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Distance-Transitive Graphs Distance-Transitive Graphs Submitted for the module MATH4081 Robert F. Bailey (4MH) Supervisor: Prof. H.D. Macpherson May 10, 2002 2 Robert Bailey Department of Pure Mathematics University of Leeds Leeds, LS2 9JT May 10, 2002 The cover illustration is a diagram of the Biggs-Smith graph, a distance-transitive graph described in section 11.2. Foreword A graph is distance-transitive if, for any two arbitrarily-chosen pairs of vertices at the same distance, there is some automorphism of the graph taking the first pair onto the second. This project studies some of the properties of these graphs, beginning with some relatively simple combinatorial properties (chapter 2), and moving on to dis- cuss more advanced ones, such as the adjacency algebra (chapter 7), and Smith’s Theorem on primitive and imprimitive graphs (chapter 8). We describe four infinite families of distance-transitive graphs, these being the Johnson graphs, odd graphs (chapter 3), Hamming graphs (chapter 5) and Grass- mann graphs (chapter 6). Some group theory used in describing the last two of these families is developed in chapter 4. There is a chapter (chapter 9) on methods for constructing a new graph from an existing one; this concentrates mainly on line graphs and their properties. Finally (chapter 10), we demonstrate some of the ideas used in proving that for a given integer k > 2, there are only finitely many distance-transitive graphs of valency k, concentrating in particular on the cases k = 3 and k = 4. We also (chapter 11) present complete classifications of all distance-transitive graphs with these specific valencies. Acknowledgements I would like to thank my supervisor, Prof. H.D. Macpherson, for his assistance and encouragement throughout the duration of the project, and for suggesting the topic in the first place. Thanks are also due to my tutor, Dr. R.B.J.T. Allenby, for (voluntarily!) reading a preliminary version of the project, to Profs. J.K. Truss and J.C. McConnell for answering my “Do you know anything about....” questions, and to the assessor for his positive comments during the preliminary assessments. I should also thank Prof. E.R. Vrscay of the University of Waterloo for supplying [26]. 3 Contents 1 Introduction 6 1.1 Basic Definitions . 6 1.2 A Little Group Theory . 7 1.3 Automorphisms . 9 1.4 Different Kinds of ‘Transitive’ . 10 2 Introducing the Distance-Transitive Graph 14 2.1 Basic Properties . 14 2.2 Distance Partitions . 17 2.3 Intersection Numbers and Intersection Arrays . 18 2.4 Distance-Regular Graphs . 24 3 Uniform Subset Graphs 27 3.1 Introduction . 27 3.2 The Johnson Graphs J(n;2;1) ................... 28 3.3 The Johnson Graphs J(n;k;k 1) . 30 − 3.4 Pretty Pictures . 33 3.5 The Odd Graphs . 33 4 Some Permutation Group Theory 39 4.1 Primitive and Imprimitive Actions . 39 4.2 Direct and Semi-direct Products . 40 4.3 Wreath Products . 41 4.4 Projective Groups . 41 5 Hamming Graphs 43 5.1 Introduction . 43 5.2 Distance-Transitivity . 44 5.3 The k-Cubes . 46 6 Grassmann Graphs 47 6.1 Introduction . 47 6.2 Distance-Transitivity . 49 6.3 Intersection Arrays . 52 4 CONTENTS 5 6.4 Linking the Grassmann and Johnson Graphs . 54 7 Linear Algebra and Distance-Transitive Graphs 56 7.1 The Spectrum and the Adjacency Algebra . 56 7.2 Distance Matrices . 60 7.3 The Intersection Matrix . 62 7.4 Algebraic Constraints on the Intersection Array . 67 8 Primitive and Imprimitive Graphs 72 8.1 Introduction . 72 8.2 Antipodal Graphs . 75 8.3 Bipartite Distance-Transitive Graphs . 77 8.4 Smith’s Theorem . 80 9 New Graphs from Old 84 9.1 Line Graphs . 84 9.2 Automorphisms of Line Graphs . 86 9.3 Eigenvalues of Line Graphs . 88 9.4 Distance-Transitive Line Graphs . 90 9.5 Bipartite Doubles . 95 10 Bounding the Diameter 99 10.1 Introduction . 99 10.2 Cubic Graphs . 99 10.3 Tetravalent Graphs . 104 10.4 Extending to Higher Valencies . 107 11 Graphs of Low Valency 110 11.1 Smith’s Program . 110 11.2 Cubic Distance-Transitive Graphs . 110 11.3 Tetravalent Distance-Transitive Graphs . 117 Chapter 1 Introduction 1.1 Basic Definitions There are many simple definitions from graph theory that the reader is probably familiar with already (if not, consult an introductory text such as Wilson [40]). However, we will include these, if only to demonstrate our terminology and nota- tion, which varies considerably between texts. Definitions 1.1.1 – Graph Theory Definitions Let Γ be a graph, with VΓ denoting the set of vertices of Γ and EΓ the set of • edges of Γ. A graph is simple if all edges join two distinct vertices, and between any pair • of vertices u;v there is at most one edge. In this project, we will always be considering simple graphs. Two vertices u;v VΓ are adjacent if there is a single edge joining them. • We write u v to2 denote this. (Note that this relation is symmetric, but not reflexive∼ or transitive.) Also, two edges are adjacent∼ if they are incident with a common vertex. The degree, or valency of a vertex v is the number of edges incident with v, • denoted deg(v). A graph Γ is said to be regular if deg(u) = deg(v) for all u;v VΓ. It is • k-regular if deg(v) = k for all v VΓ. In this case, we refer to the2valency of Γ. A 3-regular graph is frequently2 described as a cubic graph, or sometimes as a trivalent graph. A path π in Γ is a finite sequence of edges from vertex u to vertex v where • all the intermediate vertices are distinct. 6 1.2 A Little Group Theory 7 Γ is said to be connected if for any u;v VΓ there exists a path π from u to • v. Otherwise, we say Γ is disconnected.2 A maximal connected subgraph of a disconnected graph Γ is called a component of Γ. A geodesic in Γ is a path from u to v containing the least number of edges. • The distance from u VΓ to v VΓ is the least number of edges in a path • from u to v. This is denoted2 by d2(u;v). The maximum distance in a graph Γ is called the diameter of Γ. • A circuit in Γ is a path from v to v. • The girth of Γ is the length of the shortest circuit in Γ. • A graph is bipartite if VΓ = V1 ˙V2 and each edge of Γ has one end in V1 and • [ the other end in V2. It can be shown that Γ is bipartite if and only if it has no circuits of odd length. An isomorphism from a graph Γ to a graph ∆ is a bijective function ϕ : VΓ • V∆ such that for u;v VΓ, ϕ(u) ϕ(v) (in ∆) if and only if u v (in Γ).! In Γ 2 ∆ ∼ Γ ∆ ∼ this case we say and are isomorphic, denoted by ∼= . An isomorphism from a graph Γ to itself is called an automorphism of Γ. • 1.2 A Little Group Theory It is assumed that the reader is already familiar with basic group theory, for example groups, Abelian groups, subgroups, direct products, cosets, Lagrange’s Theorem, homomorphisms, isomorphisms and factor groups. (If not, see a book such as Al- lenby [2] or Gallian [18].) In this project, we are particularly concerned with the concept of a group action on a set (specifically on the set of vertices of a graph). Formally, this is defined as follows: Definition 1.2.1 A group action of a group G on a set X is a function ρ : X G X satisfying × ! for all x X and for all g;h G, x(gh) = (xg)h; • 2 2 for all x X, xe = x (where e is the identity element of G). • 2 An action is said to be faithful if the only element of G fixing all elements of X is the identity. (The basics of group actions are covered well in Slomson [31]). This definition leads us straight away to a number of others: 8 Introduction Definition 1.2.2 The orbit of an element x X is the set OrbG(x) = xg g G (i.e. all elements 2 f 2 g of X that are the image of x under some permutation g G). OrbG(x) is a subset of X. 2 Definition 1.2.3 Suppose G is a permutation group acting on a set X. We say G is transitive on X if, for all x;y X, there exists g G such that xg = y. 2 2 We can also characterise transitivity in this way: Proposition 1.2.4 G is transitive on X if and only if, for all x X, OrbG(x) = X. 2 Proof: Suppose G is transitive on X. Then, by definition 1.2.3, for any x X there exists 2 some g G such that xg = y for all y X. Thus y OrbG(x) for all x X. 2 2 2 2 Conversely, suppose OrbG(x) = X for all x X. Then, by definition 1.2.2, for any x;y X, there exists g G such that xg = y2. So G is transitive on X . 2 2 Definition 1.2.5 The stabiliser of an element x X is the set StabG(x) = g G xg = x (i.e. all 2 f 2 g permutations of x that map x to itself).
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