 = 1 (and consequently m = n) can be constructed from a finite projective plane with parameters (n2 + n + 1, n + 1, 1) by removing one of its lines and the points on it (thus obtaining a finite affine plane with parameters (n2 , n, 1)), and then removing a parallel class of lines from the obtained affine plane (for the explicit construction, see [43]). This means that every bipartite distance-regular graph G with spec2 trum [±r, (±√r − 1)r −r+1 ] (that is, the incidence graph of a finite projective plane with n = r − 1) contains an induced distance-regular subgraph with spectrum

3.8 Partially balanced incomplete block designs | 117 (r−1)(r−2) , 02(r−2) ] (that is, the incidence graph of a transversal de[±(r − 1), √r − 1 sign with parameters (r − 1, r − 1, r, 0, 1)).

Quadrangle-free regular graphs with bounded second largest eigenvalue We start with the two exercises. Exercise 3.8.22 ([196]). Prove that every connected quadrangle-free bipartite regular graph with 3 distinct non-negative eigenvalues is the incidence graph of some symmetric 2-class PBIBD. (Hint: If r is the degree of our incidence graph, then show that NN T = rI + A, where A is the adjacency matrix of a strongly regular graph of degree r(r − 1).) Exercise 3.8.23. Prove that every connected quadrangle-free bipartite r-regular graph with 5 distinct eigenvalues is distance-regular. (Hint: Show that the diameter of a graph under consideration must be 4, and then use the previous exercise and Theorem 3.8.19.) We continue by the following characterization. Theorem 3.8.24 ([194]). Let G be a connected quadrangle-free bipartite r-regular graph, with r ≥ 2, whose second largest eigenvalue satisfies λ22 ≤ r. Then G is either the incidence graph of a finite projective plane with parameters (r2 − r + 1, r, 1) or the incidence graph of a symmetric 2-class PBIBD based on a complete multipartite r(r − 1)-regular graph. Proof. Let (1.8) be the adjacency matrix of G. Since the girth is at least 6, two vertices in the same part can have at most one common neighbour, and therefore we have NN T = rI + A, where A is the adjacency matrix of an r(r − 1)-regular graph H. We next have λ22 (G) = λ2 (NN T ) = r + λ2 (H), and by the assumption λ22 (G) ≤ r, which implies λ2 (H) ≤ 0. We observe that H must be connected (otherwise, it holds λ2 (H) = r(r − 1) ≤ 0, implying r ≤ 1). Therefore, the two possibilities for H can occur. It is either the complete graph K r(r−1)+1 or a complete multipartite regular graph. For the former possibility, we have NN T = (r − 1)I + J, and then G has 4 distinct eigenvalues: ±r and ±√r − 1. Hence, it is the incidence graph of a symmetric BIBD with parameters (r(r − 1) + 1, r, 1) (see Theorem 3.8.5 and cf. (3.38)). In fact, this BIBD is a finite projective plane (recall definition from page 110). For the latter possibility, the identity (3.41) is obviously satisfied, and consequently G is the incidence graph of a symmetric 2-class PBIBD based on a complete multipartite r(r − 1)-regular graph. Corollary 3.8.25 ([194]). Let G be a connected quadrangle-free bipartite r-regular graph, with r ≥ 2, whose second largest eigenvalue satisfies λ22 < r. Then G is the incidence graph of a finite projective plane with parameters (r2 − r + 1, r, 1).

118 | 3 Particular types of regular graph Proof. Setting λ22 < r in the previous proof, we get λ2 (H) < 0. This implies that H must be the complete graph K r(r−1)+1 , and the result follows. Concerning Theorem 2.6.6 in conjunction with the last results, we immediately get another corollary. Corollary 3.8.26. Every bipartite graph that attains the bound of Theorem 2.6.6 is the incidence graph of a finite projective plane with parameters (r2 − r + 1, r, 1). We may also consider the bound of Theorem 3.8.24. Corollary 3.8.27 ([194]). Every graph that attains the equality λ22 = r in Theorem 3.8.24 is the incidence graph of a symmetric group divisible design whose parameters satisfy l1 = 0, l2 = 1 and (m − 1)n = r(r − 1). Proof. Let the notation be the same as in the proof of the corresponding theorem. It follows that the former possibility for H cannot occur under assumption λ22 = r. Thus, H ≅ mK n , for some integers m, n ≥ 2. The mn × mn incidence matrix N of this symmetric 2-class PBIBD has row and column sums equal to r (so, every point occurs in r blocks of size r). Considering the structure of H, we deduce that the points of this design can clearly be divided into m groups of size n in such a way that two points from the same group (corresponding to the vertices belonging to the same part of H) are simultaneously contained in 0 blocks (l1 = 0), and two points from different groups (corresponding to the vertices belonging to different parts of H) are simultaneously contained in a single block (l2 = 1). The last equality of this corollary follows from the third identity of (3.42). Example 3.8.28. Examples of graphs that correspond to the last corollary are the incidence graphs of symmetric group divisible designs with m = q2 + q + 1, n = q(q − 1), r = q2 , where q is a prime power. A method for constructing these designs is described by Sprott [289]. The spectrum of the incidence graph has the form 1

[(±q2 ) , (±q)q(q+1) , (±√q)

q4 −q2 −2q−1

].

Exercise 3.8.29. Let G be a connected quadrangle-free bipartite r-regular graph (r ≥ 2) with 2n vertices whose second largest eigenvalue satisfies λ22 ≤ r. Prove that n ≤ r2 holds. (Hint: Consider both possibilities for H in the proof of Theorem 3.8.24.) Dual and complement Without loss of generality we may assume that the inequality l1 > l2 holds for all symmetric 2-class PBIBDs. Namely, if these parameters are equal, then the corresponding design reduces to a BIBD (compare the identity (3.41) with the first of (3.36)). In addi-

3.8 Partially balanced incomplete block designs | 119

tion, a symmetric 2-class PBIBD with l1 < l2 is just a symmetric 2-class PBIBD obtained by interchanging the first and the second associates. Under this assumption, we write SPBIBDl1 ,l2 (H) for a symmetric 2-class PBIBD based on H. In general case, only what we can say about the dual of some (symmetric) 2-class PBIBD is that it is some block design. Hoffman [167] proved that, under certain assumptions, the dual of a symmetric 2-class (in general case, h-class) PBIBD is again a symmetric 2-class PBIBD with the same parameters l1 and l2 . In the same paper, the author constructed the SPBIBD2,0 (4K2 ) whose dual is not a PBIBD. The corresponding incidence graph is the Hoffman graph illustrated in Figure 3.7. In particular, by Theorem 3.8.18, every bipartite distance-regular graph with diameter 4 is the incidence graph of a SPBIBDl1 ,0 (H1 ), where H1 is one halved graph of G. Observe that, in this case, the dual SPBIBDl1 ,0 (H1 )∗ coincides with the SPBIBDl1 ,0 (H2 ), where H2 is the other halved graph of G. Notice also that l1 is in fact the intersection number c2 (of G). By the following result, complementing of a (not necessary symmetric) 2-class PBIBD results in a 2-class PBIBD. Theorem 3.8.30 (cf. [196]). The complement of a 2-class PBIBD is also a 2-class PBIBD based on the same strongly regular graph. Proof. The p×b incidence matrix of a design under consideration satisfies the identity NN T = rI p + l1 A + l2 (J p − I p − A) (the sizes of the corresponding matrices are given). For the complementary design with the incidence matrix J p×b − N, we compute (J p×b − N)(J p×b − N)T

=

(J p×b − N)(J b×p − N T )

=

J p×b J b×p − J p×b N T − NJ b×p + NN T

=

(n − r)I p + (n − 2r + l1 )A + (n − 2r + l2 )(J p − I p − A).

In other words, the incidence matrix of the complement satisfies the identity (3.41) (for appropriately chosen parameters), and we are done. A detailed analysis including a number of examples of bipartite regular graphs with 3 distinct non-negative eigenvalues is given Subsection 4.4.3.

4 Determinations of regular graphs In this chapter we consider regular graphs that satisfy specified spectral constraints. In some cases the exposition is based on results presented in Chapters 2 and 3. Section 4.1 is an extension of Sections 3.6 and 3.7. There we consider exceptional regular graphs and those with second largest eigenvalue at most 1. Bipartite regular graph whose second largest eigenvalue does not exceed 2 are determined in Section 4.2. We will see that there is a simple infinite family and the additional 123 connected graphs satisfying this property. Regular graphs whose spectrum consists entirely of integers are considered in Section 4.3. We show that cubic graphs with this property are completely determined and present a partial result concerning 4-regular graphs. We also use the spectrum of signless Laplacian to consider regular line graphs. We completely resolve the case of regular graphs whose degree is at most 4 and give certain partial results concerning those with larger degrees. Regular graphs with at most 6 distinct eigenvalues are investigated in Section 4.4. In that context, we present strongly regular graphs (i.e., those with 3 distinct eigenvalues) of larger order, regular graphs with 4 distinct eigenvalues and bipartite regular graphs with 5 or 6 distinct eigenvalues. Cospectrality of regular graphs are considered Section 4.5. In particular, we present a number of situations in which a regular graph under consideration has a unique spectrum, and also a number of the opposite situations. Constructions of both sorts of graphs are presented.

4.1 Exceptional regular graphs Using Theorems 2.2.9 and 3.6.13 (see also the paragraph below the latter one), we immediately arrive at the following result attributed to Cameron et al. [61]. Theorem 4.1.1 ([61]). The spectrum of a connected regular graph is bounded by −2 if and only if this graph is either (i) a line graph, (ii) a cocktail party graph or (iii) an exceptional graph (with representation in the system E8 ). The problem of determining all exceptional regular graphs was posed by Hoffman in 1968 [171], and all these graphs were determined by Bussemaker et al. in 1976 [53, 54]. They used a mixture of theoretical reasoning on the basis of root systems along with a computer search. Another way to determine the same graphs is to apply the star complement technique to appropriately chosen star complements. Clearly, an exceptional regular graph has an exceptional star complement for −2. DOI: 10.1515/9783110351347-004

4.1 Exceptional regular graphs | 121

Since, by Theorem 3.6.17, the least eigenvalue of every exceptional regular graph is equal to −2, to determine all of them one should start with exceptional graphs represented in E8 whose least eigenvalue is greater than −2. By Theorem 3.6.16, these graphs have between 6 and 8 vertices, and therefore can be determined by computer search. This is done in [116] (see also [94]); 20 out of them have 6 vertices, 110 have 7 vertices and 443 have 8 vertices (giving 573 graphs in total). Applying the star complement technique described in Subsection 3.7.2 with a restriction to regular extensions, exactly 187 exceptional regular graphs are obtained. They can be found in [94, pp. 213–225]. The following result established in [53, 54] can also be deduced by examining the mentioned 187 graphs. Theorem 4.1.2. An exceptional r-regular graph G with n vertices satisfies one of the following conditions: (i) (ii) (iii)

n = 2(r + 2) ≤ 28; ≤ 27 and G is an induced subgraph of the Schläfli graph; n = 3(r+2) 2 n = 4(r+2) ≤ 16 and G is an induced subgraph of the Clebsch graph. 3

In particular, we have r ≤ 16 and this bound is attained for the Schläfli graph.

4.1.1 Regular graphs with λ2 ≤ 1 The following facts are easily deduced. A connected regular graph of degree 2 has second largest eigenvalue not exceeding 1 if and only if it is the cycle C n (3 ≤ n ≤ 6). The second largest eigenvalue of every connected regular graph of degree n − 3 or larger does not exceed 1. In general case, by Theorem 2.2.1, a connected regular graph whose second largest eigenvalue does not exceed 1 is a complement of a (not necessary connected) regular graph whose components are graphs described in Theorem 4.1.1. A deeper analysis of connected r-regular graphs with λ2 ≤ 1, for 3 ≤ r ≤ 8, is given in [197, 292]. Here we mention the result from the latter reference treating the cases r = 3 and r = 4. Theorem 4.1.3 (Stanić [292]). A cubic graph with the property λ2 ≤ 1 is either K4 , C6 , 2C3 , the Petersen graph, the graph of Figure 1.3 or the first graph of Figure 4.1. A connected 4-regular graph with the property λ2 (G) ≤ 1 is either K5 , 3K2 , C7 , − C3 ∪ C4 , K4,4 , Line(K3,3 ), K5,5 or one of the remaining graphs of Figure 4.1.

122 | 4 Determinations of regular graphs

Fig. 4.1. Cubic and 4-regular graphs with λ2 ≤ 1 (for Theorem 4.1.3).

4.2 Reflexive bipartite regular graphs A graph whose second largest eigenvalue does not exceed 2 is called reflexive. These graphs occur in the theory of reflection groups (for more details, see [237, 238]). We reinterpret some results reported in [195, 198], where Koledin and Stanić determined all reflexive bipartite regular graphs (for short, RBR graphs). We may assume that graphs under consideration have 2n (n ≥ 1) vertices. Observe that degree of any of these graphs does not exceed n. Recalling from Theorem 2.2.4 the relation between the spectrum of a bipartite regular graph and its bipartite complement, we conclude that if a bipartite graph is reflexive, so is its bipartite complement. In particular, every bipartite graph of degree at most 2, along with its bipartite complement, is reflexive. In addition, if a RBR graph is disconnected, then its degree does not exceed 2. According to these observations, we may focus on search for connected reflexive bipartite regular graphs with 2n vertices and degree between 3 and 2n . Denote the set of these graphs by R. We first establish a better upper bound for degree of graphs in this set.

4.2 Reflexive bipartite regular graphs | 123

Theorem 4.2.1. The degree of any graph in R is at most 7. Proof. Assume to the contrary and let G be a graph in R whose degree r is greater than 7. By Theorem 2.6.2, its diameter is at most 3. In fact, the diameter must be equal to 3 because only connected bipartite regular graphs with diameter 2 are complete 2 −4 , bipartite (which do not belong to R). Using the inequality (2.11), we get 2r ≤ n ≤ rr−4 yielding r2 − 8r + 4 ≤ 0, which is impossible for r ≥ 8, and we are done. The next step is bounding the order of graphs in R. Here we give the final result in the form of the table: r= n≤

3 30

4 32

5 42

6 32

7 30

We demonstrate the proof for the second column. The proof for the first column can be found in our previous book [290], while all of them are collected in the original reference [195]. Lemma 4.2.2. Every RBR graph G of degree 4 has at most 32 vertices. Proof. Assume that 2n > 32 holds. Then G is an RBR graph of degree (n − 4) and, by Theorem 2.6.2, its diameter is 3. Let v be an arbitrary vertex of G and consider the following partition of the vertex set V of G: S1 = {v}, S2 = {u ∈ V : d(u, v) = 1}, S3 = {u ∈ V : d(u, v) = 2}, S4 = {u ∈ V : d(u, v) = 3}. We have |S2 | = n − 4, |S3 | = n − 1 and |S4 | = 4. Also, each vertex in S4 has exactly n − 4 neighbours in S3 , and then a simple counting shows that there are at least n − 13 vertices in S3 adjacent to all four vertices of S4 . If S3,1 is the set of such n − 13 vertices of S3 , we denote S3,2 = S3 \S3,1 , and then we have |S3,2 | = 12. The partition on S1 , S2 , S3,1 , S3,2 and S4 induces the quotient matrix: 0 1 (0 0 (0

n−4 0 n−8 n−7 0

0

0

n2 −21n+104 n−4

12(n−7) n−4

0 0 n − 13

0 0 9

0 0 4) . 3 0)

According to Theorem 1.3.8, we have λ2 (G) ≥ √ A contradiction.

7n − 64 > 2 (for n > 16). n−4

124 | 4 Determinations of regular graphs Using theoretical refinements and a computer search, we examine all the possibilities given in the above table. The final result reads as follows: (i) (ii)

(iii)

All bipartite regular graphs satisfying either r ≤ 2 or r ≥ n − 2 are reflexive. If an RBR graph is disconnected then its degree is at most 2. The set R consists of exactly 70 graphs. Precisely – 20 graphs of degree 3, – 30 graphs of degree 4, – 6 graphs of degree 5, – 9 graphs of degree 6 and – 5 graphs of degree 7. Exactly 53 bipartite complements of graphs in R do not belong to R, which means that there is an infinite family of RBR graphs described in (i) and additional 123 individual graphs.

The graphs of R are listed in Table 4.1. Following [195, 198], we use the LCF notation to represent all cubic graphs in this table, while for the other graphs we use the hexadecimal notation. Here are the descriptions of these notations. If G is a Hamiltonian cubic graph, then its vertices can be arranged in a cycle C which accounts for two edges per arbitrary vertex u. The third edge uv can be described by the length l of the path uv in C with a plus sign if we turn clockwise, and minus sign if we turn counterclockwise along C. If the pattern of the LCF notation repeats, it is indicated by a superscript in the notation. If the second half of the numbers of the LCF notation is the reverse of the first half but with all the signs changed, then it is replaced by a semicolon and a dash. For the hexadecimal notation we assume that the adjacency matrix of each graph obtained has the block form with all-0 blocks on the main diagonal like in (1.8), and then the binary integer obtained by concatenating the rows of the submatrix N is expressed as a hexadecimal integer. The reader may observe that, according to Theorem 3.8.5, the graphs represented in Table 4.1 having 4 distinct eigenvalues are the incidence graphs of symmetric BIBDs. Example 4.2.3. Consider in details the four reflexive bipartite 5-regular graphs with 24 vertices (G52 –G55 of Table 4.1). Let G stand for any of them. By forming a quotient matrix similar to that encountered in the proof of Lemma 4.2.2, we may conclude that two vertices of the same part of G must have either one or two common neighbours. If (1.8) is the adjacency matrix of G, then we have NN T = 5I +2A +(J − I − A), where A is the adjacency matrix of a 9-regular graph H with 12 vertices. Since λ2 (NN T ) = λ22 (G) and 3 < λ22 (G) ≤ 4, we get −1 < λ2 (A) ≤ 0. In other words, H has exactly one positive eigenvalue, which means that it must be a complete multipartite strongly regular graph. Obviously, H ≅ 4K3 , and consequently each of the graphs G52 –G55 is the incidence graph of a symmetric 2-class PBIBD based on 4K3 . Many other RBR graphs can be examined in a similar way.

4.2 Reflexive bipartite regular graphs | 125 Tab. 4.1. Graphs of R. Reprinted from [195] with permission from Elsevier. Graph G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20 G21 G22 G23 G24 G25 G26 G27 G28 G29 G30 G31 G32 G33 G34 G35 G36 G37 G38 G39 G40 G41 G42 G43

2n

Notation

12 12 12 14 14 14 16 16 16 18 18 18 20 20 20 20 20 20 24 30 16 16 16 16 16 16 16 16 18 18 18 18 18 18 20 20 20 20 24 24 24 24 24

[5, −5]6

[−5, −5, 3, −5, −5, −3; −] [−3, 3]6 [5, −5]7 [7, 7, −5, 3, 5, 7, −3]2 [5, −3, 5, 7, 5, −5, 5; −] [5, −5]8 [−5, 7, −5, 7, −5, 7, 3, 7; −] [7, −5, 3, −5; −]2 [5, 7, −7; −]3 [5, −5]9 [5, 7, −5, 7, 9, −5, 5, 7, −7; −] [−9, 5, −7, 7, −7, 7, −5, 7, −9, 7; −] [5, 9, −9, 5, −9, −5, −9, 5, −5, 9; −] [−5, 9, −9, 5, −5, 7, −9, 7, −5, 9; −] [5, 9, −5, 9, −9, −5, 5, 9, 5, 9; −] [5, −5, 9, −9]5 [5, −9, −7, 7, −7, −5, 5, −9, 5, 7; −] [5, −9, 7; −]4 [−13, −9, 7; −]5 F03A2DA3CC954B56 F01E99536CC52BA6 F08E552D66A3995A F02D1E6A93C559A6 F00F9C35A95366CA F02B1E6CC59359A6 F00FAA63C9559C36 F03C0F5AC3A56996 1E03A50D0EA9171663594 1E03A48D0F19270E99394 1E0662CA172517560A695 1E092E609C781B44D8A8B 1E019D15B0A51E42C9D92 1E02C8B93847A9495638A F00B1239544AA9211E251E309 F00B1239524969411E2A1E309 F007111F14A0CD82B10D4CA2A F030A170A925A19899855C063 F0024C43448A13829154180D2231868528E0 F0084903518393029860514C22A8864524E0 F0004D43448A138291541A0C2231868528E0 F008C423480B8311A82C150519246245820E F0014903518393029860584C22A8864524E0

Non-negative eigenvalues 3, 1.732 , 13 3, 2, 1.412 , 1, 02 3, 22 , 1, 04 3, 1.416 3, 1.932 , 1.412 , 0.522 3, 2, 1.414 , 02 3, 1.734 , 13 3, 2, 1.732 , 1.412 , 1, 02 3, 22 , 1.732 , 1, 04 3, 1.736 , 04 2 2 3, 1.97 , 1.73 , 1.292 , 0.682 3, 2, 1.734 , 12 , 02 3, 22 , 1.882 , 1.532 , 1, 0.352 3, 22 , 1.882 , 1.532 , 1, 0.352 3, 23 , 1.732 , 13 , 02 3, 23 , 1.732 , 13 , 02 3, 24 , 15 3, 24 , 15 3, 26 , 13 , 04 3, 29 , 010 4, 2, 1.852 , 1.412 , 0.772 4, 22 , 1.414 , 02 4, 22 , 1.732 , 12 , 02 4, 22 , 1.732 , 12 , 02 4, 23 , 1.412 , 04 4, 23 , 1.412 , 04 4, 24 , 06 4, 24 , 06 4, 22 , 1.882 , 1.532 , 0.352 4, 23 , 1.732 , 12 , 02 4, 23 , 1.732 , 12 , 02 4, 24 , 14 4, 24 , 14 4, 24 , 14 4, 25 , 14 4, 25 , 14 4, 25 , 14 4, 25 , 14 4, 25 , 1.416 4, 26 , 1.414 , 02 4, 26 , 1.414 , 02 4, 26 , 1.414 , 02 4, 27 , 1.412 , 04

126 | 4 Determinations of regular graphs

Table 4.1-continued. Graph

2n

Notation

G44 G45 G46

24 24 24

G47

26

G48

28

G49

30

G50

32

G51 G52 G53 G54 G55

22 24 24 24 24

G56

42

G57 (G58 ) G59 (G60 ) G61 (G62 )

24 24 24

G63

32

G64

32

G65

32

G66

30

G67

30

G68

30

G69

30

G70

30

F000C523480B8311A8AC050519246245820E F0032229814544AC841892540A6831413868 F00322294149446C8811C2C10AA831413864 1281CA03280CA03280CA0 3280CB03240CD03140C503 F0009424C408A411C05884130819 88A0606408E10582482C1 1E000501844641A02610851262024 2900B060C4340421512048248131 F000005C818812094025108684112212 09038062489025041C4006A028284340 189789389B88B88B8CB84B84B84BC4B F808A94A62CA19554C61313A943A1CC70265 F80C52A499158AA1CC3230F148766453821E F80C52A499158AA1CC1632B148766453821E F80C0EA158C39381CC1274B22A955166425A 1F0000802C1400836090A10B020410A22158 01000662831022094040B80821284844110825 0490604C1021C0061061210A8091111011901 FC065615DA2738E6693B18F44BA8CBD2CD13 FC0C5C88F2751E6969B13595E266A953A2DA FC0D1C15BA2D3876553B84E9D234B68CEA72 FC00303A0C6318D48A8A492C160D2B1 1A26446B0258691A18558C017534260C9 FC00302B17A0921C18C55152A1C82474 2B118926450DC0B10C9A4A6862868643 FC00300F1B48C22C907151A2256428B8 48C5451916940E23891684CAA3816252 1FC02A4CD8744DE03658B270D69C8 4EC4E98E2AA1D87590D4CD51E8CA 1FC02525AAB1263B32729532656C3 8B14CCCE91C65525C1FA383C348B 1FC0154D1A719878C47562399A6C5 4B85CAA4987D40E57272692734E8 1FC00AE932B8B0368C5CCD2B44D8 79314F29330E711D1D928D4B568D4 1FC014EA292B878657A26534C594D 629938D39433CE0CA6A34A8BF09A

Non-negative eigenvalues 4, 27 , 1.412 , 04 4, 28 , 06 4, 28 , 06 4, 1.7312 4, 27 , 1.416 4, 210 , 14 4, 212 , 06 5, 1.7310 5, 28 , 13 5, 28 , 13 5, 28 , 13 5, 28 , 13 5, 220 6, 29 , 04 6, 29 , 04 6, 29 , 04 6, 215 6, 215 6, 215 7, 214 7, 214 7, 214 7, 214 7, 214

4.3 Integral regular graphs A graph is called integral if its spectrum consists entirely of integers. Integral graphs have been extensively studied (see [19] for the results on up to 2004 and [281, 296, 304] for the recent developments). Certain advantages in using integral graphs as multipro-

4.3 Integral regular graphs | 127

cessor interconnection networks are discovered in [67] (see also [95, 140, 275]). In what follows we determine all connected integral cubic graphs and consider determining of connected integral 4-regular graphs as well as connected integral line graphs of certain bipartite semiregular graphs. For a non-bipartite graph G with eigenvalues λ1 , λ2 , . . . , λ n , its bipartite double (i.e., the tensor product G × K2 ) is a bipartite graph with 2n vertices and eigenvalues ±λ1 , ±λ2 , . . . , ±λ n (see page 50). Therefore, the essential part in determining integral graphs is the determining only those that are bipartite, since any non-bipartite integral graph can be extracted from the decompositions of bipartite ones in the form G × K2 . Note that a bipartite graph may or may not be a bipartite double, and if it is a bipartite double then the corresponding decomposition does not need to be unique. The following simple lemma is useful in resolving such situations. Lemma 4.3.1 ([270]). A bipartite graph with parts X and Y can be decomposed in the form G × K2 if and only if there exists a bijective map f : X 󳨀→ Y such that for all u, v ∈ X, u and f(u) are non-adjacent, and if u is adjacent to f(v) then v is adjacent to f(u).

4.3.1 Integral cubic graphs Using the final result of the previous section, we arrive at the following theorem. Theorem 4.3.2. There are exactly 13 connected integral cubic graphs. Exactly 8 of − them are bipartite: K3,3 , K4,4 , C4 ∪ C6 and the graphs G3 , G17 –G20 of Table 4.1. The 5 non-bipartite graphs are H1 ≅ K4 , H2 which is the Petersen graph and the graphs H3 –H5 illustrated in Figure 4.3. − Proof. The bipartite graphs K3,3 , K4,4 and C4 ∪ C6 belong to the infinite family of bipartite regular graphs described on page 124 (under item (i)). The remaining bipartite graphs are identified in Table 4.1 by direct check. Considering bipartite graphs, we get that K3,3 , C4 ∪ C6 , G18 and G20 are not bipartite doubles, while for the remaining graphs we obtain: − K4,4 ≅ H1 × K2 ,

G17 ≅ H2 × K2 ≅ H4 × K2

G3 ≅ H3 × K2 ,

G19 ≅ H5 × K2 ,

and the assertion follows. The last theorem appeared simultaneously in [52, 270]. The authors of the former paper combined theoretical reasoning with a computer search, while the author of the latter paper achieved the same result in a pure theoretical way. Here (and previously in [195]), we simply extracted the desired bipartite regular graphs from a (wider) family of reflexive bipartite regular graphs.

128 | 4 Determinations of regular graphs

3

19

20

Fig. 4.2. Integral bipartite cubic graphs (for Theorem 4.3.2).

H3

H4

Fig. 4.3. Integral non-bipartite cubic graphs (for Theorem 4.3.2).

H5

4.3 Integral regular graphs | 129

Note that the bipartite graphs G17 and G18 mentioned in Theorem 4.3.2 are already encountered in this book (they are the Desargues graph and its cospectral mate given in Figure 3.10). The remaining G i ’s are illustrated in Figure 4.2. The graphs G19 and G20 are known as the Nauru graph and the Tutte–Coxeter graph, respectively.

4.3.2 Integral 4-regular graphs Using a similar reasoning, we get our next result. Theorem 4.3.3. There are exactly 25 connected integral 4-regular graphs avoiding ±3 − in the spectrum. Exactly 17 of them are bipartite: K4,4 , K5,5 , 3C4 , 2C6 and the graphs G27 , G28 , G32 –G38 , G45 , G46 , G49 and G50 of Table 4.1. The 8 non-bipartite graphs are H6 ≅ K5 , H7 ≅ CP(3), H9 ≅ Line(2C3 ) and the graphs H8 , H10 –H13 illustrated in Figure 4.5. Proof. Bipartite graphs are determined as in the proof of Theorem 4.3.2. Considering these graphs, we get that K4,4 , 2C6 , G27 , G33 –G38 , G45 and G50 are not bipartite doubles, while for the remaining graphs we obtain: − K5,5 ≅ H6 × K2 ,

G32 ≅ H9 × K2 ≅ H10 × K2 ,

3C4 ≅ H7 × K2 ,

G46 ≅ H11 × K2 ≅ H12 × K2 ,

G28 ≅ H8 × K2 ,

G49 ≅ H13 × K2 ,

and the result follows. This result is obtained in a different way in [299] (see a correction in [304]). The bipartite graphs G27 and G28 mentioned in Theorem 4.3.3 are already encountered (they are the Hoffman graph illustrated in Figure 3.7 and the 4-dimensional cube illustrated in Figure 1.1, respectively). The remaining G i ’s are illustrated in Figure 4.4. Determination of all connected integral 4-regular graphs is an open problem. One way to determine some of them is to consider line graphs or NEPSes of known integral regular graphs. Using this reasoning, the reader may obtain a sequence of such graphs by solving the following exercise. Exercise 4.3.4. Use Theorem 2.2.10 and the results of Section 3.1 to prove that if G is a connected integral cubic graph, then the graphs Line(G), Line(G) × K2 and G◻K2 are connected integral and 4-regular. Show that the same conclusion holds for the NEPS with the basis {(1, 1), (0, 1)} of graphs G and K2 . A search for connected integral bipartite 4-regular graphs starts by determining feasible spectra. If such a graph has 2n vertices and the spectrum [4, 3m3 , 2m2 , 1m1 , 02m0 , (−1)m1 , (−2)m2 , (−3)m3 , −4] ,

130 | 4 Determinations of regular graphs

32

33

34

35

36

37

38

45

46

49

Fig. 4.4. Integral bipartite 4-regular graphs (for Theorem 4.3.3).

50

4.3 Integral regular graphs | 131

H8

H 11

H 10

H 13

H 12

Fig. 4.5. Integral non-bipartite 4-regular graphs (for Theorem 4.3.3).

by computing the 2kth spectral moments (0 ≤ k ≤ 3) and the numbers of closed walks traversing along 2k edges, we get the following system of Diophanitne equations: 1 2n 0 ∑λ 2 i=1 i 2n

1 ∑ λ2 2 i=1 i 2n

1 ∑ λ4 2 i=1 i 2n

1 ∑ λ6 2 i=1 i

=

3

1 + ∑ m i = n, i=0

=

3

16 + ∑ i2 m i = 4n, i=0

=

3

256 + ∑ i4 m i = 28n + q, i=0

=

3

4096 + ∑ i6 m i = 232n + 72q + 6h, i=0

where q and h stand for the numbers subgraphs isomorphic to quadrangles and hexagons, respectively. Using the Hoffman polynomial (see Theorem 2.1.6) and considering algebraic conjugates, we immediately get that the number of vertices 2n divides the sum ∑(4 − θ) that goes over all distinct eigenvalues different from 4 (the degree). This gives an additional feasibility condition. By considering all these conditions, Cvetković et al. [99] determined a list of feasible spectra containing the 1888 entries which can be found in [302]. In [300], Stevanović eliminated the 34 possible spectra. In further, using a combinatorial reasoning, Stevanović et al. [304] proved that the 8th spectral moment of a bipartite 4-regular

132 | 4 Determinations of regular graphs graph with 2n vertices is at least 2092n + 1012q + 144h, which gives an additional condition: 1 2n 8 ∑ λ = 65536 + 6561m1 + 256m2 + m3 ≥ 2092n + 1012q + 144h. 2 i=1 i This condition is not satisfied by the 1026 of the remaining spectra, which gives the list of the 828 feasible spectra reported in [304]. The largest putative graph has 560 vertices. Finally, the authors of the same paper established the existence of 31 integral bipartite 4-regular graphs (with ±3 in the spectrum) having at most 24 vertices. These bipartite graphs give additional 8 non-bipartite graphs.

4.3.3 Integral regular line graphs Almost all results of this subsection can be found in works of Simić and Stanić [280, 281] and Stanić [293, 294]. A graph is called Q-integral if its Q-spectrum (see Section 2.3) consists entirely of integers. Recall from page 20 that a line graph is regular if and only if its root graph is bipartite (r, s)-semiregular (where the equality r = s is allowed). In addition, by the formula (2.4), a regular line graph is integral if and only if its root graph is Q-integral. This observation allows us to consider the Q-integrality of bipartite (r, s)-semiregular graphs instead of integrality of their line graphs. This advantage becomes obvious if know that the order of a connected graph containing at least two cycles is smaller than the order of its line graph. Similarly to the previous section, the main tool for obtaining feasible integral Q-spectra are the Q-spectral moments defined by T k = ∑ni=1 κ ki , for k ≥ 0. In the context of the Q-matrix, the counterparts of walks are semi-edge walks. Each of them is a sequence of alternate vertices and edges v1 , e1 , v2 , e2 , . . . , e k−1 , v k such that, for 1 ≤ i ≤ k − 1, the vertices v i and v i+1 are (not necessary distinct) ends of the edge e i . This does not mean that we allow the existence of loops, but that at each step we traverse along a half of an edge and then either return along the same half or continue along the other half. Then the (i, j)-entry of the matrix Q k is equal to the number of semi-edge walks along k edges that start at vertex i and terminate at vertex j. In particular, if G has n vertices, m edges, t triangles and vertex degrees d1 , d2 , . . . , d n , then we have n

n

n

n

i=1

i=1

i=1

i=1

T0 = n, T1 = ∑ d i = 2m, T2 = 2m + ∑ d2i and T3 = 6t + 3 ∑ d2i + ∑ d3i .

4.3 Integral regular graphs | 133

If G is bipartite semiregular, then we can go further by computing Q-spectral moments of higher order. Observe first that for a bipartite (r, s)-semiregular graph with n1 vertices of degree r and n2 vertices of degree s, we have n1 =

m (r + s)m m , n2 = and n = n1 + n2 = . r s rs

(4.1)

Here is a lemma. Lemma 4.3.5 ([281, 293]). For a bipartite (r, s)-semiregular graph with m edges, q quadrangles and h hexagons, we have

T4

=

(r3 + s3 + 4(r2 + s2 ) + 2(r + s) + 4rs − 2)m + 8q,

T5

=

(r4 + 5(r3 + r2 − r) + s4 + 5(s3 + s2 − s) + 5rs(r + s + 2))m + 20(r + s)q,

T6

=

(r5 + s5 + 6(r4 + s4 ) + 9(r3 + s3 ) − 7(r2 + s2 ) − 6(r + s + rs) + 6rs(r2 + s2 +rs) + 21(r2 s + s2 r) + 4)m + 12(3(r2 + s2 ) + 2(r + s) + 4rs − 4)q + 12h.

Proof. Notice first that the matrices A, A2k , A2k+1 and the diagonal matrix of vertex degrees D have the following forms:

A=(

O NT

N ∗ ) , A2k = ( O O

O O ) , A2k+1 = ( ∗ ∗

∗ rI ), D = ( O O

O ). sI

Thus, we immediately get the identities DAD = rsA, A2k D = DA2k and tr(A2k+1 D l ) = 0,

(4.2)

where k, l ≥ 1. For k = 4, we compute T4 = tr(Q4 ) = tr(A + D)4 = tr(A4 + 4A3 D + 4A2 D2 + 4AD3 + D4 + 2ADAD),

(4.3)

where the latter follows since tr(BC) = tr(CB) holds for pair of (feasible) matrices B and C. Next, we have tr(A3 D) = tr(AD3 ) = 0 (see the third identity of (4.2)). By direct computation, we get tr(A2 D2 ) = ∑ni=1 d3i and tr(D4 ) = ∑ni=1 d4i . Since DAD = rsA, we next obtain tr(ADAD) = rs ∑ni=1 d i . By counting closed walks of length 4, we obtain tr(A4 ) = ∑ni=1 d2i + ∑ni=1 d i (d i − 1) + 8q, where the first term counts walks traversing the edges incident with the starting vertex, the second one those traversing paths of length 2 and the third one walks along a quadrangle. Substituting the values obtained in (4.3) and using (4.1), we get the first formula of the theorem.

134 | 4 Determinations of regular graphs In the remaining two situations, we get T5 = tr(Q5 ) = tr(A + D)5 = tr(5A4 D + 5A2 D3 + D5 + 5rsA2 D). and T6 = tr(Q6 ) = tr(A + D)6

=

tr(A6 + 6A4 D2 + 6A2 D4 + D6 + 6rsA4 + 6rsA2 D2 +3r2 s2 A2 + 3A2 DA2 D).

The proof is completed by computing all the terms on the right hand side of the above equalities. Certain Q-integral regular graphs Consider first the Q-integral bipartite semiregular graphs of small edge degree. Theorem 4.3.6 ([281, 293]). A connected Q-integral bipartite semiregular graph with edge degree at most 4 is either a complete graph, its subdivision, a complete bipartite graph, the quadrangle, the hexagon or one of the remaining integral bipartite cubic graphs described in Theorem 4.3.2. Proof. If the edge degree of a graph under consideration is at most 3, then its line graph is an integral cycle or an integral cubic graph, and this part is easily resolved by inspecting whether a cycle is integral or whether an integral cubic graph (all of them are given in Theorem 4.3.2) is a line graph or not. Assume that the edge degree is 4. If our graph is regular, then it is one of the 8 connected integral bipartite cubic graphs. Otherwise, it is bipartite (r, s)-semiregular with (r, s) = (1, 5) or (r, s) = (2, 4). In the former case, we get a unique Q-integral solution K1,5 . In the latter case, since the Q-spectrum of our putative graphs has the form [6, 5m5 , 4m4 , 3m3 , 2m2 , 1m1 , 0], using the Q-spectral moments T0 –T3 , we arrive at the following system: 5

2 + ∑ m i = T0 = i=1 5

3 m, 4

36 + ∑ i2 m i = T2 = 8m, i=1

6 + ∑5i=1 im i = T1 = 2m, 216 + ∑5i=1 i3 m i = T3 = 38m.

In addition, by the result of Exercise 2.3.6, we have m1 = m5 . Solving the system and bearing in mind that all the variables are non-negative integers, we get m ≤ 20. Since n = 43 m, we have m ∈ {4, 8, 12, 16, 20}. Computing the multiplicities m1 –m4 for all these possibilities, we get the feasible Q-spectra. The proof is concluded by inspecting the possible bipartite (2, 4)-semiregular candidates.

4.3 Integral regular graphs | 135

According to the previous theorem, complete bipartite graphs or subdivisions of complete graphs may be encountered as Q-integral. In what follows we will see that this is not a coincidence. Namely, every complete bipartite graph and every subdivision of a complete graph must be Q-integral. The former graphs are easily resolved by the following theorem. Theorem 4.3.7 ([296]). Every complete bipartite graph is Q-integral. Proof. The line graph of K n1 ,n2 is the lattice graph K n1 ◻K n2 (see page 50). The assertion follows by the fact that the Cartesian product of integral graphs is integral (if necessary, see Theorem 3.1.4). Concerning subdivisions of complete graphs, we prove a more general result reported in the forthcoming theorem. Observe first that Theorems 2.2.10 and 2.2.12 remain valid for regular loopless multigraphs. Now, if G is a connected s-regular loopless multigraph with n vertices, m edges (where we may assume that m ≥ n) and eigenvalues λ1 , λ2 , . . . , λ n , then its subdivision Sd(G) is a bipartite (2, s)-semiregular graph with m vertices of degree 2 and n vertices of degree s. Now, we express the characteristic polynomial of the signless Laplacian matrix of Sd(G) in the following way. Using Theorem 2.2.12, we first express the characteristic polynomial of Sd(G) in terms of the characteristic polynomial of G. In an intermediate step, using Theorem 2.2.10 we express the characteristic polynomial of Line(Sd(G)) in terms of the eigenvalues of G. Finally, using the relation (2.4) we arrive at the expression: n

QSd(G) (x) = (x − 2)m−n ∏ ((x − 2)(x − s) − (λ i + s)) . i=1

We are ready to prove the announced result. Theorem 4.3.8 (cf. [294]). Let G be a connected s-regular loopless multigraph. The graph Sd(G) is Q-integral if and only if G is a multigraph with two vertices or a complete graph. Proof. Using the above formula, we get that the Q-eigenvalues of Sd(G) are s + 2 ± √s2 + 4(λ n + 1) s + 2 ± √s2 + 4(λ1 + 1) s + 2 ± √s2 + 4(λ2 + 1) , , ..., 2 2 2 and 2m−n . Hence, Sd(G) is Q-integral if and only if all the numbers s2 +4(λ i +1), where 1 ≤ i ≤ n, are squares. Clearly, s2 + 4(λ + 1) is a square for λ = −1. The squares that are closest to s2 are (s ± 1)2 , but the numbers s2 and (s ± 1)2 do not have the same parity, and therefore (s ± 1)2 cannot be equal to s2 + 4(λ + 1) (since s2 and s2 + 4(λ + 1) have the same parity). The next closest squares are (s ± 2)2 . Setting s2 + 4(λ + 1) = (s ± 2)2 , we get λ = ±s. Observe that any other square of the form s2 + 4(λ + 1) is obtained for

136 | 4 Determinations of regular graphs λ ∉ [−s, s] in which case λ cannot be an eigenvalue of G (since the entire spectrum lies in [−s, s]). Therefore, the number s2 +4(λ +1) is a square if and only if either λ = −1 or λ = ±s. In other words, all the eigenvalues of G must belong to {−s, −1, s}. If both −s and s are the eigenvalues of G, then following the discussion upon Theorem 1.3.12 we conclude that G must be bipartite in which case its spectrum is symmetric with respect to the origin. The only possibility is that G has two vertices joined by s edges. If −s is not an eigenvalue of G, then G cannot have multiple edges and must be complete (since −1 is its eigenvalue). If a graph G from the previous theorem has two vertices, then its subdivision is a complete bipartite graph with two vertices in one part. Thus, the theorem can be rephrased as follows. Corollary 4.3.9. A connected bipartite (2, s)-semiregular graph is Q-integral if and only if it is complete bipartite or the subdivision of a complete graph. Q-integral bipartite (r, s)-semiregular graphs with edge degree 5 If these graphs are connected, then we have that r + s = 7. For r = 1 we get a unique solution K1,6 , while for r = 2 according to the previous corollary, we have exactly two solutions: K2,5 and Sd(K6 ). For the remaining case, we can offer a partial result starting with the following lemma. Lemma 4.3.10 ([281]). A connected Q-integral bipartite (3, 4)-semiregular graph has one of the Q-spectra listed in Table 4.2. Each row contains the number of vertices, the number of edges, the multiplicities of the Q-eigenvalues 7, 6, . . . , 0, the number of quadrangles (q) and the number of hexagons (h). Proof. Using the spectral moments T0 –T6 , we arrive at the following system: 6

2 + ∑ m i = T0 = i=1

6 7 m, 7 + ∑ im i = T1 = 2m, 12 i=1

6

6

i=1

i=1

49 + ∑ i2 m i = T2 = 9m, 343 + ∑ i3 m i = T3 = 46m, 6

2401 + ∑ i4 m i = T4 = 251m + 8q, i=1 6

16807 + ∑ i5 m i = T5 = 1422m + 140q, i=1 6

117649 + ∑ i6 m i = T6 = 8251m + 1596q + 12h, i=1

4.3 Integral regular graphs | 137 Tab. 4.2. The feasible Q-spectra and some additional data related to connected Q-integral bipartite (3, 4)-semiregular graphs. Reprinted from [281] with permission from Elsevier.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

n

m

7

6

5

4

3

2

1

0

q

h

7 14 14 21 21 21 28 28 35 35 35 42 42 49 49 70

12 24 24 36 36 36 48 48 60 60 60 72 72 84 84 120

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 2 1 4 3 2 4 5 5 6 7 7 8 9 10 15

0 0 3 0 3 6 6 3 9 6 3 9 6 9 6 9

2 3 1 4 2 0 1 3 0 2 4 1 3 2 4 5

3 5 3 7 5 3 5 7 5 7 9 7 9 9 11 15

0 0 3 0 3 6 6 3 9 6 3 9 6 9 6 9

0 2 1 4 3 2 4 5 5 6 7 7 8 9 10 15

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

18 18 12 18 12 6 6 12 0 6 12 0 6 0 6 0

24 12 40 0 28 56 44 16 60 32 4 48 20 36 8 0

where m i is the multiplicity of the eigenvalue i (1 ≤ i ≤ 6). Solving this system and using the result of Exercise 2.3.6 (giving m1 = m6 and m2 = m5 ), we get m ≤ 120. By using the Hoffman polynomial, we find that m divides 7! = 5040, which gives m ∈ {12, 24, 36, 48, 60, 72, 84, 120}. Computing the remaining variables for every possible m, we obtain the 16 feasible Q-spectra listed in Table 4.2. We consider the smallest graphs of Table 4.2 (by hand or by computer search). The results are summarized in the following theorem. Theorem 4.3.11 ([294]). There exists one graph with data as in the 1st row of Table 4.2 (K3,4 ), one graph with data as in the 3rd row (F1 of Figure 4.6), one graph with data as in the 4th row (F2 ), three graphs with data as in the 5th row (F3 –F5 ). There is no graph with data as in the 2nd nor 6th row. If the Q-spectrum of a bipartite (3, 4)-semiregular graph has the form [7, 6m1 , 5m2 , 4m4 , 3m3 , 2m2 , 1m1 , 0], then we easily deduce that the spectrum (of the adjacency matrix A) of the same graph has the form [±√12, (±√6)m1 , (±√2)m2 , 0m0 ] , (clearly, m0 does not appear as a multiplicity of any Q-eigenvalue). Following [281], in the next two theorems we consider the graphs G whose Q-spectrum is given in rows 9, 12, 14 and 16 (in other words, these are the rows with q = 0). Since q = 0 (and also since G is bipartite), the multigraph corresponding to A2 has two components which are both graphs with additional loops. The first

138 | 4 Determinations of regular graphs

F1

F2

F3

F4

F5

Fig. 4.6. Q-integral bipartite (3, 4)-semiregular graphs (for Theorem 4.3.11).

component has n1 vertices and 3 loops at each vertex. The second has n2 vertices and 4 loops at each vertex. So, the eigenvalues of the first (resp. second) component with loops excluded is contained in the following set (these eigenvalues are equal to the eigenvalues of A2 decreased by the number of loops at each vertex): {9, 3, −1, −3} (resp. {8, 2, −2, −4}). Now, the existence of G implies the existence of two regular graphs whose distinct eigenvalues belong to the above sets. Moreover, the above components cannot have

4.3 Integral regular graphs | 139

less than 4 distinct eigenvalues (the only critical case appears when the number of distinct eigenvalues is 3, but this can easily be resolved by inspecting small strongly regular graphs). Theorem 4.3.12 ([281]). There is no graph with data as in the 9th, 12th nor 14th row of Table 4.2. Proof. If we take the 9th row, then we get n = 35 and m = 60. So, n1 = 20 and n2 = 15 (by (4.1)). But then the second graph (with spectrum containing the numbers 8, 2, −2 and −4) does not exist (see [112]). Similarly, if we take the 12th or 14th row, we get (n, m) = (42, 72) (with n1 = 24 and n2 = 18) and (n, m) = (49, 84) (with n1 = 28 and n2 = 21), respectively. But then, by the same reference, the second graph in both cases does not exist. Finally, we consider the last row. Theorem 4.3.13 ([281]). There exists a unique graph G with data as in the last row of Table 4.2. Proof. First, we have n = 70 and m = 120. So, n1 = 40 for the first component, while n2 = 30 for the second component (denote it H). In addition, we can find that the spectrum of H is [8, 215 , (−2)9 , (−4)5 ] (as follows easily by considering the spectral moments up to the second order). Next, by inspecting the data from [112] we find exactly 11 graphs having the required spectrum. Since G has neither quadrangles nor hexagons, each vertex of H is adjacent to 4 disjoint copies of K2 . This local property fails to hold for 10 out of 11 candidates for H. The remaining candidate passes this test. Thus, the closed neighbourhood of each of its vertices induces K1 ∇4K2 , that is we have that each vertex of H is incident with 4 triangles having no other vertices in common. Note next that the vertices of each triangle of H are adjacent to the same vertex of G. In other words, each triangle of H corresponds to a star K1,3 in G (its centre has degree 3 in G, while the pendant vertices have degree 4 in G). This suggests that we can reconstruct from H a unique candidate for G in the following way: (i) (ii) (iii)

Take the triangles of H as vertices of G for the first part; Take the vertices of H as vertices of G for the second part; Insert an edge between a vertex from (i) and a vertex from (ii) if the corresponding triangle from (i) contains a vertex under consideration from (ii).

By direct computation, we obtain that G has the required Q-spectrum. Note that the graph G of the last theorem is the incidence graph of a block design with points 1, 2, . . . , 30 blocked into triples:

140 | 4 Determinations of regular graphs

1 1 1 1

2 4 6 8 2 10

3 5 7 9 15

2 2 3 3 3

11 12 16 17 18

14 13 20 21 19

4 4 4 5 5

10 11 16 12 17

24 23 22 26 27

5 6 6 6 7

18 10 13 19 14

25 28 25 22 26

7 7 8 8 8

29 23 29 27 22

9 9 9 10 11

15 18 20 17 18

26 28 24 30 30

12 12 13 14 14

19 21 20 16 21

23 24 30 25 28

15 15 22 23 24

16 19 26 27 25

27 29 30 28 29

17 20 11 13 21

(4.4)

The first component (which arises in the same way as H) is also interesting: It is an example of a 9-regular graph with 40 vertices having 4 distinct eigenvalues (its spectrum is [9, 315 , (−1)9 , (−3)15 ]). Remark 4.3.14. The spectra of regular line graphs of the Q-integral graphs determined in this subsection can easily be computed from the corresponding Q-spectra. The remaining possible Q-spectra of Table 4.2 are unresolved so far. More results treating Q-integral bipartite (3, s)-semiregular graphs can be found in [280]. We conclude by the following observation unmentioned in our references. Theorem 4.3.15. The graphs Line(F2 ) and Line(G), where F2 is one of graphs obtained in Theorem 4.3.11 and G is a unique graph obtained in Theorem 4.3.13, are determined by their spectra. Proof. Assume that there exists a graph cospectral with some of these two graphs. Then, by Corollary 2.1.3, this putative graph is connected and 5-regular. By Theorem 4.1.2, it cannot be exceptional (since it has more than 28 vertices), and thus by Theorem 2.2.9 it must be a line graph. Consequently, its root graph shares the same Q-spectrum with one of F2 or G, but by Theorems 4.3.11 and 4.3.13 this may occur only if the root graph is isomorphic to one of these graphs. A contradiction. The graph Line(F2 ) is illustrated in Figure 4.7. Its spectrum is [5, 44 , 24 , 17 , (−1)4 , (−2)16 ] . The graph Line(G) can be reconstructed from the scheme (4.4): The vertices are indexed by the 120 elements of the scheme, and two vertices are adjacent precisely if the corresponding elements are identical or belong to the same triple. The spectrum of Line(G) is [5, 415 , 39 , 25 , 115 , 09 , (−1)15 , (−2)51 ] .

4.4 Regular graphs with a small number of distinct eigenvalues |

141

Fig. 4.7. A 5-regular graph with a unique spectrum (for Theorem 4.3.15).

4.4 Regular graphs with a small number of distinct eigenvalues A connected regular graph has 2 distinct eigenvalues if and only if it is a non-trivial complete graph. By Theorem 3.4.7, it has 3 distinct eigenvalues if and only if it is strongly regular. Strongly regular graphs of larger order are considered in Subsection 4.4.1. In the sequel, we increase the number of distinct eigenvalues: Constructions of connected regular graphs with 4 distinct eigenvalues are considered in Subsection 4.4.2, while constructions of bipartite regular graphs with 5 or 6 distinct eigenvalues are considered in Subsection 4.4.3.

4.4.1 Strongly regular graphs of larger order The problem of existence and determination of strongly regular graphs with given parameters has been studied by a number of mathematicians for over the decades. We recall that the complete list of primitive strongly regular graphs with at most 16 vertices is given on page 75. Particular strongly regular graphs (like conference graphs, generalized Latin squares, partial geometry graphs, etc.) of larger order are often considered separately and some results can be encountered in Section 3.4.

142 | 4 Determinations of regular graphs Tab. 4.3. Data on strongly regular graphs with n vertices (17 ≤ n ≤ 64). (n, r, a, b) (17, 8, 3, 4) (21, 10, 3, 6) (25, 8, 3, 2) (25, 12, 5, 6) (26, 10, 3, 4) (27, 10, 1, 5) (28, 12, 6, 4) (29, 14, 6, 7) (35, 16, 6, 8) (36, 10, 4, 2) (36, 14, 4, 6) (36, 14, 7, 4) (36, 15, 6, 6) (37, 18, 8, 9) (40, 12, 2, 4) (41, 20, 9, 10) (45, 12, 3, 3) (45, 16, 8, 4)

θ 2 m2

θ 3 m3

1.565

(−2.56)8

114 38 212 213 120 47 2.1914 220 410 221 58 315 2.5418 224 2.7020 320 69

(−4)6 (−2)16 (−3)12 (−3)12 (−5)6 (−2)20 (−3.19)14 (−4)14 (−2)25 (−4)14 (−2)27 (−3)20 (−3.54)18 (−4)15 (−3.70)20 (−3)24 (−2)35

# 1 1 1 15 10 1 4 41 3854 1 180 1 32 548 28 78 1

(n, r, a, b) (45, 22, 10, 11) (49, 12, 5, 2) (49, 18, 7, 6) (49, 24, 11, 12) (50, 7, 0, 1) (50, 21, 8, 9) (53, 26, 12, 13) (55, 18, 9, 4) (56, 10, 0, 2) (57, 24, 11, 9) (61, 30, 14, 15) (63, 30, 13, 15) (64, 14, 6, 2) (64, 18, 2, 6) (64, 21, 8, 6) (64, 27, 10, 12) (64, 28, 12, 12)

θ 2 m2

θ 3 m3

2.8522

(−3.85)22

512 418 324 228 325 3.1426 710 235 518 3.4130 335 614 245 521 336 428

(−2)36 (−3)30 (−4)24 (−3)21 (−4)24 (−4.14)26 (−2)44 (−4)20 (−3)38 (−4.41)30 (−5)27 (−2)49 (−6)18 (−3)42 (−5)27 (−4)35

# 1

1

1 1

1 167

In general case, constructions of strongly regular graphs of larger order depend on specified parameters, additional skills and computer search. A building block of combinatorial constructions of strongly regular graphs (like that of Fon-Der-Flaass [128]) consists of strongly regular graphs of comparatively small order. Then following fixed rules these graphs are extended to larger, but also strongly regular, graphs. In the algebraic context, a search for strongly regular graphs usually starts with a set of feasible parameters (in sense of the restrictions obtained in Subsection 3.4.1). Then, such parameters are considered by various combinatorial and algebraic methods combined with a computer search. For sets of feasible parameters, the reader may consult the books [48, 78], but it is much convenient to visit the web pages devoted to strongly regular graphs, like [285, 307]. In Table 4.3 we list the data on primitive strongly regular graphs with at least 17 and at most 64 vertices. We avoid the presentation of complementary graphs, and so every graph corresponding to a single data is either self-complementary (like in the case of the first row of the table) or has a non-isomorphic strongly regular complement (like in the case of the second row). Each row contains the parameters, the two non-simple eigenvalues with multiplicities and the number of existing graphs (if this number is known). For every row the existence of at least one strongly regular graph is confirmed (cf. [81]). Some graphs can be described on the basis of the previous results. For example, all graphs with equal multiplicities of non-simple eigenvalues (in particular, all graphs with irrational eigenvalues) are the conference graphs, the graph with 27 vertices is

4.4 Regular graphs with a small number of distinct eigenvalues |

143

the complement of the Schläfly graph, the graph with parameters (50, 7, 0, 1) is the Hoffman–Singleton graph, etc. Hoffman–Singleton graph The fact that the Hoffman–Singleton graph is a unique strongly regular graph with the above parameters can be proved in different ways. In the original reference [174], Hoffman and Singleton used the fact that the graph they are looked for is a cage of girth 5 (in particular a Moore graph with diameter 2) to show that it must contain exactly 1260 pentagons which can be partitioned into the three sets with respect to their neighbourhoods. Then, the graph itself is constructed in a pure combinatorial way. Rowlinson and Sciriha [261] used star complements to show that a strongly regular graph with these parameters must contain a specified induced subgraph: A (rooted) tree with 7 neighbours of a fixed vertex and additional distinct pairs of neighbours for each of these 7 vertices. In further, they proved that the only regular graph with specified tree as the star complement for the eigenvalue 2 is the Hoffman–Singleton graph. A simple construction described by Robertson [258] reads as follows. Take the 10 copies of the pentagon C5 , denote the first five of them by H0 –H4 , and the remaining five by S0 –S4 . If the vertices of all the pentagons are considered in a natural order, then label the vertices of all H i ’s by 0, 1, 2, 3, 4 and those of S j ’s by 0, 2, 4, 1, 3. Insert an edge between the vertex k of H i and the vertex ij + k (mod 5) of S j , where 0 ≤ i, j, k ≤ 4. Other constructions of the same graph can be found in [33, 161]. Families with common parameter sets Feasible parameters that correspond to more than one strongly regular graph are often considered by exhaustive computer search. Complete enumerations of the 15 12-regular graphs with 25 vertices and the 10 10-regular graphs with 26 vertices are given by Paulus and Rozenfel’d [248, 263]. One of the graphs with 28 vertices is the line graph of K8 , while the remaining three graphs are identified by Chang [64]. By [49, p. 125], the Chang graphs can be obtained by the Seidel switching in Line(K8 ) with respect to the sets that arise from (i) (ii) (iii)

a perfect matching in K8 , 8 edges forming edge-disjoint triangle and pentagon in K8 and 8 edges forming an octagon in K8 .

Next, the 41 graphs with 29 vertices appeared in works of several authors, while the total number is confirmed by Bussemaker and independently by Spence (see [78, p. 856]). All the 3854 graphs with 35 vertices, the 180 14-regular graphs with 36 vertices and the 32 548 15-regular graphs with 36 vertices are enumerated by McKay and Spence [227]. All the 28 graphs with 40 vertices are determined by Spence [286], while the 78 12-regular graphs with 45 vertices can be found in a work of Coolsaet et

144 | 4 Determinations of regular graphs al. [81]. Finally, the 167 18-regular graphs with 64 vertices are enumerated by Haemers and Spence [160]. Following [227], we describe how the authors obtained all strongly regular graphs with 35 vertices, and consequently the two large families of strongly regular graphs with 36 vertices (the reader may also consult [284]). If A is the adjacency matrix of a putative strongly regular graph with parameters (35, 16, 6, 8), then the matrix 2J − 5(A − 2I) is positive semidefinite with the eigenvalues 30 (with multiplicity 14) and 0 (with multiplicity 21). We may assume that A has the form 0 A = (j O

jT A1 B

OT BT ) , A2

(4.5)

where the 16 × 16 submatrix A1 corresponds to the neighbourhood of a fixed vertex. Moreover, since we are dealing with a strongly regular graph in which two adjacent vertices have 6 common neighbours, we conclude that A1 must be the adjacency matrix of a 6-regular graph. In addition, the eigenvalues of A1 interlace those of A, and so apart from the spectral radius, the spectrum of A1 lies in the segment [−4, 2]. The procedure has the following four steps. Step 1. To generate all 6-regular graphs with 16 vertices, the authors took an advantage of programs library nauty [226] which enables generating graphs by adding vertices at induced subgraphs. In this way, for each intermediate graph a test for eigenvalue feasibility is applied. The computer search resulted in 137 668 graphs. Step 2. This step employs another test for the submatrices A1 . Namely, if A1 can be embedded into (4.5), then for each column x of B, the matrix 0 (j 0

jT A1 xT

0 x) 0

must be feasible in sense that exactly 8 entries of x are equal to 1, and again the eigenvalues of the above matrix interlace those of A. In addition, A1 and x need to be compatible in sense that for each vertex of a 6-regular graph there must be at least 9 vectors x that include that vertex. Exactly 12 699 out of 137 668 graphs passed this test. Finally, for each admissible A1 , compatible columns x are generated in an lexicographical order. Step 3. Once the columns x have been generated, the 16 rows of the submatrix B of (4.5) are generated in such a way that the corresponding columns are ordered lexicographically and the inner product of any two rows is either 6 or 8 according to strong regularity. Step 4. Now, for each A1 obtained in Step 2 and each B formed (for that A1 ) in Step 3, the adjacency matrix A is obtained by determining the remaining submatrix A2 . This step proceeds quickly, since if A1 and B are fixed then the submatrix A2

4.4 Regular graphs with a small number of distinct eigenvalues | 145

needs to satisfy very restrictive conditions. As a result, exactly 7440 pairs of matrices (A1 , A2 ) can be embedded into (4.5). These embeddings give rise to the 3854 non-isomorphic strongly regular graphs with required parameters. Now, if G is one of the 3854 strongly regular graphs with 35 vertices, then the Seidel spectrum of K1 ∪ G is [715 , (−5)21 ] (which is not difficult to be seen). Considering the feasible parameters (36, 14, 4, 6) and (36, 20, 10, 12), we conclude that all putative graphs corresponding to both sets of parameters would share the Seidel spectrum with K1 ∪ G. A fuller analysis of the results given before and after Theorem 3.6.3 shows that this is not a coincidence. As a conclusion, every putative graph belongs to a switching class (with respect to the Seidel switching) represented by K1 ∪ G for some of the 3854 candidates for G. For the first case, considering the 14-regular classes for all the 3854 graphs, exactly 180 non-isomorphic strongly regular graphs are obtained. For the second case, a similar consideration of the 20-regular classes resulted in exactly 32 548 non-isomorphic strongly regular graphs giving the same number of complementary graphs (with parameters (36, 15, 6, 6)). Remark 4.4.1. Using strongly regular graphs with 35 vertices and following Example 3.6.5, we can construct a set of 36 equidistant lines that are embedded into ℝ15 (see also [56]). Strongly regular graphs with least eigenvalue −2 Using the results of Section 4.1, we may establish all these graphs. Theorem 4.4.2 (Seidel [273]). A strongly regular graph G with least eigenvalue −2 is one of the following graphs: (i) Line(K n ) (n ≥ 4), (ii) Line(K n,n ) (n ≥ 2), (iii) CP(n) (n ≥ 2), (iv) the Petersen graph,

(v) (vi) (vii) (viii)

the Clebsch graph, the Shrikhande graph, the Schläfli graph or one of the three Chang graphs.

Proof. By Theorem 4.1.1, G is either a line graph Line(H), a cocktail party graph or an exceptional regular graph. The second possibility gives (iii). For the first possibility, H is either regular or bipartite semiregular. Since G has 3 distinct eigenvalues, using Theorem 2.2.10 we get the following possibilities for H: It is bipartite regular with 3 distinct eigenvalues (giving (ii)), non-bipartite regular with 2 distinct eigenvalues (giving (i)) or bipartite semiregular (giving no solutions, by Theorem 2.2.11). Finally, by inspecting all of the 187 exceptional regular graphs, we get the remaining solutions.

146 | 4 Determinations of regular graphs 4.4.2 Regular graphs with 4 distinct eigenvalues Almost all results in this subsection can be found in works of Van Dam [103] and Van Dam and Spence [112]. The former reference contains certain characterizations of connected regular graphs with 4 distinct eigenvalues, feasibility conditions for their eigenvalues and lot of examples encountered in specified sets of graphs. The complete list of the feasible spectra for connected regular graphs with 4 distinct eigenvalues and at most 30, together with additional data on known graphs, can be found in the latter reference. By Theorem 3.3.5, every distance-regular graph with diameter 3 has 4 distinct eigenvalues. Therefore, by setting D = 3 for all graphs considered in Section 3.3, we may obtain many examples of such graphs. In what follows we give a detailed analysis. For this purpose, we denote the spectrum of a connected r-regular graph G under consideration by [r, θ2 m2 , θ3 m3 , θ4 m4 ] .

(4.6)

Feasibility conditions Here is a closer characterization of the eigenvalues and their multiplicities. Lemma 4.4.3 ([103, 112]). Under the notation from the above, if k = (i) (ii)

n−1 3

then

m2 = m3 = m4 = k and r ∈ {k, 2k} or G has 2 or 4 integral eigenvalues. If G has exactly 2 non-integral eigenvalues, then √ they are algebraic conjugates of the form a±2 b (a ∈ ℤ, b ∈ ℕ), with equal multiplicities.

Proof. (i) If m2 = m3 = m4 , then all of these multiplicities must be equal to k (specified in the theorem). Since r + k ∑4i=2 θ i = 0, we get that ∑4i=2 θ i is an integer, and so r is a multiple of k. The equality n = 3k + 1 leads to the assertion. (ii) If m j is the largest multiplicity, while the remaining two are equal to p, then m −p (x − θ j ) j can be expressed as (x − θ j )

m j −p

=

(x − r)p−1 P G (x) ((x − r) ∏4i=2 (x − θ i ))

p

.

Since the polynomials on the right hand side are monic with integral coefficients, we conclude that θ j is integral. Next, it follows from the basic theory of polynomials with integral coefficients (see also page 9) that the remaining two of distinct eigenvalues are both integral or they are algebraic conjugates of the above form. Finally, considering m j as the smallest multiplicity and expressing the product ∏i=j̸ (x − θ i )m i −m j as

4.4 Regular graphs with a small number of distinct eigenvalues |

∏ (x − θ i )m i −m j = i=j̸

(x − r)m j −1 P G (x) ((x − r) ∏4i=2 (x − θ i ))

mj

147

,

we arrive at the desired conclusion. The three eigenvalue conditions arise immediately from the spectral moments: 4

4

4

i=2

i=2

i=2

1 + ∑ m i = n, r + ∑ m i θ i = 0 and r2 + ∑ m i θ2i = nr.

(4.7)

Observe that m i ’s are uniquely determined by the above equations. To get more feasibility conditions, we need to take into consideration the next simple characterization. Theorem 4.4.4 (cf. [103]). Every connected regular graph with 4 distinct eigenvalues is walk-regular. Proof. Using the Hoffman polynomial (see Theorem 2.1.6), we get 4

∏ (A − θ i I) = i=2

1 4 (∏ (r − θ i )) J, n i=2

which gives 4

A3 = ( ∑ θ i ) A2 − (θ2 θ3 + θ2 θ4 + θ3 θ4 ) A + θ2 θ3 θ4 I + i=2

1 4 (∏ (r − θ i )) J. n i=2

Since all the matrices A2 , A, I and J have constant main diagonal, the same holds for A3 . A successive multiplying of both sides by A leads to the conclusion that A k has a constant main diagonal for all k ≥ 1, and then the assertion follows by Lemma 3.2.1. Next, the number of triangles through a fixed vertex in a walk-regular graph is independent of that vertex (see Lemma 3.2.1), and so by computing the number of such triangles we get that it is equal to 4 tr(A3 ) 1 = (r3 + ∑ m i θ3i ) , 2n 2n i=2

which gives a feasibility condition for the eigenvalues since the right hand side must be a non-negative integer. In general case, 1 k 4 (r + ∑ m i θ ki ) n i=2

(4.8)

148 | 4 Determinations of regular graphs must be a non-negative integer. If k is odd, the integer (4.8) cannot be odd (since the number of closed walks of odd length is even). If k = 4, we get that the number of quadrangles containing a fixed vertex is equal to 1 n

(r4 + ∑4i=2 m i θ4i ) − r(2r − 1)

, (4.9) 2 and so this number must also be a non-negative integer. Using the feasibility conditions of (4.7)–(4.9), Van Dam and Spence [112] obtained the feasible spectra for connected regular graphs with 4 distinct eigenvalues when these graphs have at most 30 vertices. This resulted in a list of the 244 entries given in the original reference. The non-existence of any graph for the 68 of them is confirmed. Bipartite graphs According to Theorem 3.8.5, every connected bipartite regular graph G with 4 distinct eigenvalues is the incidence graph of a symmetric BIBD. If such a BIBD has the parameters (p, r, l), then the spectrum of G is given by [r, √r − l

p−1

, (−√r − l)

p−1

, −r] .

(4.10)

The next question is on whether a symmetric BIBD exists or not. It is not difficult to see that a symmetric BIBD with parameters (p, p −1, p −2) exists for all p ≥ 3. The corresponding incidence graph is just K −p,p (with spectrum [p − 1, 1p−1 , (−1)p−1 , 1 − p]). According to [112], some other sets of parameters for which the corresponding symmetric BIBDs exist are: (7, 3, 1), (7, 4, 2), (11, 5, 2), (11, 6, 3), (13, 4, 1), (13, 9, 6), (15, 7, 3) and (15, 8, 4). Observe that these parameter sets are paired, which is not surprising: If a symmetric BIBD is determined, then its pair follows easily (this can be an exercise). The first 6 sets give a unique design each, and consequently a unique incidence graph. The last 2 correspond to the 4 different designs each, and together give the 8 non-isomorphic incidence graphs. These 8 graphs are integral and they can also be found in [160]. In Figure 4.8 we illustrate the incidence graphs of BIBDs with parameters (7, 3, 1) and (7, 4, 2), respectively. The first of them is known as the Heawood graph. We will meet them again in Subsection 4.4.3. Graphs with θ4 ≥ −2 Since a cocktail party graph does not have 4 distinct eigenvalues, by Theorem 4.1.1, it remains to examine regular line graphs and exceptional regular graph as possible connected candidates. By Doob [115] (see also [116]), if a connected regular graph has 4 distinct eigenvalues, least eigenvalue greater than or equal to −2 and is the line graph, then its root graph is either K n1 ,n2 (n1 > n2 ≥ 2), the incidence graph of a symmetric BIBD, a strongly regular graph or the cycle C7 .

4.4 Regular graphs with a small number of distinct eigenvalues |

149

Fig. 4.8. Heawood graph and its bipartite complement.

Consider now the spectra of these graphs. The spectrum of Line(K n1 ,n2 ) is computed on page 51. If the spectrum of the incidence graph of a symmetric BIBD is given by (4.10), then (by Theorem 2.2.10) the spectrum of its line graph is [2(r − 1), (r + √r − l − 2)

p−1

, (r − √r − l − 2)

p−1

, (−2)(r−2)p+1 ] .

(4.11)

If a strongly regular graph with n vertices has the spectrum [r, θ2 m2 , θ3 m3 ],

(4.12)

then the spectrum of its line graph is [2(r − 1), (r + θ2 − 2)m2 , (r + θ3 − 2)m3 , (−2)

n(r−2) 2

].

Finally, the spectrum of C7 ≅ Line(C7 ) is computed easily (see Example 2.2.14). Observe that this is the only connected regular line graph with 4 distinct eigenvalues and least eigenvalue greater than −2 (by Theorem 3.6.17, the least eigenvalue of any exceptional regular graph is equal to −2). Inspecting the list of the 187 exceptional regular graphs from [94], we find that exactly 12 out of them have the required number of distinct eigenvalues. Precisely, there is one graph for each of the spectra [4, 23 , 03 , (−2)5 ] and [7, 43 , 15 , (−2)10 ], exactly 8 graphs with spectrum [10, 44 , 23 , (−2)16 ] and one graph for each of the spectra [10, 42 , 14 , (−2)11 ] and [14, 44 , 22 , (−2)17 ] (see also [103, 112]). Graphs from strongly regular graphs If G is a disjoint union of k (k ≥ 2) cospectral strongly regular graphs whose common spectrum is given by (4.12), then G is a connected regular graph with spectrum

150 | 4 Determinations of regular graphs

[n − r − 1, (−θ3 − 1)m3 , (−θ2 − 1)m2 , (−r − 1)k−1 ] , where n is its order. Assume now that the n × n adjacency matrix of a strongly regular graph H can be written in the form A=(

A1 B

BT ), A2

where all the submatrices have the size 2n × 2n and the induced subgraphs determined r−θ3 3 by A1 and A2 are regular of degrees r+θ 2 and 2 , respectively. If none of them is complete nor totally disconnected graph, then both are strongly regular. (We say that H admits a regular partition into halves.) Applying the Seidel switching with respect to the set of 2n vertices that correspond to A1 , we get the graph with the adjacency matrix (

A1 J−B

J − BT ). A2

Its spectrum follows from the spectrum of the corresponding Seidel matrix and reads as follows [θ3 +

n n , θ2 m2 , θ3 m3 −1 , r − ] . 2 2

According to [103], if a regular graph has the above spectrum, then it is necessarily obtained from a strongly regular graph in the described way. For an illustration, consider the lattice graph Line(K n,n ) whose spectrum is given 2 by [2(n − 1), (n − 2)2(n−1) , (−2)(n−1) ]. If we label the vertices in each part of K n,n by 1, 2, . . . , n, then the vertices of Line(K n,n ) are determined by the ordered pairs (i, j), where 1 ≤ i, j ≤ n. We may form a regular partition of Line(K n,n ) by taking the set of the vertices {(i, j) : 1 ≤ i, j ≤ 2n } ∪ {(i, j) : 2n + 1 ≤ i, j ≤ n} for the first half and the set of the remaining vertices for the second. Applying the Seidel switching, we get a connected regular graph with spectrum [

2 n2 n2 − 2, (n − 2)2(n−1) , (−2)(n−1) −1 , − + 2(n − 1)] . 2 2

Forming another partition by taking the vertices {(i, j) : 1 ≤ i ≤ n, 1 ≤ j ≤ 2n } for the first half, we arrive at the connected regular graph with spectrum [

2 n2 n2 + n − 2, (n − 2)2n−3 , (−2)(n−1) , − + 2(n − 1)] . 2 2

It has 4 distinct eigenvalues whenever n ≥ 6. Exercise 4.4.5 ([103]). Prove that for n ≥ 6 there exist exactly one graph with n2 vertices having the above spectrum.

4.4 Regular graphs with a small number of distinct eigenvalues | 151

We conclude the paragraph with the two more exercises. Exercise 4.4.6 (cf. [103]). Let H be a connected non-bipartite strongly regular graph with spectrum (4.12). Prove that if K is a Delsarte clique (recall definition on page 67) of size k in H, then the induced subgraph obtained by removing K is connected regular with spectrum [r + θ2 + 1 − k, θ2 m2 −k , (θ2 + θ3 + 1)k−1 , θ3 m3 +1−k ] . This exercise may be useful if H is a generalized quadrangle graph. Exercise 4.4.7. Show that removing a clique of size 3 from the generalized quadrangle graph GQ(2, 4) (see Figure 3.9) results in a unique connected regular graph with spectrum [9, 117 , (−3)2 , (−5)4 ]. Show that the graph H mentioned in the proof of Theorem 4.3.13 is obtained by removing a co-clique of size 10 from GQ(3, 3). Kronecker products Recall from page 49 the definition of a Kronecker product of two matrices. If H and H 󸀠 are graphs with n and n󸀠 vertices and adjacency matrices A and A󸀠 respectively, then the graph with the adjacency matrix A ⊗ I n󸀠 + I n ⊗ A󸀠 is nothing else but the Cartesian product H◻H 󸀠 . We also consider the graphs H ⊗ determined by the adjacency matrix A ⊗ J (where J is of size k) and the graphs H ⊙ determined by the adjacency matrix (A + I) ⊗ J − I. Since the eigenvalues of H◻H 󸀠 are obtained as all possible sums (with multiplicities) of the eigenvalues of H and H 󸀠 , if H is a strongly regular graph whose spectrum is given by (4.12) and H 󸀠 is the complete graph K n then the spectrum of their Cartesian product is given by [r + n − 1, (r − 1)n−1 , (θ2 + n − 1)m2 , (θ2 − 1)(n−1)m2 , (θ3 + n − 1)m3 , (θ3 − 1)(n−1)m3 ] . Hence, if r − θ2 = θ2 − θ3 = n, then the resulting graph has 4 distinct eigenvalues. For the remaining matrix operations, let H be a connected regular graph whose spectrum is given by (4.6), where θ3 ∈ {−1, 0}. For θ3 = 0, the spectrum of H ⊗ is [rk, (θ2 k)m2 , 0(k−1)n+m3 , (θ4 k)m4 ], while for θ3 = −1, the spectrum of H ⊙ is [(r + 1)k − 1, ((θ2 + 1) k − 1)m2 , (−1)(k−1)n+m3 , ((θ4 + 1) k − 1)m4 ] (for both spectra, the reader may compare Theorems 3.1.3 and 3.1.4). Observe that the resulting product has 4 distinct eigenvalues even if m3 = 0, that is if H is strongly regular.

152 | 4 Determinations of regular graphs Exercise 4.4.8. Compute the spectrum of C⊗5 , C⊙5 and K −n,n ⊙ in terms of k (the size of J). Exercise 4.4.9 ([103]). If A is the adjacency matrix of a conference graph whose parameters are given in Theorem 3.4.16, then prove that a graph determined by the adjacency matrix (

A I

I ) J−I−A

has the spectrum [

−1 + √n + 4 n+1 n−3 , ,( ) 2 2 2

n−1

,(

−1 − √n + 4 ) 2

n−1

].

4.4.3 Bipartite regular graphs with 3 distinct non-negative eigenvalues Here we reinterpret some results of Koledin and Stanić reported in [196]. According to the convention introduced in Remark 1.3.2, whenever we say that a graph has 3 distinct non-negative eigenvalues, we always assume that 3 is the exact number of distinct non-negative eigenvalues. We start with feasibility conditions for the spectra of graphs under consideration, continue with their connections with PBIBDs, give a sequence of examples and conclude by the lists of connected bipartite regular graphs with 6 distinct eigenvalues which are either cubic or have at most 20 vertices. Recall from Subsection 3.8.2 that every incidence graph of a 2-class PBIBD has 3 distinct non-negative eigenvalues (so, 5 or 6 in total). A weaker converse that includes distance-regularity is given in Theorem 3.8.18 stating that every bipartite distance-regular graph with 3 distinct non-negative eigenvalues is the incidence graph of a symmetric 2-class PBIBD based on its halved graph. Therefore, if we intend to determine some of graphs described in the title of this subsection, the first natural direction indicates to PBIBDs. Conversely, if we have a bipartite regular graph with required number of distinct eigenvalues, we may check whether it is the incidence graph of some PBIBD by checking its distance-regularity. Before we proceed with PBIBDs, we give some eigenvalue conditions. Assume in further that a graph under consideration is connected. The bipartiteness (in conjunction with regularity) assures that it has an even order, say 2n, while its spectrum can be divided into the two symmetric halves. Therefore, it is convenient to consider the non-negative half containing distinct eigenvalues r, θ2 and θ3 (ordered decreasingly). As before, we denote the multiplicities of the two latter eigenvalues by m2 and m3 , but now with an additional convention for m3 : If θ3 = 0, then its multiplicity in the entire spectrum is 2m3 (in other words, m3 stands for the multiplicity in the considered half). We also denote the squares of θ2 and θ3 by ϑ2 and ϑ3 , respectively. This notation will be used until the end of the subsection.

4.4 Regular graphs with a small number of distinct eigenvalues | 153

Feasibility conditions We first consider the eigenvalues and their multiplicities. Lemma 4.4.10. For a connected bipartite graph with 3 distinct non-negative eigenvalues, we have (i) (ii)

m2 ≠ m3 and both ϑ2 and ϑ3 are integral or m2 = m3 and both ϑ2 and ϑ3 are either integral or algebraic conjugates of the √ form a±2 b (a ∈ ℤ, b ∈ ℕ).

Proof. The proof is similar to that of Lemma 4.4.3. For (i), if m2 > m3 then by expressm −m ing (x2 − ϑ3 ) 2 3 as (x2 − ϑ3 )

m2 −m3

=

∏3i=2 (x2 − ϑ i )m5−i ∏3i=2 (x2 − ϑ i )m3

,

we get the assertion (since both products are monic polynomials with integral coefficients). The case m2 < m3 is considered by interchanging the roles of ϑ2 and ϑ3 . For (ii), if m2 = m3 then we have ϑ2 + ϑ3 , ϑ2 ϑ3 ∈ ℤ, and the result follows. The two eigenvalue conditions arise immediately from the spectral moments: 3

3

i=2

i=2

1 + ∑ m i = n, r2 + ∑ m i ϑ i = nr,

(4.13)

giving m i ’s in terms of distinct non-negative eigenvalues. Acting as in the proof of Theorem 4.4.4, that is by using the Hoffman polynomial to express A4 as A4 = (ϑ2 + ϑ3 )A2 − ϑ2 ϑ3 I +

(r2 − ϑ2 )(r2 − ϑ3 ) J ( n 0

O ), J

(4.14)

we arrive at an unsurprising result. Theorem 4.4.11. Every connected regular graph with 3 distinct non-negative eigenvalues is walk-regular. Using the last result, we deduce that the number of quadrangles containing a fixed vertex is independent of that vertex. Clearly, the number of quadrangles containing a fixed vertex is half the number of closed walks of length 4 along a quadrangle starting at that vertex. Since the total number of closed walks of length 4 starting at a vertex is equal to the number of such walks that traverse along a quadrangle and those that do not traverse along a quadrangle, and since the latter number is equal to r(2r − 1), we get that the number r4 + m2 ϑ22 + (n − m2 − 1)ϑ23 − r(2r − 1) n

(4.15)

154 | 4 Determinations of regular graphs must be a non-negative integer and cannot be odd, which is another condition. (Compare the expression (4.9) for graphs with 4 distinct eigenvalues.) Similarly, the number of closed walks of length 6 starting at a fixed vertex and traversing along a hexagon or a quadrangle cannot be odd, neither. Since the number of such walks that do not traverse along a hexagon nor a quadrangle is equal to r(5r2 − 6r + 2), we conclude that the non-negative integer r6 + m2 ϑ32 + (n − m2 − 1)ϑ33 − r(5r2 − 6r + 2) n

(4.16)

cannot be odd. Concerning the conditions from the above, Stevanović [303] obtained a number of feasible spectra for connected bipartite regular graphs with 5 distinct eigenvalues. In particular, the proof of the next exercise can be found in the same reference. Exercise 4.4.12. Prove that there exist exactly 3 non-isomorphic regular graphs with the common spectrum [±6, (±2)9 , 06 ]. We conclude the paragraph by the two more observations. Lemma 4.4.13. The inequality ϑ3 < r holds. Proof. Using the second identity of (4.13), we get that nr = r2 + m2 ϑ2 + (n − m2 − 1)ϑ3 . Thus, we have n(r − ϑ3 ) = r2 + m2 (ϑ2 − ϑ3 ) − ϑ3 > r2 − ϑ3 > 0, and we are done. Lemma 4.4.14. If ϑ2 and ϑ3 are non-integral algebraic conjugates of the form (a ∈ ℤ, b ∈ ℕ), then a = 2 (n−r)r n−1 .

a±√b 2

Proof. By Lemma 4.4.10, we have n = 2m2 + 1 which yields nr = r2 + m2 a, and so (n−r)r a = (n−r)r m2 = 2 n−1 . If our graph has 5 distinct eigenvalues (i.e., ϑ3 = 0), then Lemma 4.4.13 holds trivially, while the assumptions of Lemma 4.4.14 cannot occur. So, these results are in fact interesting in the case of graphs with 6 distinct eigenvalues. We will return to them in the last two paragraphs of this subsection. Relations with symmetric 2-class PBIBDs Consider now some connected bipartite r-regular graphs with 3 distinct non-negative eigenvalues that are incidence graphs of symmetric 2-class PBIBDs. Lemma 4.4.15. If (ϑ2 + ϑ3 − 2r + 1)r − ϑ2 ϑ3 +

(r2 − ϑ2 )(r2 − ϑ3 ) = 0, n

4.4 Regular graphs with a small number of distinct eigenvalues | 155

then we are dealing with the incidence graph of a 2-class symmetric PBIBD. Proof. Since the value on the left hand side represents the number of closed walks of length 4 starting at a fixed vertex and traversing along a quadrangle (compare the identity (4.14)), the assertion follows by the result of Exercise 3.8.22. Here is another result. Theorem 4.4.16. Assume that the 2n × 2n adjacency matrix of a connected bipartite r-regular graph G with 3 distinct non-negative eigenvalues is given by (1.8). If there exist distinct integers l1 and l2 such that

(ϑ2 +ϑ3 )r−ϑ2 ϑ3 +

(r2 − ϑ2 )(r2 − ϑ3 ) 2 −r +l2 ((n−1)l2 −2r(r−1)) = (l1 −l2 )(r(r−1)−l2 (n−1)) n

holds and every entry of the matrix B = NN T − rI − l2 (J − I) is a multiple of l1 − l2 , then G is the incidence graph of a symmetric 2-class PBIBD. Proof. Consider the matrix B2 = (NN T )2 + r2 I + l22 (J − I)2 − 2rNN T − 2l2 NN T (J − I) + 2rl2 (J − I). It follows that the diagonal entries are exactly the left hand side of the equality of the theorem, and so every diagonal entry of B2 is (l1 − l2 ) (r(r − 1) − l2 (n − 1)) . Since, by the assumption, every entry b ij of B is a multiple of l1 − l2 , it holds b2ij ≥ (l1 − l2 )b ij . We compute tr (B2 )

=

n(l1 − l2 ) (r(r − 1) − l2 (n − 1)) =

∑ 1≤i 0 if and only if G is connected. In addition, if G is connected and 0 ≠ S ⊂ V(G) where |S| = cn and c ≤ 12 , then we must remove at least hcn edges from G do separate S from the rest of the graph. Proof. Both claims follow directly from the definition of h. Here is an easy exercise (recall that δ stands for the minimal vertex degree). Exercise 5.1.5. Prove that the expansion parameter of a connected graph satisfies the inequalities 2n ≤ h ≤ δ. If v is a vertex of a graph G, then the metric ball B v (k), k ≥ 0, is the set of vertices of G that are on distance at most k from v, that is B v (k) = {u ∈ V(G) : d(u, v) ≤ k}. Let G have n vertices. There is a simple combinatorial result: n h k |B v (k)| ≥ min { , (1 + ) } , 2 ∆

(5.1)

1 In some references one can find slightly different approaches to the isoperimetric problem for graphs, and also different definitions of isoperimetric parameter.

176 | 5 Expanders where ∆ stands for the maximal vertex degree in G. In other words, the number of vertices in an arbitrary metric ball is controlled by the expansion parameter. Unsurprisingly, the same holds for the diameter D. Theorem 5.1.6 (cf. [205]). For a connected graph G, D≤

2 log

n 2

log (1 + ∆h )

+ 3.

Proof. Take u, v ∈ V(G) and denote c = 1 + ∆h . Let k ≥ 1 be the smallest integer that satisfies ck ≥

n . 2

(5.2)

Applying the inequality (5.1) to u and v, we get |B u (k)| ≥ 2n and |B v (k)| ≥ 2n . Furthermore, we have either |B u (k)| > 2n or |B u (k)| = 2n , and consequently |B u (k + 1)| > 2n . Hence, there must be B u (k + 1) ∩ B v (k) ≠ 0, which implies d(u, v) ≤ 2k+1. Since u and v are chosen arbitrarily, we have D ≤ 2k+1. Using the inequality (5.2), we get k≤

log 2n + 1, log c

which gives the assertion. Expander family We are now ready to consider the announced combinatorial definition of expanders. This is an asymptotical definition, that is it refers to an infinite family of graphs. An expander family (or a family of expanders) {G i }, i ∈ ℕ, is a collection of graphs with the following properties: – G i is a connected graph of order n i , where n i is a monotone increasing sequence; – For each i, ∆(G i ) ≤ r, where r is a fixed constant; – For each i, h(G i ) ≥ ε > 0, where ε is a fixed constant. The second condition is a combinatorial interpretation of sparsity. Clearly, it assures that the number of edges is controlled by the number of vertices and a constant r. The third condition is an interpretation of connectivity (according to the definition of the expansion parameter and Theorem 5.1.4). Finally, once we have settled the sparsity and connectivity, the first condition is an interpretation of expansion, that is graphs under consideration remain sparse and highly connected even their order grows. In the sequel, we do not make any distinction between ‘expander family’ and ‘expanders’ (i.e., both terms refer to the same family of graphs).

5.1 Combinatorial approach | 177

A direct consequence of the inequality (5.1) is the following result that measures the sizes of metric balls in an expander family. Corollary 5.1.7 (cf. [205]). For an expander family {G i }, i ∈ ℕ, there exists a constant c greater than 1 such that, for any graph G i , we have |B v (k)| ≥ min {

ni k , c }, 2

for all v ∈ V(G i ) and k ≥ 0. We may take c = 1 +

h(G i ) r .

The last property is known as the growth of metric balls in an expander family. Following the definition of a neighbourhood of a vertex in a graph G (page 1), we may proceed to define the neighbourhood of a set S ∈ V(G), and so let N(S) denote the set of vertices of a graph G that have at least one neighbour in S. Clearly, N(S) may include some vertices belonging to S. To exclude this possibility, we define the vertex ̇ to be the set of vertices outside S such that each of them boundary of S, denoted ∂S, is adjacent to at least one vertex in S. In this way we arrive at an alternative to the expansion parameter usually called a vertex expansion parameter: ̇ |∂S| ̇ h(G) = min n . 1≤|S|≤ 2 |S| Apparently, we may proceed with the definition of a family of expanders on the basis of the vertex expansion parameter ḣ (instead of h), but the following lemma says that there is no essential difference between these two approaches. Lemma 5.1.8 (cf. [205]). For any graph, 1 ̇ ≤ |∂S|, |∂S| ≤ |∂S| ∆ where S ⊂ V(G). Proof. Consider the map which sends an edge of ∂S to its end which is not in S. Since ̇ we get the first inequality, there are at most ∆ edges which map to any vertex of ∂S, while the second inequality follows from the surjectivity of the map. Consequently, we have that a family {G i }, i ∈ ℕ, is an expander family if there eẋ i ) ≥ ε.̇ ists ε̇ > 0 such that h(G We proceed with regular graphs. A family of regular expanders {G i }, i ∈ ℕ, is a collection of r-regular graphs, where r is a fixed constant, with the following properties: – G i is a connected graph of order n i , where n i is a monotone increasing sequence; – For each i, h(G i ) ≥ ε > 0, where ε is a fixed constant. Clearly, this definition does not deviate from the general one. Observe that regular expanders are extremal in sense that all their vertices attain maximal vertex degree. We set a simple exercise.

178 | 5 Expanders Exercise 5.1.9. Let {G i }, i ∈ ℕ, be a family of r-regular expanders. Show that there must be r ≥ 3. Recall that the existence of (regular) expanders is proved by Pinsker [249] (see also [119] or [217, Chapter 1]). More precisely, he used the methods of probability to prove that the expansion parameter h of almost all r-regular graphs is at least cr, where c is a constant depending on r. Namely, it can be checked that if an r-regular graph is chosen uniformly at random, then the expected number of edges in the , which is at least r|S| boundary ∂S is r|S|(n−|S|) n 2 . Standard probability arguments are then used to prove that, for any fixed set S (where 1 ≤ |S| ≤ 2n ), the probability that the number of edges in ∂S is significantly different from its expected number r|S|(n−|S|) n is very small. Thus, with high probability, for all sets S there are at least c|S|, where c = c(r), edges in their boundary. In other words, if G is an r-regular graph with n vertices, then the probability of h(G) ≥ cr tends to 1, if n → ∞. Note that the described proof does not give an explicit construction of any expander family. The first explicitly constructed family of expanders is given by Margulis [223]. In this section we restrict ourselves on the two examples of regular expanders. A deeper analysis (but without proofs) of constructions is given later in Section 5.4. Example 5.1.10 (Margulis Expanders [223], cf. [133, 177, 205, 217]). Define the vertex set V i of an 8-regular graph G i by V i = ℤi × ℤi . Let the neighbours of a vertex (a, b) ∈ ℤi × ℤi be (a, a + b), (a, −a + b), (a + b, b), (a − b, b), (a, a + b + 1), (a, −a + b + 1), (a + b + 1, b) and (a − b + 1, b), where all operations are mod i. The constructed graphs make a family of expanders. Observe that multiple edges and loops are allowed, so in fact we deal with multigraphs.² Constructions of almost all expander families are easy to describe, while the proofs are far from elementary. Here is another example. Example 5.1.11 (cf. [177, 217]). Let the vertex set V p of a 3-regular graph G p be defined by V p = ℤp , where p is an arbitrary prime. Let the neighbours of a vertex a ∈ ℤp be a + 1, a − 1 and a−1 . As in the previous example, all operations are mod i. We also define 0−1 to be 0. The graphs G p make another family of expanders. Observe that any G p contains a loop (at least at vertex labelled 0). Multiple edges can also be encountered. The multigraph G11 is illustrated in Figure 5.1 (it has loops at 0, 1 and 10, and a double edge between 3 and 4, and 7 and 8).

2 Note that the contribution of a loop in the adjacency matrix (and consequently, to the degree of the corresponding vertex) is a matter of convention. In the context of expanders, it usually contributes a value of 1, which is also in accordance with our definition of a vertex degree on page 1. In this spirit, the multigraphs constructed by Margulis are indeed 8-regular.

5.2 Algebraic approach | 179

0 10

1

2

9

8

3

7

4

5

6

Fig. 5.1. Multigraph G11 .

5.2 Algebraic approach The main connection between the expansion parameter and the second largest eigenvalue of a regular graph is given in the first theorem of this section. In the continuous case, the theorem is proved by Cheeger [65] and Buser (cf. [177]). The discrete case, which is of interest for us, is proved in different ways by a number of mathematicians [8, 10, 11, 114]. The corresponding inequalities are known as the Cheeger inequalities. Theorem 5.2.1 (Cheeger Inequalities [65], cf. [11, 114, 177]). For a connected r-regular graph G with expansion parameter h, r − λ2 ≤ h ≤ √2r(r − λ2 ). 2

(5.3)

Proof. Throughout the proof, let S ⊂ V(G), 1 ≤ |S| ≤ 2n and S = V(G)\S. Consider the first inequality. Take the vector y = |S|1S − |S|1S , where 1S is the characteristic vector of the set S, that is its ith entry is equal to 1 if the ith vertex in a fixed labelling belongs to S, and 0 otherwise. Observe that y ⊥ j, and then by the Rayleigh T principle we get λ2 ≥ yyTAy , which yields y yT Ay ≤ λ2 yT y = λ2 (|S| + |S|) |S||S|.

180 | 5 Expanders We also compute yT Ay = 2|S|2 m(S) + 2|S|2 m(S) − 2|S||S|m(S, S), where m(S) stands for the number of edges with both ends in S (i.e., m(S) = m(S, S)). The last two expressions give m(S, S) ≥

|S|m(S) |S|m(S) λ2 (|S| + |S|) + − . |S| |S| |S||S|

Since G is r-regular, we have m(S) = 12 (r|S| − m(S, S)) and m(S) = Setting this in the last inequality, we get |∂S| = m(S, S) ≥

1 2

(r|S| − m(S, S)).

(r − λ2 )|S||S| (r − λ2 )|S| ≥ , n 2

and the first inequality of (5.3) follows. Following [11], we prove the second inequality by considering the Laplacian of G. Since G is regular, by Theorem 2.3.3, we have μ n−1 = r − λ2 , and so it is sufficient 2 to prove that μ n−1 ≥ h2r . Let x = (x1 , x2 , . . . , x n )T be an eigenvector corresponding to μ n−1 . Recall that there are at least h|S| edges in ∂S. We also have x ⊥ j, that is ∑ni=1 x i = 0. In addition, we may assume that k (k ≤ 2n ) of the entries of x are positive and that the vertices are labelled in such a way that the following chain of inequalities x1 ≥ x2 ≥ ⋅ ⋅ ⋅ ≥ x k > 0 ≥ x k+1 ≥ ⋅ ⋅ ⋅ ≥ x n holds. Let y = (y1 , y2 , . . . , y n )T be the vector defined by y i = x i for i ≤ k, and y i = 0 otherwise. We compute n 󵄨 󵄨 ∑ 󵄨󵄨󵄨󵄨y2i − y2j 󵄨󵄨󵄨󵄨 = ∑ (y2i − y2j ) ≥ ∑ (y2i − y2i+1 ) hi = h ∑ y2i . i∼j

i< 2n

i∼j,i 0 holds. Therefore, the lower bound of (5.3) is crucial for us since it gives the opportunity to consider regular expanders from the algebraic perspective (in particular, using methods of the theory of graph spectra). On may immediately observe an advantage of the algebraic approach (concerning η) with respect to the combinatorial one (concerning h): Given a regular graph, the eigenvalues are easily computable, while the expansion is not. The situation is quite different in the case of random regular graphs: The expansion parameter h can be analyzed by different methods (cf. [177]), while the behaviour of their spectrum is harder to understand. Remark 5.2.4. Theorems 3.4.7 and 5.2.1 (see also Remark 3.4.8) make some of the most remarkable bridges between the spectrum of a graph and its structural properties. Finally, how large can the spectral gap be? An adequate answer comes from the theorem of Alon and Boppana (mentioned on page 45). Observe that the inequality (2.17) covers exactly the regular graphs we are interested in, that is those with given degree and unbounded order. Eigenvalue with second largest modulus Set Λ = maxi {|λ i | : λ i ≠ ±r}. Clearly, Λ = |λ2 | or Λ = |λ n |. We consider a variant of the expander mixing lemma. Lemma 5.2.5 (Expander Mixing Lemma [9], cf. [89, Lemma 3.5.2]). Given a connected r-regular graph G, let Λ = maxi { |λ i | : λ i ≠ ±r} and S, T ⊆ V(G), with |S| = s, |T| = t. Then 󵄨󵄨 󵄨 󵄨󵄨m(S, T) − rst 󵄨󵄨󵄨 ≤ Λ√st (1 − s ) (1 − t ) ≤ Λ√st. 󵄨󵄨 󵄨󵄨 n n n 󵄨 󵄨

(5.6)

182 | 5 Expanders The proof can be found in any of cited references and also in our previous book [290]. This lemma can be viewed as relating Λ to the question of ‘randomness’ of a graph. The left hand side of (5.6) compares the expected number of edges between S and T in a random graph and the exact number of these edges. A smaller value of Λ implies that this difference is smaller, and so the graph is nearly random in this sense. Use the expander mixing lemma to solve the following exercise. Exercise 5.2.6 ([157]). Let G be a connected r-regular graph with n vertices. If H denotes its induced subgraph with s vertices and the average vertex degree d H , then show that dH −

rs n−s ≤ λ2 (G) . n n

Recently, a sort of converse of the previous lemma is obtained. Lemma 5.2.7 (Bilu and Linial [36]). Given a connected r-regular graph G, let us denote Λ = maxi {|λ i | : λ i ≠ ±r}, and let S, T ⊆ V(G), with S ∩ T = 0, |S| = s, |T| = t. If, for a positive constant c, 󵄨 󵄨󵄨 󵄨󵄨m(S, T) − rst 󵄨󵄨󵄨 ≤ c√st 󵄨 󵄨󵄨 n 󵄨󵄨 󵄨 holds, then Λ ≤ O (c (1 + log cr )). Recall from page 45 the result of Friedman stating that, for a fixed r, the second largest eigenvalue of an r-regular graph satisfies λ2 ≤ 2√r − 1 + o(1) with probability tending to 1 as n tends to infinity. A connected r-regular graph is called a Ramanujan graph if Λ ≤ 2√r − 1 holds. It is not difficult to find the graphs satisfying the last property. Say, each non-trivial complete graph does. A more difficult task is the construction of infinite families of Ramanujan graphs with given degree r and order n tending to infinity. The existence of such graphs is provided by the following result. Theorem 5.2.8 ((Lubotzky et al. [218], Margulis [224], Morgenstern [234]). For every prime power q, there exist infinitely many r-regular Ramanujan graphs with r = q + 1. According to [89, p. 68], the first infinite family of Ramanujan graphs was constructed by Lubotzky et al. [218] in 1988. We say more about this construction in Subsection 5.4.3. In the previous section we have seen how the graph diameter D can be controlled by the expansion parameter h. It is natural to expect some similar result with a spectral invariant instead of h. In fact, we have the two upper bounds expressed as D≤ and

arcosh(n − 1) +1 arcosh ( Λr )

5.2 Algebraic approach | 183

arcosh ( D≤

n

g−1 ⌊ 2 ⌋−1

− 1)

r(r−1)

arcosh ( Λr )

+ 2⌊

g−1 ⌋ + 1, 2

where g is the girth of a graph. The former bound is obtained by Chung [69] and Sarnak [268], while the latter one is attributed to Quenell [251]. In his discussion, Quenell noticed that these bounds are incomparable, but the latter one is usually stronger. We continue with another inequality that involves Λ. The set N(S) was introduced upon the definition of a vertex expansion parameter (page 177). The quantity |N(S)| |S| is ̇ |∂S| sometimes used instead of |S| in the same definition. Theorem 5.2.9 (Tanner [310], cf. [89, Theorem 3.5.3])). Given a connected r-regular graph G, let Λ = maxi {|λ i | : λ i ≠ ±r} and 0 ≠ S ⊆ V(G). Then |N(S)| r2 ≥ . 2 2 |S| Λ2 + (r −Λn )|S| In other words, min1≤|S|≤n |N(S)| |S| is controlled by Λ. The proof is obtained by applying the expander mixing lemma to S and S. In the second part of the proof of Theorem 5.2.1 we used the connection between λ2 and the algebraic connectivity μ n−1 to derive the corresponding inequality for λ2 . In this context we note that, according to [89, p. 208], ‘the differences between the various measures of expansion which are used is largely superficial in that all conform to the general principle that expansion in graphs of bounded degree is controlled by algebraic connectivity’. For example, it is not restricted to regular graphs. In the case of the expansion parameter h, we mention the following inequalities obtained by Mohar [233] (see also [89, 290]): μ n−1 ≤ 2h ≤ 2√μ n−1 (2∆ − μ n−1 ), where the first inequality holds for any non-trivial graph, while the second inequality holds if G ∉ {K1 , K2 , K3 }. Alon [8] obtained the two analogous properties for the vertex expansion parameter h.̇ Theorem 5.2.10 (Alon [8], cf. [89, 290]). Let G be a non-trivial graph with n vertices and vertex expansion parameter h.̇ If 0 ≤ c ≤ μ n−1 , then ḣ ≥

2c . ∆ + 2c

If 0 < ε̇ ≤ h,̇ then μ n−1 ≥

ε̇ 2 . 2(ε̇ 2 + 2)

184 | 5 Expanders Exercise 5.2.11. Use the inequality k2 ≤

∆ μ n−1

(

1 1 + ) (1 − c1 − c2 ), c1 c2

where S and T are disjoint non-empty subsets of V(G) at distance k > 1 with |S| = c1 n, |T| = c2 n [10], to prove the first inequality of the last theorem. (Hint: If G is ̇ = n, the proof is easy. Otherwise, set T = V(G)\(S ∪ ∂S).) ̇ connected and |S| + |∂S|

5.3 Random walks A random walk in a graph G is a sequence of vertices v1 , v2 , . . . such that v i+1 is selected uniformly at random from the neighbours of v i , independently for every i. Usually, this procedure is initiated by selecting the vertex v1 ∈ V(G) randomly.³ As we will see, a crucial property of random walks in connected non-bipartite expanders is that they converge rapidly to their limit distribution. This property has an enormous amount of applications, say in constructing complex networks, combinatorial optimization or statical physics [177]. Let us see an example. Example 5.3.1. Let the vertices of the triangle C3 be labelled 1, 2, 3 (in any order) and take a random walk starting at 1. Then, at the first step t = 1, we walk to either 2 or 3 each with probability 21 . Depending on this choice, for t = 2 we arrive at either 1 or 3, or at 1 or 2, again with equal probability, and so on. Thus, the probability that we are at vertex 1 at step 2 is 21 . At the same step, we will be at any of the remaining vertices with probability 14 . Using an elementary logic, we conclude that the probability that we are at any of the three vertices at step t tends to 31 , if t → ∞. This simple example describes an important problem in the random walks domain: Given a graph G and the initial vertex, determine the probability that we are at vertex labelled i at step t ≥ 1 and compute the limit probability distribution (if it exists) when t → ∞. Let π(t) = (p1 (t), p2 (t), . . . , p n (t))T denote a probability vector, that is a vector with non-negative coordinates which sum to 1, where the coordinate p i (t) is the probability that we are at vertex i (1 ≤ i ≤ n) at step t, that is p i (t) = Pr(v t = i). To describe a random walk, we need to explain how the probability vector π(t+1) is obtained from the previous one π(t) . For this purpose we introduce the transition matrix T = (t ij ) of a graph without isolated vertices. This matrix is defined by t ij = d1j a ij , where d j is the degree of the vertex j and a ij is the corresponding entry of the adjacency matrix. Remark 5.3.2. Here are some basic properties of the matrix T. If we write D for the diagonal matrix diag(d1 , d2 , . . . , d n ), then we have T = AD−1 . Observe that, in general 3 A random walk is in fact a Markov chain where the set of states of the chain is the set of vertices of a graph. For more details on Markov chains, see [245].

5.3 Random walks | 185

̂ = D− 21 (D − A)D− 12 (where D 21 = diag(√d1 , √d2 , . . . , √d n ) case, T is not symmetric. If L 1 and D− 2 is its inverse) stands for the normalized Laplacian matrix,⁴ then we have 1 ̂ − 12 = D 12 (D− 12 AD− 12 )D− 12 = AD−1 = T, D 2 (I − L)D

̂ and T are similar. In other words, if μ ̂ is an eigenvalue and therefore the matrices I − L ̂ ̂ lie in the seĝ of L, then 1 − μ is an eigenvalue of T. Since all the eigenvalues of L ment [0, 2] (see [290, Theorem 9.2]), we have that the spectrum of T lies in [−1, 1]. ̂ we conclude that 1 is the largest eigenMoreover, since 0 is an eigenvalue of L, value of T. Finally, the largest eigenvalue of T is simple whenever the corresponding graph G is connected, while −1 belongs to the spectrum of T if and only if G is bipartite (cf. [290, Theorem 9.2], again). If G is r-regular, then T = 1r A, so it is symmetric and an eigenvector associated with 1 is just j. Consider now the vector

Tπ(t)

∑nj=1 t1j p j (t) ∑nj=1 t2j p j (t) =( ). .. . ∑nj=1 t nj p j (t)

The ith coordinate of this vector is just the probability that we go from the vertex j to the vertex i (at step t) in a random walk. Summing up over all i, we get p i (t + 1) which is the probability that we are at the vertex i at step t + 1. Therefore, π(t+1) = Tπ(t) or π(t) = T t π(0) , expressing the state of the random walk by the transition matrix. Setting π(0) = (1, 0, 0)T in Example 5.3.1, we get π(t) = T t π(0)

1+ 1 = (1 − 3 1−

2 (−2)t 1 (−2)t ) , 1 (−2)t

T

that is π(t) → ( 31 , 31 , 13 ) , if t → ∞. Now, does the limit distribution of a random walk exist for every graph? The answer is simple: No, as the following example shows. Example 5.3.3. Let the vertices of the quadrangle C4 be labelled 1, 2, 3, 4 (in natural order) and take a random walk starting at 1. For t ∈ ℕ, at step 2t − 1 we arrive at 2 or 4 (both with probability 21 ), while at step 2t we arrive at 1 or 3 (with the same probabilities). In other words, the probability vector does not have a limit. Obviously, the same conclusion holds for any bipartite graph. Conversely, it is not difficult to see that such graphs are the only exceptions.

4 For more details on the spectrum of the normalized Laplacian matrix, we recommend the book [70].

186 | 5 Expanders Assume now that G is a graph for which π = limt→∞ π(t) exists for any initial distribution π(0) . Then we have Tπ = T limt→∞ π(t) = limt→∞ Tπ (t) = limt→∞ π(t+1) = π. In other words, the limit probability distribution for a random walk is an eigenvector associated with the eigenvalue 1 of the transition matrix T. If the limit probability distribution does not exists and the eigenvector to 1 is scaled so that the sum of its coordinates is 1, then each of these coordinates may be considered as the proportion of time spent at the corresponding vertex in a random walk. T According to this, if G is r-regular then this eigenvector is ( 1n , 1n , . . . , 1n ) , that is we may expect that we will spend an equal amount of time at any of its vertices (for any initial distribution). di Let p i = 2m (1 ≤ i ≤ n). The following theorem is a crucial link between random walks and expanders (see also the discussion after Corollary 5.3.5). Theorem 5.3.4 (cf. [215]). For a connected graph G with n vertices, let ̂λ1 > ̂λ2 ≥ ⋅ ⋅ ⋅ ≥ ̂λ n be the eigenvalues of the transition matrix T, and set ̂ Λ = max{|̂λ2 |, |̂λ n |}. For every starting vertex i, any vertex j and every step t of a random walk v1 , v2 , . . ., we have |Pr(v t = j) − p j | ≤ √

p j ̂t Λ. pi

Proof. Following the original reference, since i is the starting vertex, we first observe ̂ share the same spectrum and the second mathat Pr(v t = j) = (T t )ji . Since T and I − L ̂ = ∑n ̂λ k xk xT , where trix is symmetric, we may consider it instead of T. We have I − L k=1 k xk = (x1k , x2k , . . . , x nk )T (1 ≤ k ≤ n) are the corresponding eigenvectors. We may choose x1i = √p i , and then compute n

1 ̂ t D− 12 ei = ∑ ̂λ t (eT D 21 xk ) (eT D− 12 xk ) , Pr(v t = j) = eTj D 2 (I − L) j i k

k=1

1

where ei , ej are the eigenvectors of the canonical basis of ℝn . Since eTj D 2 xk = x kj √p j 1 x ki and eTi D− 2 xk = √p , we get i Pr(v t = j) = p j + √

pj n ̂t ∑ λ x ki x kj . p i k=2 k

Finally, 󵄨󵄨 󵄨󵄨 n n n 󵄨󵄨 ̂ t 󵄨 t t ̂ 󵄨󵄨 ∑ λ x ki x kj 󵄨󵄨󵄨 ≤ ̂ Λ |x x | ≤ Λ |x ki x kj | ≤ ̂ Λt , ∑ ∑ ki kj 󵄨󵄨 󵄨󵄨 k 󵄨󵄨k=2 󵄨󵄨 k=2 k=1 where the last follows from the Cauchy–Schwarz inequality. The next corollary is an immediate consequence of the last theorem and the discussion given in Remark 5.3.2. Corollary 5.3.5. If G is a connected non-bipartite graph, then

5.4 Constructions | 187

π(t) → π = (

dn T d1 d2 , ,..., ) 2m 2m 2m

for any initial distribution π(0) . The distribution π is called the stationary distribution of a random walk. The motivation for this name arises from the last corollary. In addition, since for connected non-bipartite r-regular graphs it holds ̂ Λ = Λr (because Λ = max{|λ2 |, |λ n |}), we conclude that smaller Λ implies a faster convergence of the probability distribution to the stationary one. Furthermore, we may exclude the least eigenvalue by considering semi-edge random walks. According to their definition given on page 132, at each step we stay at the same vertex with probability 12 or move to the next vertex with equal probability. The stationary distribution of a semi-edge random walk is the same, while the transition matrix is replaced by the matrix 12 (I + T) with non-negative eigenvalues and the difference between two largest eigenvalues equal to half of the original. In this way, expander graphs have a rapid convergence of a semi-edge random walk probability distribution. Remark 5.3.6. An observant reader must have noticed that a random walk considered in this section is just a walk defined on page 3 (i.e., a sequence of alternate vertices and edges), and similarly for a semi-edge random walk. The word ‘random’ emphasizes the randomness in the choice of the next vertex visited. A semi-edge walk is also called a lazy walk.

5.4 Constructions We restrict ourselves to proofless descriptions of several constructions. We often consider Λ instead of the second largest eigenvalue λ2 . In this way, by considering families of r-regular graphs having the property r − Λ ≥ ε > 0, we simultaneously consider graphs with η ≥ ε > 0. It is also convenient to use the following short terminology. An r-regular graph with n vertices is called an (n, r)-graph, and for c > 0 it is called an (n, r, c)-graph if Λ ≤ cr. Clearly, a family of (n i , r, c)-graphs (where n i is monotonically increasing) is an expander family whenever c < 1 − ε, where ε > 0 is a fixed constant. The construction of an expander family is an intriguing non-trivial task with extraordinary results based on different mathematical disciplines. Previously we gave the two examples of such families. Here we continue in the same direction by considering more ways to construct them. Before we start, we distinguish the two types of construction depending on complexity. We say that a family of graphs is explicitly constructible if there is a polynomial time algorithm that generates the ith graph in this family (that is, the ith graph is constructed in a time less than c1 i c2 , for non-negative

188 | 5 Expanders constants c1 , c2 ). A family of graphs is strongly explicitly constructible if there is a polynomial time algorithm on the number of bits representing the triple (i, v, k) (where i, k are integers and v is a vertex of a graph G i ) that returns the kth neighbour of v in G i . Observe that the family of Margulis expanders given in Example 5.1.10 is strongly explicitly constructed, while the family from Example 5.1.11 is only explicitly constructed since it deals with (large) primes. Assume that we have constructed an expander family (in any way), then we may immediately get another family by taking their line graphs. Namely, by virtue of Theorem 2.2.10, the spectral gap of a regular graph is equal to the spectral gap of its line graph. Of course, it would be more convenient if we could start with one (or just few) regular graphs with comparatively large expansion parameter and then construct an (infinite) expander family.

5.4.1 Covers of graphs We proceed with expander families based on covers of graphs. For two graphs, G connected and H arbitrary, we say that f : V(H) 󳨀→ V(G) is a covering map if for every v󸀠 ∈ V(H), f maps N(v󸀠 ) injectively onto N(f(v󸀠 )). In this case, we say that H is a cover of G. In particular, G is obtained from its cover H by identifying the vertices mapped in the same vertex v = f(v󸀠 ) ∈ V(G). The sets of such vertices in a cover have fixed size k ≥ 1, and accordingly we say that H is a k-cover of G. If so, then H has the vertex set V(G) × {1, 2, . . . , k}, where f maps (v, 1), (v, 2), . . . , (v, k) in v. If we associate a permutation Permuv of {1, 2, . . . , k} to every edge uv of G, then the edges of H induced by uv have the form ((u, i), (v, Permuv (i)) , i ∈ {1, 2, . . . , k}. In this context, the edges are considered as ordered pairs of their ends. Remark 5.4.1. Another name for a cover (of a graph) is a lift. Any 2-cover is usually called a double cover. Example 5.4.2. The cycles C6 and C9 represent a double cover and a 3-cover of C3 , respectively. To see this, we label the vertices of all these cycles by 0, 2, . . . , n − 1 (in natural order). Then a covering map for both covers is defined by f(i) = i (mod 3), where 0 ≤ i ≤ n − 1. Clearly, a cover of a graph is not determined uniquely. For example, the disconnected graphs 2C3 and 3C3 are also a double cover and a 3-cover of C3 (with a covering map defined in an obvious way). Exercise 5.4.3. Use the tensor product (see page 49) with K2 , to show that every graph G has a bipartite double cover. Show next that if G is already bipartite, then the graph G × K2 consists of two disjoint copies of G. This exercise explains why the tensor product G × K2 is named a bipartite double (of a graph G).

5.4 Constructions | 189

Using the eigenvalue equation (1.2) for both G and its double cover H, we easily conclude that the eigenvalues of G are eigenvalues of H. Usually, these eigenvalues are called the old eigenvalues of H. If every edge of G is assigned by weight 1 or −1 depending on whether the associated permutation is (1, 2) or (2, 1), then there is a bijective correspondence between the set of double covers of G and the set of signed matrices A󸀠 obtained from the adjacency matrix of G by the corresponding permutations. Again, using the eigenvalue equation we get that the new eigenvalues of H are the eigenvalues of A󸀠 . The property that the eigenvalues of a double cover may be computed from the adjacency matrix of the original graph enabled Bilu and Linial [36] to make computer experiments producing a conjecture that every r-regular Ramanujan graph has a double cover whose new eigenvalues are in the range [−2√r − 1, 2√r − 1]. If this conjecture is true, then the plan for constructing the corresponding expander families is clear: Start with any Ramanujan graph and construct the family by successive applying of double covering. Observe that, even it is true, the conjecture does not provide the answer which double cover satisfies our spectral property, so for any such construction additional examinations are needed. The same authors obtained the following result. Theorem 5.4.4 ([36]). For every r ≥ 3, there is an explicitly constructible family of (n, r, c)-graphs, where cr = O (√r log3 r) .

5.4.2 Zig-zag product Here we focus on constructions based on a graph operation called a zig-zag product. To arrive at this product we need a function known as a rotation and another product called a replacement. Given an r-regular graph G, let the neighbours of each of its vertices be labelled 1, 2, . . . , r (in any order). The rotation acts on a vertex and its neighbour in the following way: RotG (u, i) = (v, j), where v is the ith neighbour of u and u is the jth neighbour of v. r is For an (n, r)-graph G and an (r, k)-graph H, the replacement product G○H the (nr, k + 1)-graph obtained by replacing each vertex of G by the graph H (usually r are the union of the edges of G (which now join called a cloud). The edges of G○H clouds) and n copies of the edges of H (one copy for each cloud). So, the set of verr is the Cartesian product V(G) × V(H), the edges of H are copied to each tices of G○H r cloud, while the remaining edges (that in fact make a perfect matching in G○H) are as follows. We enumerate the vertices of H in any order, and then let (u, i) denote

190 | 5 Expanders the ith vertex in the uth cloud. If uv is an edge in G such that RotG (u, i) = (v, j), then r the vertices (u, i) and (v, j) are adjacent in G○H. Computational experiments, supported by theoretical considerations, say that if G and H have large expansion parameters, or more algebraically if they have large spectral gaps, then the same would hold for their replacement product [11, 135, 177]. On the contrary, graphs constructed by successive applying of this product do not have constant degree. So, to produce expander families with a constant degree this product is usually combined with other graph modifications. Such a construction is given in [11], while an additional modification is described as follows. z is the For an (n, r)-graph G and an (r, k)-graph H, the zig-zag product G○H (nr, k2 )-graph with the same vertex set as the replacement product. Two vertices z are adjacent precisely if there exist the ver(u, i), (v, j) ∈ V(G) × V(H) of G○H 󸀠 󸀠 tices i , j ∈ V(H) such that {(u, i), (u, i󸀠 )}, {(u, i󸀠 ), (v, j󸀠 )} and {(v, j󸀠 ), (v, j)} are the r edges of G○H. These three edges correspond to the three steps in determining the z After an exercise, we will say more about this. edges of G○H. Exercise 5.4.5. If G is a disconnected (n, r)-graph containing an (n󸀠 , r)-component G󸀠 and H is an (r, k)-graph, then prove that z = G○H| z V(G󸀠 )×V(H) . G|V(G󸀠 ) ○H In other words, the zig-zag product operates separately on each component of G (which is an expected property). The three steps for determining an edge make a heart of the zig-zag product. The first and the third are realized in the cloud H (zig moves), and the second is a step between the neighbouring clouds (zag move). The zig moves are random but realized in a comparatively small cloud, while the zag move is deterministic. Due to these facts, the probability distribution of a random walk in a graph obtained by the zig-zag product is more deterministic and converges rapidly to the stationary distribution which makes them good expanders (in sense of large expansion parameter). Moreover, the following result holds. Theorem 5.4.6 (Zig-Zag Theorem [177, 257]). If G is an (n, r, c1 )-graph and H is an z (r, k, c2 )-graph, then their zig-zag product G○H is an (nr, k2 , c3 )-graph, where c3 = c3 (c1 , c2 ) and the following statements hold: – If c1 < 1 and c2 < 1, then c3 < 1; – c3 ≤ c1 + c2 ; (1−c1 )(1−c22 ) . – c3 ≤ 1 − 2 The first bound follows easily, while the remaining two are used in constructing expander families. A simplified proof can be found in [256]. Exercise 5.4.7. Prove the first bound of the zig-zag theorem.

5.4 Constructions | 191

Usually, any construction based on the zig-zag product starts with a small regular graph H with large expansion parameter. If H 2 is the graph obtained by doubling each edge of H, then the following iterative procedure gives the required family {

G1 ≅ H 2 , z G i+1 ≅ G2i ○H, i ≥ 1,

(5.7)

which is provided by the following theorem. Theorem 5.4.8 (cf. [177]). If H is an (r4 , r, 14 )-graph, then the graph G i obtained by the iterative procedure (5.7) is an (r4i , r2 , 12 )-graph, for all i ≥ 1. Proof. For i = 1, the claim follows directly. For i > 1, if G i is an (r4i , r2 , 12 )-graph, then G2i is an (r4i , r2 , 14 )-graph, its degree is equal to the size of H, and therefore z is an (r4(i+1) , r2 )-graph. The last step follows from the second bound G i+1 ≅ G2i ○H for c3 form the zig-zag theorem. The graph H from the previous theorem (more generally, from the iterative procedure (5.7)) is usually found by a computer search. Its existence is provided by the probabilistic methods discussed on page 178. Observe that the construction assumes two recursive computations on G i , each of them being logarithmic in the order of G i+1 , and so the whole construction is realized in a polynomial time. As a conclusion, the iterative procedure (5.7) is an explicit construction of expander families. Following [146], we explain a modification of this procedure making a strongly z we set explicit construction. Namely, instead of G i+1 ≅ G2i ○H z G i+1 ≅ (G i × G i )2 ○H, where G i × G i stands for the tensor product of G i with itself. If H is an (r8 , r, 14 )-graph, i+1 then by letting G1 ≅ H 2 we arrive at the family of ((r8 )2 −1 , r2 , 12 )-graphs. The latter follows from the second bound of the zig-zag theorem applied to G i × G i and H, where the constant c1 is computed on the basis of Theorem 3.1.4. Computing the neighbourhood of a vertex in G i+1 reduces to the same computations in G i which are, due to the tensor product, logarithmic in the length of the description of vertices of the resulting graph.

5.4.3 Examples of Cayley expanders Given a group F, let S be its (usually finite) subset that is closed under inversion. The Cayley graph C(F, S) has the vertex set F, while there is an edge between the vertices u and v if and only if us = v for some s ∈ S. The definition is suggested by the famous Cayley theorem stating that every group is isomorphic to a permutation group (see, for example, [58, Theorem 3.24]).

192 | 5 Expanders If S is considered as a multiset, the corresponding Cayley graph has multiple edges. If S is symmetric (i.e., s ∈ S ⇒ s−1 ∈ S), C(F, S) is undirected and |S|-regular. We say that S generates F if every v ∈ F may be represented as v = ∏s∈S s. Observe that C(F, S) is connected if and only if S generates F. Example 5.4.9. The Cayley graph C(F, F) is obtained from the complete graph K|F| by adding a loop at each of its vertices. The Cayley graph C(ℤn , {1, −1}) is just the cycle C n . Exercise 5.4.10. Show that: (i) (ii)

The complete bipartite graph K n,n is a Cayley graph; The r-dimensional cube is a Cayley graph obtained from the group ℤ2r and its subset constituted by the r-dimensional canonical basis.

Exercise 5.4.11. Prove that the Cayley graph of the group ℤ2 with the generating set S = {(1, 0), (−1, 0), (0, 1), (0, −1)} is the infinite grid on the plane ℝ2 . Various constructions of Cayley expander families are mostly based on the Kazhdan property of the corresponding group or, relative of the spectral gap, the Kazhdan constant. Remark 5.4.12. In this book we do not give a detailed analysis of Cayley expanders constructed, but for the sake of completeness we mention a brief definition of a Kahzdan constant. We say that a group F generated by a finite set S has the Kazhdan property if and only if there exists a constant ε = ε(S) such that for every unitary representation τ : F 󳨀→ Hτ on a Hilbert space Hτ without non-zero invariant vectors, we have max ||τ(s)ξ − ξ|| ≥ ε||ξ||, s∈S

for every ξ ∈ Hτ . A positive constant ε for which the above inequality is satisfied is called a Kazhdan constant for F with respect to S [217]. It occurs that the existence of a Kazhdan constant is sufficient to establish the lower bound for the spectral gap in a constructed family [177, 205, 207, 217]. As an example of Cayley expanders, we describe the first family of Ramanujan graphs mentioned on page 182. We start with primes p and q both congruent to 1 mod 4. Let F be the group of the 2 × 2 non-singular matrices over ℤp where proportional matrices are identified. For an integer t, t2 ≡ −1 (mod p), we define the set S by a1 + ta2 S = {( −a3 + ta4

4 a3 + ta4 ) : ∑ a2i = q; a1 > 0 odd, a1 , a2 , a3 even} . a1 − ta2 i=1

Observe that S is closed under inversion and that there are exactly q + 1 solutions (a1 , a2 , a3 , a4 ), and so C(F, S) is (q + 1)-regular. Required Ramanujan graphs are ob-

5.4 Constructions | 193

tained by taking a component of every graph obtained (it can be seen that C(F, S) is either connected or consists of two isomorphic components). Consider the group SL3 (p) of the 3 × 3 matrices of determinant 1 over the field with p elements (p being a prime), and let the (generating) set S consists of all the 3 × 3 matrices with 1’s on the main diagonal and one extra ±1 outside. The 12-regular Cayley graphs C(SL3 (p), S) are proved to be expanders with second largest eigenvalue at most 12 − ε, for fixed positive ε that does not depend on p. The proof is based on the fact that the considered group has the Kazhdan property (cf. [177, 207]). Recall once again the first constructed expander family, that is the Margulis expanders described in Example 5.1.10. Alternatively, these graphs may be constructed in the following way. The vertex set V i of an 8-regular graph G i is still defined by V i = ℤi × ℤi . For the edges, we introduce the following matrices and vectors A1 = (

1 0

2 1 ) , A2 = ( 1 2

0 1 0 ) , e1 = ( ) , e2 = ( ) . 1 0 1

Now, each vertex (u, v) ∈ V(G i ) is adjacent to u u u u A1 ( ) , A2 ( ) , A1 ( ) + e1 , A2 ( ) + e2 , v v v v and to vertices obtained by the four inverse transformations. As in Example 5.1.10, all operations are mod i. It is easy to verify that the obtained families of graphs coincide. Margulis’ proof in [223] is based on the Kazhdan property and does not give an explicit lower bound for the spectral gap. Later on, Gabber and Galil [133] used a method of the harmonic analysis to show that Λ = max{|λ2 |, |λ n |} ≤ 5√2 < 8. A simplification of their proof, based on the Fourier analysis, can be found in a work of Boppana (cf. [177, 205, 314]). Observe that the zig-zag product of two Cayley graphs does not necessary produce a Cayley graph, but under some restrictions on the basis of the semidirect product of two matrices, the resulting graph will be a Cayley graph. Rozenman et al. [264] used this fact in their iterative construction of Cayley expanders. If F is a group and X is a set, then an action of F on X is a group homomorphism f : F 󳨀→ Perm(X), sending each element of F to a permutation of elements of X. According to the above, Margulis expanders may be viewed as the action of the group SL2 (i) of the 2 × 2 matrices of determinant 1 generated by the two linear transformations. If S is a subset of F, then the Schreier graph Sch(F, X, S) has the vertex set X, while the neigbours of v ∈ X are all vertices of the form f(s)v, s ∈ S. Due to Gross [151], every regular graph of even degree is a Schreier graph determined by some finite group acting on some finite set. Unsurprisingly, every Cayley graph C(F, S) is the Schreier graph Sch(F, F, S). In addition, we may associate the Cayley graph C(F, S) with the corresponding Schreier graph Sch(F, X, S), and in this case every eigenvalue of Sch(F, X, S) is eigenvalue of C(F, S) [177]. Consequently, if a family

194 | 5 Expanders of Cayley graphs is an expander family, the same holds for the corresponding connected Schreier graphs. In this context, if SL2 (i) is the group of the 2 × 2 matrices with determinant 1 over the ring of integers mod i, then the set consisting of 1 ( 0

±1 1 ) and ( 1 ±1

0 ) 1

1 [217]. This group acts on generates a Cayley graph whose spectral gap is at least 2000 2 the set ℤi . The corresponding Schreier graphs Sch(SL2 (i), ℤ2i , S) are 4-regular subgraphs of Margulis expanders, and so every connected component of such a Schreier graph has spectral gap bounded by the same constant.

5.5 Codes from graphs In the coding theory we usually deal with a system of rules to convert some data into another form of representation called a code. This procedure is known as an encoding. The reverse process that converts code symbols back into the corresponding data is known as a decoding. One of the crucial points in the coding theory is constructing of codes that can be encoded and decoded in an efficient⁵ way. It appears that some significant codes are based on specified graphs. For example, recall from literature that a Gray code is a binary numeral system in which two successive numbers differ in only one binary digit. For example, a 2-binary Gray code (including all 2-binary words) is given by 00, 01, 11, 10. Observe that all other 2-binary Gray codes are obtained from this one by reversing the order and/or making circular shifts. By identifying all such codes, the reader may arrive at a generalization stated in the following exercise. Exercise 5.5.1 ([232]). Prove that there is a bijection between the set of r-binary Gray codes and the set of Hamiltonian cycles in the r-dimensional cube Q r . So, each Gray code may be identified with the corresponding Hamiltonian cycle. These codes are widely used to facilitate error corrections in digital communications. An error correction is a technique used for controlling errors in data transmission over an unreliable noisy communication channel. The main idea is the sender to encodes the message by using an error correcting code (for short, ECC). An ECC allows the receiver to detect and correct all or a limited number of errors that may occur anywhere in the message transmitted.

5 As we will see, there is a precise meaning of ‘efficient’.

5.5 Codes from graphs | 195

In this chapter we meet a curious role of expanders in obtaining such ECCs. We start with an example concerning random walks in expanders encountered in Section 5.3. Subsequently, we give a strict definition of ECC together with related results. Finally, we consider a family of ECCs based on graphs with comparatively large expansion parameters.

5.5.1 A probabilistic algorithm The following algorithm is a modification of the Miller–Rabin primality test [252] (a polynomial time probabilistic algorithm that determines whether a given number is prime or not). Given a language X consisting of binary words, let Alg be a probabilistic polynomial time algorithm that uses d random bits to decide whether a given input x ⁶ belongs to X or not. Precisely, Alg takes a d-binary word s and computes a boolean function f(x, s) as follows. If x ∈ L, then f(x, s) = 1. For otherwise, if x ∉ L then the probability that f(x, s) = 1 (over the choice of s) is at most b. In other words, the algorithm never fails for x ∈ X, while for every x ∉ X the probability of false affirmative answer (that occurs if Alg returns 1) does not exceed b. Decreasing the probability of false affirmative answer is naturally expected, and a simple way to do this is the repetition of the same procedure. So, consider the same algorithm Alg but applied k times, each time with random and independent d-binary word si (1 ≤ i ≤ k). Let this new algorithm (Alg repeated k times) return 1 if f(x, si ) = 1 for all i, and 0 otherwise. From the independence in choice of si ’s, if x ∈ X the algorithm will return 1, while if x ∉ X the probability that it will return 1 is b k . In this way, we decreased the probability of false answer to b k at linear cost dk. Using expanders, we can do even more. Choose a strongly explicit (2d , r, c)-graph G whose vertex set consists of d-binary words for which c is sufficiently smaller than b from the previous algorithm (while r is not strictly specified). The new algorithm chooses a d-binary word (which is simultaneously a vertex of G) s1 , then form a random k-walk s1 , s2 , . . . , sk and for each vertex si it computes the same boolean function as before. Similarly to the above, if x ∈ X the algorithm will return 1, while if x ∉ X the probability that it will return 1 is (b + c)k [6, 177]. Here, we decreased the probability of false affirmative answer to (b + c)k at cost d + k⌈log r⌉ = d + O(k) (since the graph is strongly explicit, the walk is constructed in specified time).

6 All binary words may be considered as vector columns.

196 | 5 Expanders

noisy channel sender

encoder

decoder

receiver

Fig. 5.2. Data transmission.

5.5.2 Error correcting codes Here is a short introduction to codes specified in the title. For more details, the reader can consult some of books [181, 316]. Consider a transmission of some data from one individual (sender) to another (receiver) over an unreliable medium (noisy channel). Before transmission the corresponding data is encoded into its binary code, and upon reception the code is decoded to retrieve the original data (see Figure 5.2). But, due to the nature of noisy channel, some of bits transmitted may change their value (from 0 to 1 or vice versa) in which case we say that these bits are flipped. ECCs use additional bits (apart from those for encoding the data) to allow the receiver to correct an amount of flipped bits. Formally, a code C is a set of r-binary words called codewords which are considered as r × 1 vectors. If so, then we say that r is the block length (usually abbreviated to length) of C. The weight of a codeword is the number of 1’s contained. The distance distH (x, y) between the codewords x and y is the Hamming distance, that is the number of coordinates in which these codewords differ. The distance of C is dist(C) = min{distH (x, y) : x, y ∈ C}. We may simultaneously deal with the relative distance defined by reldist(C) = dist(C) r . Usually the data for transmission is divided into a finite number of separated messages, where each of them is encoded by a single codeword. Another parameter is a rate of C, denoted rate(C), which determines the number of bits in each codeword used for encoding the corresponding message, that is if a codeword has r bits then the rate(C)r out of them are message bits and the remaining (1 − rate(C))r bits are determined by the choice of message bits. A simple way to decode the received message y is to check whether the message sent was encoded by the code x that is closest to y. This method is correct whenever the number of bit errors does not exceed ⌊ dist(C)−1 ⌋. 2 Consider now a family C of codes of fixed length. In practice, all these codes are expected to have efficient encoding, decoding and error correcting procedure. Therefore, it is natural to expect that all codewords of a single code C ∈ C are well separated so they assure a quick error correcting. In other words, dist(C) should not be too small and preferably bounded from below by a constant. Moreover, the number of additional bits in each codeword (those that do not correspond to the message) should be mini-

5.5 Codes from graphs | 197

mized, that is the rate should be as largest as possible and again preferably bounded from below by a constant. Strictly speaking, a family C of codes with r-binary codewords is asymptotically good if there exist fixed constants δ and ρ such that dist(C) > δr and rate(C) > ρ, for all C ∈ C. The same family is said to be efficient if both encoding and decoding with at most δr 2 bit errors can be realized in a polynomial time in r. Constructing asymptotically good efficient codes is one of the most significant problems in the coding theory. If k = rate(C)r, a code C is called linear of dimension k and length r if it represents a k-dimensional linear subspace of {0, 1}r . Encoding in linear codes can efficiently be performed in a polynomial time by using the corresponding basis. Moreover, under some constraints it may be performed linearly [177]. Note that, as long as we are dealing with the efficiency of codes, the polynomial time is sufficient. Exercise 5.5.2. Prove that the distance of a linear code is equal to the minimal weight of a non-zero codeword in it. A construction of expander codes We give one construction of ECCs that is based on expanders. Such codes belong to the family of linear low complexity codes introduced by Tanner [309]. The construction deal with a family of linear codes and produce a new asymptotically good and efficient family of codes called expander codes. For similar results, the reader may consult [63, 134, 177, 205, 282, 314, 320]. Start with a single linear code C of length r (belonging to the specified family) and an (n, r)-graph G having a comparatively large expansion parameter. Using these ingredients, Spielman [287, 288] constructed a code C(G, C) of length nr 2 in the following way. Each bit of the code corresponds to an edge in G, and for each vertex of G we require that all the edges incident must form a codeword in the initial code C (for this purpose, all the edges incident with the same vertex must be labelled in any but fixed order). This requirement yields (1 − rate(C))r linear constraints for each of the n vertices of G, which gives (1 − rate(C))nr constraints in total. Observe that each codeword x in C(G, C) may be considered as the characteristic vector of a set of edges in G. Note that these edges are exactly those which induce the codewords of the initial code at each vertex of G. Observe that a code C(G, C) is not unique, that is it depends on the mentioned edge labelling. So far so good, but why the code C(G, C) is interesting to us? First, according to the construction, the rate is 2rate(C) − 1. In addition, the code distance is bounded from below. Indeed, a codeword of C(G, C) with low weight corresponds to a small set of edges. Since G has a large expansion parameter (algebraically, η(G) ≥ ε > 0), we conclude that a small set of edges is incident with a large number of vertices. Thus, at least one of these vertices is incident with fewer than reldist(C)r of these edges, but then these edges cannot induce a codeword of C at that vertex. In other words, bearing

198 | 5 Expanders in mind Exercise 5.5.2, we conclude that dist(C(G, C)) cannot be too small. Formally, after the straightforward computation based on the last arguments, we get reldist(C(G, C)) > (

reldist(C) − 1−

r−ε r

r−ε r

2

) ,

which means that the relative distance is bounded from below. Apparently, the same conclusion holds for the distance, and so this family of expander codes is asymptotically good. Since all codes are linear, the encoding is efficient. Considering decoding, Spielman provided a linear time algorithm that maps any r-binary vector received to a codeword of C whenever the distance between them is at most a given positive constant. The parallel algorithm consists of a logarithmic number of steps. In each of them, every vertex the incident edges of whose are at distance dist(C) from a codeword sends a 4 flip message to every edge that differs from that codeword. Next, the edges that receive at least one flip message flip their bits. A detailed analysis of this algorithm based on computing the number of corrupted edges at the end of each step, shows its linearity (for more details, see the corresponding references). Observe also that this algorithm allows a correct edge to receive a flip message, and so to become corrupted in a single step but nevertheless the number of corrupted edges decreases over the algorithm.

6 Distance matrix of regular graphs Section 6.1 is reserved for the definition of the distance matrix and the corresponding D-spectrum, basic properties and a Hoffman-like polynomial related to transmission regular graphs. In Section 6.2 we compute the D-spectrum of certain particular regular graphs. More precisely, we show that the D-spectrum of any graph with diameter 2 can be expressed in terms of its spectrum and main angles. Using this expression, we compute the D-spectrum of a strongly regular graph in terms of its parameters. We also consider regular graphs with diameter 3 or 4. The D-spectrum of distance-regular graphs is considered in Section 6.3. In particular, we show that the D-spectra of these graphs are determined by their spectra. We also show that any distance-regular graph with diameter D has at most D + 1 distinct D-eigenvalues and present a sequence of graphs attaining (or not attaining) this bound. Section 6.4 deals with regular graphs that have exactly one positive D-eigenvalue and those with more positive than negative D-eigenvalues.

6.1 Definition and a Hoffman-like polynomial The distance matrix D of a connected graph G with n vertices is defined by D = (d ij ), where d ij = d(i, j). In other words, the (i, j)-entry is equal to the distance between vertices i and j. We denote the eigenvalues of the distance matrix by ϱ1 , ϱ2 , . . . , ϱ n and, as before, we assume that ϱ i ≥ ϱ j holds whenever i < j. The matrix D is obviously symmetric, but also irreducible (this can easily be checked), and so its largest eigenvalue is simple. Following adopted terminology, throughout the chapter we use abbreviations like D-eigenvalues, D-spectrum, and so on. There are some graph invariants that are easily accessible from the distance matrix (pairwise distances between vertices, the diameter, vertex degrees, etc.). A study on the D-spectrum dates back to the early 1970s when Graham and Pollack [149] addressed a problem in data communication system to the number of negative D-eigenvalues of the corresponding graph. They also proved that any tree with at least two vertices has one positive and all other negative D-eigenvalues. For a branch of applications in computer science or chemistry the reader may consult [18, 155, 222]. We just mention that, by the last reference, the difference between DOI: 10.1515/9783110351347-006

200 | 6 Distance matrix of regular graphs the smallest positive and the largest negative D-eigenvalue can be used as a measure of network complexity. Certain properties of the D-spectrum are reversed with respect to the spectrum of the adjacency matrix. By definition, an entry of D is smaller if two vertices are closer. In other words and in an informal sense, a larger connectivity of a graph produces smaller entries in the corresponding distance matrix. In light of this observation, the following result can be expected. For any connected graph G with n vertices, we have ϱ1 (K n ) ≤ ϱ1 (G) ≤ ϱ1 (P n ), with equality if and only if G ≅ K n or G ≅ P n . See [45] for the first inequality, [265] for the second and compare the inequalities (1.7). The transmission of a vertex i is the sum of the distances from i to all other vertices of a graph. Clearly, it is equal to the sum of the ith row (or column) of D. A graph is called r-transmission regular if the transmission of all its vertices is equal to r. The reader may observe that a transmission regular graph does not need to be regular. It is also easy to see that the largest D-eigenvalue of an r-transmission regular graph is r. The following result is a counterpart of Theorem 2.1.6 concerning the Hoffman polynomial. The proof is similar. Theorem 6.1.1 (Indulal and Gutman [184]). For an r-transmission regular graph G with the distance matrix D there exists a polynomial P such that P(D) = J, where J is all-1 matrix. In this case we have k

P(x) = n ∏ i=2

x − ϱ̇ i , r − ϱ̇ i

where n is the number of vertices and r, ϱ̇ 2 , . . . , ϱ̇ k are all distinct D-eigenvalues of G.

6.2 D-spectrum of certain regular graphs The distance matrix of a graph is more complex than the (0, 1)-adjacency matrix, and therefore computing D-spectra is a much more intense problem. The D-spectrum of some non-regular graphs (like paths, stars, some other trees or various graph compositions) can be found in [15, 130, 182, 184, 255, 265, 290]. Here we pay more attention to regular graphs. In particular, whenever possible, we try to determine a connection between the D-spectrum and the spectrum of a regular graph under consideration. Observe first that the adjacency matrix and the distance matrix coincide in the case of any complete graph, and thus the D-spectrum of K n consists of n − 1 and n − 1 numbers all equal to −1. In addition, concerning the D-spectrum of a graph having 2 distinct D-eigenvalues and using eigenvalue interlacing, we easily conclude that the diameter of such graphs must be 1 making them non-trivial complete [183].

6.2 D-spectrum of certain regular graphs |

201

Caporossi et al. (cf. [15]) proved that the D-spectrum of a complete multipartite graph with n i vertices in the part i (1 ≤ i ≤ k) contains the eigenvalue j − 2 with multiplicity at least p j − 1 and the eigenvalue −2 with multiplicity at least ∑i≥2 (j − 1)p j , where p j = |{i : n i = j}| and p j ≥ 2 holds. Consequently, one can derive a conclusion for complete multipartite regular graphs. In particular, since the cocktail party graph CP(n) can be considered as a complete n-partite regular graph with two vertices in every part, we get that its D-spectrum consists of 2n, 0 with multiplicity n−1 and −2 with multiplicity n. Moreover, the D-spectrum of the complete bipartite graph K n1 ,n2 consists of n1 + n2 − 2 ± √n21 − n1 n2 + n22 and −2 with multiplicity n1 + n2 − 2. Setting n1 = n2 = n, we get that the D-eigenvalues of K n,n are 3n − 2, n − 2 and −2 with multiplicity 2(n − 1). The D-spectrum of a cycle is left as an exercise. Exercise 6.2.1 ([147, 179]). Prove that the D-eigenvalues of an even cycle are given by ϱ1 =

n2 4

and, for 2 ≤ i ≤ n, 0, { ϱi = { π 2 { −cosec ( n i) , while for an odd cycle they are given by ϱ1 =

if i is even, if i is odd,

n2 − 1 4

and, for 2 ≤ i ≤ n, 1 2 π { − 4 sec ( 2n i) , ϱi = { 1 2 π { − 4 cosec ( 2n i) ,

if i is even, if i is odd.

Stevanović and Indulal [305] considered the D-spactrum of the join of two regular graphs. Theorem 6.2.2 ([305]). The D-spectrum of the join of two regular graphs G1 and G2 consists of the eigenvalues −λ ij − 2 (1 ≤ i ≤ 2, 2 ≤ j ≤ n i ) and two more eigenvalues of the form n1 + n2 − 2 −

r1 + r2 √ r1 − r2 2 ± (n1 − n2 − ) + n1 n2 , 2 2

where r i = λ i1 ≥ λ i2 ≥ ⋅ ⋅ ⋅ ≥ λ in i are the eigenvalues (of the adjacency matrix) of G i (1 ≤ i ≤ 2). Proof. The distance matrix of this join has the form

202 | 6 Distance matrix of regular graphs

2(J − I) − A(G1 ) D=( J n2 ×n1

J n1 ×n2 ). 2(J − I) − A(G2 )

The D-eigenvalues that correspond to the eigenvalues −λ ij − 2 (1 ≤ i ≤ 2, 2 ≤ j ≤ n i ) are obtained in the following way. If λ is an eigenvalue of (say) G1 with an eigenvector x satisfying jT x = 0, then −λ − 2 is the D-eigenvalue of the join with the D-eigenvector whose first n1 coordinates coincide with the coordinates of x, while the remaining coordinates are 0’s. And similarly for G2 (there the first n1 coordinates are 0’s, while the remaining n2 coordinates coincide with the coordinates of the corresponding eigenvector). Since all D-eigenvectors obtained in this way are orthogonal to the vectors of the form (jT | OT )T and (OT | jT )T , we conclude that the remaining two D-eigenvectors have the form (ajT | bjT )T , for some real numbers a and b. Solving D(ajT | bjT )T = ϱ(ajT | bjT )T in ϱ (where we use the identities A(G i )j = r i j, 1 ≤ i ≤ 2), we get the remaining pair of D-eigenvalues. Observe that the result holds even if some of regular graphs considered is disconnected. If so, then the D-spectrum of K n1 ,n2 also follows from this result. (The last proof technique is similar to that of Theorem 2.3.4.) Regular graphs with a small diameter Notice that, apart from cycles, all regular graphs considered so far have small diameter (1 or 2). In order to obtain some generalizations, we reproduce some of our results reported in [13]. The first of them concerns all graphs (not necessary regular) with diameter 2 and also can be deduced from a work of Cvetković and Rowlinson [88] (see also [246]). The following proof can also be found in a modified form in [89, Proposition 2.3.1]. Theorem 6.2.3. The characteristic polynomial of the distance matrix of a graph G with n vertices and diameter 2 is given by k

D G (x) = (−1)n P G (−x − 2) (1 − 2n ∑ i=1

β2i

x + 2 + θ̇ i

),

where θ̇ i and β i (1 ≤ i ≤ k) are all distinct main eigenvalues of the adjacency matrix and the corresponding main angles. Proof. Since the diameter of G is 2, we conclude that its distance matrix can be expressed as D = 2(J − I) − A, and so we have D G (x) = det ((x + 2)I + A − 2J). Denote the matrix (x + 2)I + A by B and let, for U ⊆ ℕn = {1, 2, . . . , n}, B U stand for the matrix obtained from B by replacing all its columns indexed by U with −2j. Using the multilinearity of a determinant, we get det ((x + 2)I + A − 2J) = ∑ det(B U ), U⊆ℕn

6.2 D-spectrum of certain regular graphs |

203

where the summation goes over 2n summands. Since the corresponding determinant is equal to 0 whenever |U| > 1, the last term is equal to n

det (B) + ∑ det(B i ), i=1

where we write B i instead of B{i} . Using the Laplacian development over the ith column ̂ ij for the (i, j)-cofactor of B, we get of B i and writing B n

n

n

̂ ij = −2jT adj(B)j = −2jT det(B)B−1 j. ∑ det(B i ) = −2 ∑ ∑ (−1)i+j B i=1

i=1 j=1

Setting this in the previous expression and using the spectral decomposition of B (recall the identity (1.5) on page 11), we get k

D G (x) = det(B) (1 − 2jT B−1 j) = (−1)n P G (−x − 2) (1 − 2 ∑ i=1

jT P i j

x + 2 + θ̇ i

),

where P i represents the orthogonal projection of ℝn onto the eigenspace of θ̇ i . The assertion follows by jT P i j = nβ2i . Since a connected regular graph has exactly one main eigenvalue, we immediately obtain the following corollary. Corollary 6.2.4. The characteristic polynomial of the distance matrix of a connected r-regular graph G with n vertices and diameter 2 is given by D G (x) = (−1)n P G (−x − 2) (1 −

2n ). x+2+r

This result is also obtained in [184]. Since the diameter of any connected strongly regular graph is 2, using the expressions (3.17) and (3.18) together with the last corollary, we get another result. Corollary 6.2.5. The D-spectrum of a connected strongly regular graph with parameters (n, r, a, b) consists of 2(n − 1) − r and the numbers 1 − (a − b ± √(a − b)2 + 4(r − b)) − 2 2 with multiplicity 1 2r + (n − 1)(a − b) (n − 1 ∓ ). 2 √(a − b)2 + 4(r − b) Exercise 6.2.6 ([184]). Prove that the D-spectrum of the extended bipartite double (defined on page 50) of an r-regular graph with n vertices and diameter 2 consists of the numbers 5n − 2r − 4, 2r − n, 2λ i and −2(λ i + 2) for 2 ≤ i ≤ n.

204 | 6 Distance matrix of regular graphs Theorem 6.2.3 provides the D-spectrum in terms of the spectrum of a connected (not necessary regular) graph with diameter 2. The corresponding formula is convenient in the case of graphs with a small number of main eigenvalues. For larger diameter, we first consider bipartite semiregular graphs. Theorem 6.2.7 ([13]). The characteristic polynomial of the distance matrix of a bipartite (r, s)-semiregular graph G with diameter 3 is determined by

D G (x)

=

(−1)n1 +n2

P G (− 12 (x + 2)) 1 4 (x

+ 2)2 − rs

⋅ (x2 − 2(n1 + n2 − 2)x − 5n1 n2 − 4(n1 + n2 − rs + 3rn1 + 1)) , where n1 (resp. n2 ) is the number of vertices of degree r (resp. s). Proof. The distance matrix of G can be expressed as D = 2J − 2I − 2A + A(K n1 ,n2 ) (where A is the adjacency matrix of G), and thus we have D G (x) = det ((x + 2)I + 2A − A(K n1 ,n2 ) − 2J). Denoting the matrix (x + 2)I + 2A − A(K n1 ,n2 ) by B and using the fact that the matrices I, A and A(K n1 ,n2 ) commute, we arrive at det(B) = (−1)n1 +n2 (x + 2 + (2λ1 − √n1 n2 ))(x + 2 − (2λ1 − √n1 n2 ))

P G (− 12 (x + 2)) 1 4 (x

+ 2)2 − rs

.

Assume that r ≠ s. Recall that every bipartite semiregular graph has 2 main eigenvalues ±√rs (if necessary, see [260]). The eigenspaces of the eigenvalues ±√rs of A coincide with the eigenspaces of the eigenvalues ±√n1 n2 of A(K n1 ,n2 ), and so we have D G (x)

=

det(B) (1 − 2jT B−1 j)

=

det(B) (1 − 2 (

jT P 2 j jT P 1 j + )) , (6.1) x + 2 + 2λ1 − √n1 n2 x + 2 − 2λ1 + √n1 n2

where P1 (resp. P2 ) represents the orthogonal projection of ℝn onto the eigenspace 2 of √rs (resp. −√rs). The assertion follows by the equalities jT P1 j = n1 +n 2 + √n 1 n 2 and n1 +n2 T j P2 j = 2 − √n1 n2 . The special case r = s implies n1 = n2 when G reduces to a regular graph with a unique main eigenvalue λ1 = r. The last summand in (6.1) is lost and the assertion follows by jT P1 j = 2n1 . The last result can be rephrased in the following way. Corollary 6.2.8. Let G be a graph described in the previous theorem. If its eigenvalues are √rs, λ2 , . . . , −√rs, then its D-eigenvalues are n1 + n2 − 2 ± √n21 + n22 + 7n1 n2 + 4rs − 12rn1 and −2λ i − 2 for 2 ≤ i ≤ n1 + n2 − 1.

6.2 D-spectrum of certain regular graphs |

205

If, in addition, G is regular then we have the following assertion. Corollary 6.2.9. The characteristic polynomial of the distance matrix of a bipartite r-regular graph with 2n vertices and diameter 3 is determined by

D G (x) =

P G (− 12 (x + 2)) 1 4 (x

+ 2)2 − r2

(x2 − 4(n − 1)x − 8n − 5n2 − 4r2 + 12rn + 4) .

(6.2)

In other words, if the eigenvalues of G are r, λ2 , . . . , −r, then its D-eigenvalues are 5n − 2r − 2, 2r − n − 2 and −2λ i − 2 for 2 ≤ i ≤ 2n − 1. We now increase the diameter by 1 and consider the connected incidence graphs of symmetric 2-class PBIBDs. These graphs must be bipartite and regular. If G is such a graph, then recall from Subsection 3.8.2 that G is based on a strongly regular graph, say H. It also follows from the identity (3.41) that G has 3 distinct non-negative eigenvalues. If its diameter is 4, then the second parameter l2 in (3.41) is 0, and so according to the notation introduced on page 119, we may briefly say that G is the incidence graph of a SPBIBDl1 ,0 (H). Here is a theorem. Theorem 6.2.10 ([13]). Let G be a bipartite r-regular graph with 2n vertices and diameter 4. If G is the incidence graph of a SPBIBDl1 ,0 (H) and the dual SPBIBDl1 ,0 (H)∗ is also a symmetric 2-class PBIBD with the same parameters, then: (i)

If the spectrum of G is [ ± r, (±θ2 )m2 , 0m3 ], then its D-spectrum is [7n − 4 − 2r − m2 m3 2θ22 2r l1 − 4 ± 2θ 2 ) , ( l1 ) ]; If the spectrum of G is [ ± r, (±θ2 )m2 , (±θ3 )m3 ] (where θ3 ≠ 0), then its D-spectrum m2 2θ22 2r is [7n − 4 − 2r − l21 (r2 − r), n + 2r − 4 − l21 (r2 − r), ( 2r l1 − l1 − 4 ± 2θ 2 ) , ( l1 − 2 m 3 2θ3 l1 − 4 ± 2θ 3 ) ]. 2 2 l1 (r

(ii)

− r), n + 2r − 4 −

2 2 l1 (r

− r), ( 2r l1 −

Proof. Consider the blocking of the distance matrix induced by the two parts in our bipartite graph G: D=(

D1 D3

D2 ). D4

The adjacency matrix induced in the same way has the form (1.8), and so we immediately get D2 = 3J − 2N and D3 = 3J − 2N T . Next, since G is the incidence graph of SPBIBDl1 ,0 (H), by (3.41) we have that NN T = rI + l1 A H , and similarly since SPBIBDl1 ,0 (H)∗ is also a symmetric 2-class PBIBD with the same parameters, we have N T N = rI + l1 A(F) where A(F) is the adjacency matrix of some strongly regular graph F. Observe now that two vertices belonging to the same part are at distance 2 in G if the corresponding entry of NN T

206 | 6 Distance matrix of regular graphs (or N T N) is l1 or at distance 4 in G if this entry is 0. Hence, using the above expressions for NN T and N T N, we get D1 =

1 2 (NN T − rI) + 4 (J − I − (NN T − rI)) l1 l1

and 2 T 1 (N N − rI) + 4 (J − I − (N T N − rI)) . l1 l1 Accordingly, the matrix D may be rewritten as D4 =

2r 2 ) I − 2A − A2 , l1 l1 where A is the adjacency matrix of G, as before. Since all the matrices in this identity commute, the proof follows by direct computation. D = 4J − A(K n,n ) − (4 −

By the discussion on page 119, every bipartite distance-regular graph with diameter 4 is the incidence graph of a SPBIBDl1 ,0 (H) where H is one halved graph of G, while the dual is also a symmetric 2-class PBIBD with the same parameters based on the other halved graph of G. Thus, the previous theorem can be applied whenever a graph under consideration is distance-regular. In the next section we give more results on these graphs.

6.3 D-spectrum of distance-regular graphs Let G be a distance-regular graph with diameter D and A i the adjacency matrix of the distance-i graph G i (1 ≤ i ≤ D), that is the (0, 1)-matrix in which an entry is 1 precisely when the corresponding vertices are at distance i. Observe that the distance matrix of G can be expressed as D = ∑D i=1 iA i . By the recurrence relation (3.7) and the subsequent text in the proof of Theorem 3.3.5, we have that, for all i (1 ≤ i ≤ D), the matrix A i can be written as a polynomial of degree i in A, so A i = P i (A). Therefore, the matrix D can be written as a polynomial of degree D in A, that is D = P(A) where P is a linear combination of P i ’s. In addition, for every eigenvalue λ of G, P(λ) is its D-eigenvalue. Since the coefficients of all P i ’s can be derived from the intersection array of G, we arrive at the following conclusion. Theorem 6.3.1. The D-spectrum of a distance-regular graph is determined by its intersection array (and consequently, by its spectrum). Since, by Theorem 3.3.5, a distance-regular graph with diameter D has D + 1 distinct eigenvalues, it follows from the above discussion that it has at most D + 1 distinct D-eigenvalues. Namely, they are P(θ i ) (1 ≤ i ≤ D + 1) where θ i ’s are all distinct eigenvalues. It may happen that P(θ i ) = P(θ j ) holds for some distinct i and j, and so the upper bound for the number of distinct D-eigenvalues does not need to be attained.

6.3 D-spectrum of distance-regular graphs |

207

The reader may observe that a distance-regular graph with diameter D is simultaneously s-transmission regular with s = ∑D i=1 ik i , where k i ’s are given by (3.5), and so Theorem 6.1.1 can be applied to these graphs. Atik and Panigrahi [16] gave an explicit way to compute the D-eigenvalues. For i, j, k ∈ {0, 1, . . . , D}, let u and v be the vertices at distance i. Then we denote by p ikj the number of vertices at distance j from u and k from v. Clearly, these numbers can be derived from the intersection array. Theorem 6.3.2 ([16]). The set of all distinct D-eigenvalues of a distance-regular graph with diameter D is equal to the set of all distinct eigenvalues of the (D + 1) × (D + 1) i matrix B = (b ij ), where b ij = ∑D k=0 kp kj . Example 6.3.3. Using Theorem 6.3.2, we conclude that the distinct D-eigenvalues of a Taylor graph with intersection array (r, s, 1; 1, s, r) are given by 3(r + 1), 0 and 1 2 2 √ 2 (2s − r − 5 ± (r + 1) + 4(r − 1)s + 4s ) (see its distinct eigenvalues on page 63). Using more sophisticated methods, the authors of [16] determined the D-spectrum of a Johnson graph J(k, d) as k ς(k, d) [ς(k, d), 0(d)−k , (− ) k−1

k−1

],

with d d k−d ς(k, d) = ∑ i( )( ). i i i=1

(6.3)

It is proved in [182] that the D-spectrum of a Hamming graph H(k, d) is given by [(k − 1)k d−1 d, 0k

d

−(k−1)d−1

, (−k d−1 )(k−1)d ] .

As mentioned, a nice property of the spectrum of a distance-regular graph with diameter D is that it has exactly D+1 distinct eigenvalues without any exception. In addition, a connected regular graph has 3 distinct eigenvalues if and only if it is strongly regular (by Theorem 3.4.7), and if a connected bipartite regular graph has 4 distinct eigenvalues then it is necessarily distance-regular (by Theorems 3.8.3 and 3.8.5). All these properties fail to hold in the case of the D-spectrum. Here is a detailed analysis. Considering first the upper bound for the number of D-eigenvalues, we can easily find graphs that do not attain it as well as those that attain. For example, concerning cycles and their D-eigenvalues given in Exercise 6.2.1, we conclude that any odd cycle has D + 1 distinct D-eigenvalues, while any even cycle has ⌈ D 2 ⌉ + 2 distinct D-eigenvalues. In other words, the upper bound D + 1 for the number of distinct D-eigenvalues is attained for all odd cycles, the quadrangle and the hexagon. The same bound is trivially attained in the case of complete graphs. By Corollary 6.2.5, the same conclusion holds in the case of connected strongly regular graphs. By the previous example, a Taylor graph has 4 (equal to D+1) distinct D-eigenvalues whenever s ≠ 3r +1,

208 | 6 Distance matrix of regular graphs and 3 distinct D-eigenvalues otherwise. By the same example, apart from complete graphs, all Johnson and Hamming graphs have 3 distinct D-eigenvalues regardless of their diameter (which does not occur in the case of the spectrum of the adjacency matrix). Next, all strongly regular graphs have 3 distinct D-eigenvalues which is nice, but the converse is not true (consider again Johnson or Hamming graphs). Finally, examples of bipartite regular but not distance-regular graphs with 4 distinct D-eigenvalues can be obtained by examining some of graphs that we already met. It is not difficult to show that none of the bipartite 6-regular graphs with 24 vertices and spectrum [±6, (±2)9 , 04 ] (the graphs G57 –G62 of Table 4.1) is distance-regular, but by Corollary 6.2.9, all of them have 4 distinct D-eigenvalues (their common D-spectrum is [46, 29 , (−2)5 , (−6)9 ]). Distance-regular graphs with diameter 3 and 3 distinct D-eigenvalues In what follows we reinterpret more results from [7, 13] concerning distance-regular graphs with 3 distinct D-eigenvalues. We first explicitly compute the D-eigenvalues of distance-regular graphs with diameter 3 in terms of intersection numbers and eigenvalues. Theorem 6.3.4 ([7]). Let G be a distance-regular graph with n vertices, intersection array {b0 , b1 , b2 ; c1 , c2 , c3 } and spectrum [b0 , θ2 m2 , θ3 m3 , θ4 m4 ] . Then the D-eigenvalues of G belong to the set

{3n −

b20 c2

− (2 −

b0 −b1 −c1 ) b0 c2

− (3 −

b0 c2 ) ,

θ2

− c2i − (2 −

b0 −b1 −c1 ) θi c2

− (3 −

b0 c2 )} ,

where 2 ≤ i ≤ 4. Proof. From the recurrence relation (3.7), we easily obtain A1 = A, A2 = b1 − c1 )A) − b0 I and A3 = J − I − A − A2 . Since D = ∑3i=1 iA i , we have D = 3J −

1 2 c2 (A

− (b0 −

1 2 b0 − b1 − c1 b0 A − (2 − ) A − (3 − ) I. c2 c2 c2

The matrices I, J, A and A2 commute, so they can be simultaneously diagonalized, and then the result follows by direct computation. In relation to the last result, the D-eigenvalues can also be computed by Theorem 6.3.2, where the parameters mentioned in that theorem are determined by intersection numbers. The proof of the last result is more elegant and the eigenvalues θ i (2 ≤ i ≤ 4), can

6.3 D-spectrum of distance-regular graphs |

209

also be eliminated (by Theorem 3.3.5). Observe that some numbers located in the mentioned set can be mutually equal. Our next result describes distance-regular graphs with diameter 3 having 3 distinct D-eigenvalues. Theorem 6.3.5 ([7]). Under the assumptions of the previous theorem, if a3 − b2 > −1 then G has 3 distinct D-eigenvalues if and only if θ2 is equal to b0 − b2 + c2 − c3 , while if a3 − b2 ≤ −1 then G has 3 distinct D-eigenvalues if and only if one of θ2 or θ3 is equal to the same value. Proof. Consider the function f(x) = −

b0 − b1 − c1 b0 x2 − (2 − ) x − (3 − ). c2 c2 c2

Let i and j be fixed distinct numbers from {2, 3, 4} for which f(θ i ) = f(θ j ) holds, and let k be the remaining number from the same set. The condition f(θ i ) = f(θ j ) is equivalent to θ i + θ j = b0 − b1 − c1 − 2c2 , while from Exercise 3.3.7 we get that the eigenvalues of G distinct from the degree are the eigenvalues of the matrix −1 (1 0

b1 b0 − b1 − c2 c2

0 ). b2 b0 − b2 − c3

Considering the trace of this matrix, we get θ k = b0 − b2 + c2 − c3 = a3 − b2 + c2 > a3 − b2 . Using the results from page 66 that culminate with the inequality (3.10), we get θ2 > a3 − b2 > θ4 and that θ3 lies between −1 and a3 − b2 . So, we arrive at k ≠ 4. Thus, if a3 − b2 > −1 then θ2 > a3 − b2 ≥ θ3 , and so k = 2. Similarly, if a3 − b2 ≤ −1 then θ2 > −1 ≥ θ3 ≥ a3 − b2 , and so in this case we have that either k = 2 or k = 3. Altogether, if θ2 = b0 − b2 + c2 − c3 (resp. θ3 = b0 − b2 + c2 − c3 ) then θ3 + θ4 = b0 − b1 − c1 − 2c2 (resp. θ2 + θ4 = b0 − b1 − c1 − 2c2 ), so it holds f(θ3 ) = f(θ4 ) (resp. f(θ2 ) = f(θ4 )), and we are done. We note in passing that examples of graphs corresponding to the last theorem are the halved 7-dimensional cube with intersection array (21, 10, 3; 1, 6, 15) or the Doro graph with intersection array (12, 10, 3; 1, 3, 8) [48]. This theorem also enables us to determine the incidence graphs of symmetric BIBDs having 3 distinct D-eigenvalues. By Theorem 3.8.3, if a bipartite distance-regular graph is an incidence graph of such a design with parameters (p, r, l), then its intersection array is given by (r, r − 1, r − l; 1, l, r).

(6.4)

Corollary 6.3.6. Let G be a bipartite distance-regular graph with intersection array (6.4). Then G has 3 distinct D-eigenvalues if and only if r(r − 4l − 1) + l(4l + 1) = 0.

(6.5)

210 | 6 Distance matrix of regular graphs Proof. Since a3 − b2 = l − r ≤ −1, by Theorem 6.3.5, we have that G has 3 distinct D-eigenvalues if and only if √r − l = |2l − r| holds, which gives the assertion. Using Theorem 3.8.4, we may include bipartite complements. Corollary 6.3.7. Let G be a bipartite distance-regular graph with intersection array (6.4), where l ≤ r − 2. Then G has 3 distinct D-eigenvalues if and only if G has 3 distinct D-eigenvalues. Proof. Using the intersection array of G given in Theorem 3.8.4 together with Corollary 6.3.6, we get that G has 3 distinct D-eigenvalues if and only if 4(p − 2r + l)2 − + 1 (this fol4(p − 2r + l)(p − r) + (p − r)2 − r + l = 0 holds. After substituting p = r(r−1) l (r−l)(r−l−1)(r(r−4l−1)+l(4l+1)) lows from the second identity of (3.34)), we get = 0, which is l2 (since l ≤ r − 2) equivalent to the condition (6.5), and then the assertion follows by the corresponding corollary. In the last two corollaries we started with a distance-regular graph and its intersection array, and then we deduced whether such a graph has 3 distinct D-eigenvalues. We continue by stating a bit general result that does not impose distance-regularity, but still requires the diameter equal to 3. Recall from page 112 the definition and parameters of a Menon design. Theorem 6.3.8 ([7, 13]). Let G be a bipartite regular graph with diameter 3. Then G has 3 distinct D-eigenvalues if and only if it is the incidence graph of a Menon design. Proof. Assume that G has 3 distinct D-eigenvalues. Since the diameter is 3, by Theorem 1.3.3, it has at least 4 distinct eigenvalues. If this number is 5 or more, then the relation (6.2) contradicts the existence of (exactly) 3 distinct D-eigenvalues. It next follows that G is the incidence graph of a symmetric BIBD (by Theorem 3.8.5), and consequently G is distance-regular (by Theorem 3.8.3). Without loss of generality we may assume that its intersection array is given by (6.4), where the intersection numbers satisfy the equation (6.5). Solving this equa√ tion in r, we get r = 4l+1±2 4l+1 . This implies that 4l + 1 is a square, say 4l + 1 = k2 . 2 Since k is odd, we may write k = 2t + 1 where t ∈ ℤ. The case t = 0 does not correspond to a graph satisfying our assumptions. We next have l = t(t + 1), which implies that either r = t(2t + 1) or r = 2t2 + 3t + 1. In the former case, the intersection array of G is given by (t(2t + 1), 2t2 + t − 1, t2 ; 1, t(t + 1), t(2t + 1)) ,

(6.6)

and thus G is the incidence graph of a symmetric BIBD with parameters (4t2 , t(2t + 1), t(t + 1)) , that is the incidence graph of a Menon design.

(6.7)

6.3 D-spectrum of distance-regular graphs | 211

In the latter case, by replacing t with t − 1 in the corresponding expression for r, in the same way we get the parameters that can be obtained from those of (6.7) by reversing the sign of t. Thus, the first implication follows. If G is the incidence graph of a Menon design, then its intersection array is given by (6.6). Applying Corollary 6.3.6 to this intersection array, we conclude the proof. Inspecting the previous proof, we can deduce that bipartite distance-regular graphs with diameter 3 and 3 distinct D-eigenvalues are always paired (i.e., obtained by changing the sign of the parameter t) with exception of the incidence graph of the − Menon design with parameters (4, 3, 2) (this is K4,4 ). In addition, not every Menon design has the incidence graph that is necessary connected. Such designs are eliminated by the assumption of the theorem (regarding the diameter). For example, by setting t = −1 in (6.7), we get a disconnected incidence graph 4K2 . (We had a similar discussion on page 112.) Exercise 6.3.9. Apply Theorem 6.3.4 to show that the D-spectrum of a Menon design (with parameters (4t2 , t(2t + 1), t(t + 1)) is given by 2

[2(8t2 − t − 1), (2(t − 1))4t , (−2(t + 1))4t

2

−1

].

Bipartite distance-regular graphs with diameter 4 and 3 distinct D-eigenvalues Imitating the proof of Theorem 6.3.4, we get the following result. Theorem 6.3.10 ([7]). Let G be a bipartite distance-regular graph with 2n vertices, intersection array {b0 , b1 , b2 , b3 ; c1 , c2 , c3 , c4 } and spectrum [b0 , θ2 m2 , 0m3 , (−θ2 )m2 , −b0 ] . Then the D-eigenvalues of G belong to the set

{8n − (4 −

b30 c2 c3

−

2b0 c2 ),

2b20 c2

− (3 −

−(4 −

b0 +b1 c2 c2 c3 )b 0 3

θ2 2b0 c2 ), c2 c3

(3 −

−

2θ22 c2

θ3

− (4 −

2b0 c2 ),

+ (3 −

b0 +b1 c2 c2 c3 )θ 2

b0 +b1 c2 c2 c3 )b 0

− (4 −

− c2 c2 3 −

2θ22 c2

− (4 −

− (3 −

b0 +b1 c2 c2 c3 )θ 2 −

b30 2b0 c2 ), c2 c3

−

2b20 c2 +

2b0 c2 )}.

Obviously, the D-spectrum of G contains between 3 and 5 distinct D-eigenvalues. There are more restrictions on intersection numbers caused by bipartiteness of G. Since a i = 0 (see Exercise 3.3.1), we have b i + c i = b0 . Even more, since the diameter is 4, we also have b1 = b0 − 1, b2 = b0 − c2 , b3 = b0 − c3 and c4 = b0 . In other words, the parameters b i (1 ≤ i ≤ 3) and c4 are determined by b0 , c2 and c3 . Therefore, from Theorem 3.3.5, we can deduce that

212 | 6 Distance matrix of regular graphs

θ22 = b0 + c2 b0 − c2 − c2 c3 .

(6.8)

We now determine whether G has less than 5 distinct D-eigenvalues. Theorem 6.3.11. Under the assumptions of the previous theorem, G has less than 5 distinct D-eigenvalues if and only if G ≡ C8 or at least one of the equalities 󵄨󵄨 2c2 c3 󵄨󵄨󵄨󵄨 󵄨 θ2 = 󵄨󵄨󵄨b0 − 󵄨 or θ2 = c2 . 󵄨󵄨 2c3 − b0 󵄨󵄨󵄨 holds. The proof (whose details can be found in [7]) follows by defining the function f(x) = −

x3 2x2 b0 + b1 c2 2b0 − − (3 − ) x − (4 − ), c2 c3 c2 c2 c3 c2

and considering the following six cases that may occur: f(θ2 ) = f(0), f(θ2 ) = f(−θ2 ), f(−b0 ) = f(0), f(−b0 ) = f(−θ2 ), f(−b0 ) = f(θ2 ) and f(−θ2 ) = f(0). And now, we show whether the number of distinct D-eigenvalues is 3. In fact, there is exactly one solution. Theorem 6.3.12 ([13]). The 4-dimensional cube is a unique bipartite distance-regular graph with diameter 4 and 3 distinct D-eigenvalues. Proof. We easily conclude that if G is the required graph, then there must hold f(−b0 ) = f(−θ2 ) = f(0) (see the above). These two equalities next imply θ2 = c2 and b0 = 2c3 − c2 . Substituting this in (6.8), we get 2c22 − c2 c3 − 2c3 + 2c2 = 0, and so we 2 +1) have c3 = 2c2c(c < 2c2 . 2 +2

Observe that f(−b0 ) = f(−θ2 ) is possible only if 2c3 > b0 , thus c2 > b40 . Since by Theorem 3.3.3(iv), bc20 must be an integer, we have either c2 = b30 or c2 = b20 . The former case leads to the conclusion that the equality b0 = 2c3 − c2 is not satisfied. In the latter case, using the conditions upon the equality (6.8), we obtain the intersection array (4, 3, 2, 1; 1, 2, 3, 4) that corresponds to the desired graph. The D-spectrum of the 4-dimensional cube is [32, 011 , (−8)4 ]. Recall that this graph is recognized as a 4-Hadamard graph (in Example 3.8.12), and so its intersection array can be derived from that of Exercise 3.8.13.

6.4 Regular graphs constrained by D-spectrum Distance-regular graphs with one positive eigenvalue Recall from page 9 that the graphs with exactly one positive eigenvalue are non-trivial complete multipartite graphs with possible isolated vertices. This question is more in-

6.4 Regular graphs constrained by D-spectrum

| 213

triguing in the case of the D-spectrum. We mentioned at the beginning of the chapter that all non-trivial trees have exactly one positive D-eigenvalue. Koolen and Shpectorov [204] made a further progress by determining all distance-regular graphs with the same property. Here is a proofless result. Theorem 6.4.1 ([204]). A distance-regular graph has exactly one positive eigenvalue if and only if it is one of the following graphs: (i) (ii) (iii) (iv) (v) (vi) (vii)

a cocktail party graph, a cycle, a Hamming graph, a Doob graph, a Johnson graph, a double odd graph, a halved r-dimensional cube,

(viii) (ix) (x) (xi) (xii) (xiii)

the Petersen graph, the icosahedron, the dodecahedron, the Schläfli graph, one of the three Chang graphs or the Gosset graph,

The graphs from (iv), (vi) and (xiii) are unmentioned so far. A Doob graph D(k, d) is the Cartesian product of the k copies of the Shrikhande graph and the Hamming graph H(4, d). The vertices of a double odd graph DO(k) are identified with the subsets of size k or k + 1 of a set of size 2k + 1. Two vertices are adjacent precisely if one of the corresponding sets is a (proper) subset of the other. Finally, the vertices of the Gosset graph are identified with the vectors of length 8 consisting of either two 1’s and six 0’s or two − 12 ’s and six 12 ’s. Two vertices are adjacent precisely if the inner product of the corresponding vectors is 1. It is a 27-regular graph with 56 vertices. Items (i)–(vii) contain infinite families of graphs. The D-spectra of graphs (i)–(iii) and (v) are already given in this chapter, while the remaining are given in the next table (for double odd graphs, the function ς is defined by (6.3)) [1]: Graph Doob D(k, d)

D-spectrum [42k+d−1 3(2k

2k+d −6k−3d−1

+ d), 04

3(2k+d)

, (−42k+d−1 )

]

2k 2((2k+1 k )−k−1) , (− 2ς(2k+1,k) ) Double odd DO(k) [(2k + 1)(2k+1 , −(2k + 1)(2k+1 k ), 0 k ) + 4ς(2k + 1, k)] k

Halved cube 21 Q r Petersen Icosahedron Dodecahedron

[2r−3 r, 02

r−1 −r−1

r

, (−2r−3 ) ] , r ≥ 3 [15, 04 , (−3)5 ]

[18, 05 , (−3 ± √5)3 ] [50, 09 , (−7 ± 3√5)3 , (−2)4 ]

Schläfli

[36, 020 , (−6)6 ]

Chang (all three)

[42, 020 , (−6)7 ]

Gosset

[84, 048 , (−12)7 ]

Observe that all strongly regular graphs with one positive D-eigenvalue can easily be deduced from the last theorem.

214 | 6 Distance matrix of regular graphs

Fig. 6.1. Three forbidden induced subgraphs (for Exercise 6.4.2).

Exercise 6.4.2 ([255]). Prove that if none of the graphs illustrated in Figure 6.1 is an induced subgraph of an r-regular graph G with diameter 2, then Line(G) has exactly one positive D-eigenvalue equal to (n − 2)r. Optimistic strongly regular graphs A graph is called optimistic if it has more positive than negative D-eigenvalues. In 1978, Graham and Lovász [148] asked a question on whether such graphs even exist. The question was not attracted much attention, and it was answered affirmatively 36 years later by Azarija [17]. In what follows, we consider optimistic strongly regular graphs. As we will see, some of them can easily be determined. Concerning the D-spectrum of a strongly regular graph (computed in Corollary 6.2.5), we deduce that every conference graph with at least 13 vertices is optimistic. The same conclusion holds for every strongly regular graph with parameters (n2 , 3(n − 1), n, 6), where n ≥ 5 [17]. The next result follows by the same corollary. Theorem 6.4.3 ([1]). A connected strongly regular graph with parameters (n, r, a, b) is optimistic if and only if b−

2r r+b−4 ≤a< . n−1 2

Consequently, whether or not a strongly regular graph is optimistic depends only on its parameters. Here is an exercise. Exercise 6.4.4 ([1]). Prove that a connected strongly regular graph with parameters (n, r, a, a) is optimistic if and only if r ≥ a + 4. We conclude by the result concerning optimistic strongly regular graphs with optimistic complements. Theorem 6.4.5 ([1, 17]). If G is a primitive strongly regular graph with parameters (n, r, a, b), then both G and G are optimistic if and only if G is a conference graph with at least 13 vertices. Proof. The parameters (n, r, a, b) of G are given by (3.12). The previous theorem gives 2r 2r b − n−1 ≤ a. Applying the same theorem to G, we compute b − n−1 ≤ a. Using (3.12), we 2(n−r−1) 2r get n − 2r + a − n−1 ≤ n − 2(r + 1) + b, which after simplifying leads to b− n−1 ≥ a.

6.4 Regular graphs constrained by D-spectrum

| 215

Altogether, we have 2r = a, n−1 giving the equality between the multiplicities of the two non-simple eigenvalues (of the adjacency matrix) of G (cf. (3.19)). Consequently, G is a conference graph. By the above discussion, it is optimistic for n ≥ 13, and the assertion follows. b−

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Graph index antipodal 56 antipodal cover 62 Biggs–Smith 69 bipartite 3 bipartite complement 4 bipartite double 50 bipartite Kneser 87 bipartite semiregular 4 cage 41 Cayley 191 Clebsch 75 cloud 189 cocktail party 1 complement 4 complete 1 complete bipartite 4 complete multipartite 4 conference 76 connected 3 cover 188 cube 63 cubic 1 cycle 3 Desargues 87 determined by the spectrum 10 disconnected 3 distance-i 56 distance-regular 55 distance-transitive 62 dodecahedron 69 Doob 213 Doro 209 double odd 213 double star 102 Dyck 161 edge-regular 2 edge-transitive 79 Erdős–Rényi random 46 exceptional 97 expander 176 extendability 101 extended bipartite double 50 extremal strongly regular 93

F26A 161 folded 62 Foster 70 Franklin 162 generalized Latin square 82 generalized line 5 generalized odd 69 generalized quadrangle 83 geometric 67 Gosset 213 Grassmann 64 H-free 2 halved 61 Hamiltonian 3 Hamming 64 Heawood 148 hexagon 63 Higman–Sims 104 Hoffman 68 Hoffman–Singleton 41 icosahedron 63 incidence (of a block design) 105 integral 126 Johnson 63 k-colourable 3 k-partite 4 Kneser 86 Latin square 81 lattice 50 lift see cover line 5 Möbius–Kantor 162 McLaughlin 93 Moore 40 n-crossed prism 54 n-Paley 78 Nauru 129 octagon 113 octahedron 63

232 | Graph index odd 57 optimistic 214 partial geometry 83 path 3 pentagon 41 Petersen 21 Q-integral 132 quadrangle 3 r-dimensional cube 1 Ramanujan 182 random regular 47 reflexive 122 regular 1 represented in a line system 94 root 5 Schläfli 84 Schreier 193 self-complementary 18

Shrikhande 63 simple 1 star 4 strongly regular 70 subdivision 20 symmetric 79 Taylor 63 totally disconnected 3 transmission regular 200 tree 3 triangle 3 trivial 1 Tutte–Coxeter 129 twisted Grassmann 61 undirected 1 vertex-transitive 54 walk-regular 51 Wells 70

Index angles – main 13 array – orthogonal 81 ball – metric 175 block – length 196 block design 105 – 2-class partially balanced incomplete, 2-class PBIBD 113 – balanced incomplete, BIBD 107 – blocks 105 – complement 105 – dual 105 – incidence matrix 105 – partially balanced incomplete, PBIBD 106 – points 105 – symmetric 105 bound – absolute 92 – fundamental 65 – relative 89 – Terwilliger tree 60 boundary – edge 174 – vertex 177 bounds – Krein 74 chain – Markov 184 clique 2 – Delsarte 67 closure – star- 95 co-clique 2 code 194 – completely regular 67 – distance 196 – Gray 194 – linear 197 – rate 196 – relative distance 196 codewords 196 – distance (between) 196

– weight 196 complement – star 98 connectivity – algebraic 22 – edge 3 – vertex 3 constant – Kazhdan 192 cospectrality 28 cycle – Hamiltonian 3 – length 3 decoding 194 decomposition – spectral 11 degree – average 2- 40 – edge 2 – maximal 1 – minimal 1 – vertex 1 design – group divisible 114 – Hadamard 112 – Menon 112 – t- 113 – transversal 116 – triangular 115 – Witt 113 distance – between vertices 3 – Hamming 196 – spectral 28 distance-regular graph – classical parameters 65 – head 59 – imprimitive 61 – intersection array 55 – intersection numbers 55 – primitive 61 distribution – Kesten–McKay see distribution, McKay – McKay 47 – semicircle 46

234 | Index – stationary 187 – Wigner 47 edges – adjacent 1 eigenvalue 7 – eigenspace 10 – Laplacian, L- 22 – local 75 – main 13 – multiplicity 9 – Seidel, S- 26 – signless Laplacian, Q- 22 encoding 194 endvertex 1 equation – eigenvalue 11 excess – spectral 60 expanders – Margulis 178 extension – good 100 family – expander 176 family of codes – asymptotically good 197 – efficient 197 – expander 197 formula – Biggs 59 gap – spectral 9 geometry – partial 82 graph – automorphism 1 – component 3 – diameter 3 – divisor 13 – eigenvalue 8 – eigenvector 10 – energy 85 – extension 100 – girth 3 – ith subconstituent 56 – local 56

– oddgirth 45 – order 1 – regular two- 90 – size 1 – spectrum 9 – toughness 36 – two- 90 graphs – cospectral 10 – explicitly constructible 187 – isomorphic 1 – strongly explicitly constructible 188 group – action 193 inequalities – Cauchy 12 – Cheeger 179 inequality – Fisher 107 – Schwenk 40 – Stanley 40 join 4 k-walk see walk lemma – expander mixing 181 line system – 1-line extension 96 – decomposable 95 – indecomposable 95 – maximal 94 – star 94 – star-closed 94 lines – equiangular see lines, equidistant – equidistant 88 map – covering 188 matching 2 – perfect 2 matrices – simultaneously diagonalizable 8 matrix – adjacency 8 – admittance see matrix, Laplacian

Index

– all-0 7 – all-1 7 – co-Laplacian see matrix, signless Laplacian – conference 76 – distance 199 – divisor 13 – Hadamard 111 – incidence 8 – irreducible 7 – Kirchhoff see matrix, Laplacian – Laplacian 22 – non-negative 7 – normalized Laplacian 185 – permutation 7 – quotient 13 – real 7 – reducible 7 – regular Hadamard 112 – Seidel 26 – signless Laplacian 22 – symmetric 7 – transition 184 – unit 7 measure – cospectrality 28 moment – spectral 9 multigraph 4 neighbourhood – closed 1 – open 1 notation – hexadecimal 124 – λ- 9 – LCF 124 – θ- 9 number – chromatic 3 – clique 2 – independence 2 – of spanning trees 23 p-sum – complete 49 – incomplete 49 – non-complete extended, NEPS 49 parameter – expansion 174

– isoperimetric 175 – vertex expansion 177 partition – equitable 13 partners – good 100 path – end 3 – length 3 petal 5 plane – Fano 110 – finite affine 111 – finite projective 110 point – replication 105 polynomial – characteristic 8 – Hoffman 16 – Hoffman-like 200 – minimal (of a graph) 9 polynomials – predistance 58 principle – Rayleigh 7 product – Cartesian (of graphs) 49 – Kronecker (of matrices) 49 – replacement 189 – tensor (of graphs) 49 – zig-zag 190 property – Kazhdan 192 quadrangle – generalized 83 quotient – Rayleigh 7 radius – covering 67 – L-spectral 22 – Q-spectral 22 – spectral 9 rotation 189 set – good 100 – independent 2

| 235

236 | Index – star 98 sets – compatible 100 simplex 88 spectrum 9 – D- 199 – Laplacian, L- 22 – main part 13 – Seidel, S- 26 – signless Laplacian, Q- 22 spread – Laplacian spectral 43 square – Latin 81 star – centre 4 strongly regular graph – parameters 70 – primitive 71 – subconstituent 74 subgraph 2 – edge-deleted 2 – induced 2 – spanning 2 – vertex-deleted 2 submatrix – principal 7 switching – GM 172 – Seidel 26 system – line 88 – root 96 – Steiner 113 technique – star complement 100

theorem – Alon–Boppana 45 – Erdős–Ko–Rado 87 – Fermat two squares 76 – interlacing 12 – matrix-tree 23 – Pólya enumeration 6 – reconstruction 99 – Serre 43 – Smith 61 – spectral excess 60 – zig-zag 190 theory – Perron–Frobenius 7 tournament 158 – Hadamard 158 union – disjoint 4 vector – all-0 7 – all-1 7 – probability 184 vertex – good 100 – isolated 3 – pendant see endvertex – transmission 200 vertices – adjacent 1 walk 3 – closed 3 – lazy 187 – length 3 – random 184 – semi-edge 132

De Gruyter Series in Discrete Mathematics and Applications Volume 3 Stelios Georgiou, Christos Koukouvinos Combinatorial Designs ISBN 978-3-11-029408-8, e-ISBN (PDF) 978-3-11-029434-7, e-ISBN (EPUB) 978-3-11-038192-4, Set-ISBN 978-3-11-029435-4 Volume 2 Nikos Tzanakis Elliptic Diophantine Equations. A Concrete Approach via the Elliptic Logarithm, 2013 ISBN 978-3-11-028091-3, e-ISBN 978-3-11-028114-9, Set-ISBN 978-3-11-028115-6 Volume 1 Petrică C. Pop Generalized Network Design Problems. Modeling and Optimization, 2012 ISBN 978-3-11-026758-7, e-ISBN 978-3-11-026768-6, Set-ISBN 978-3-11-026769-3 www.degruyter.com