This book is a substantially enlarged version of the Cambridge Tract with the same title published in 1974. There are two major changes.

• The main text has been thoroughly revised in order to clarify the exposition, and to bring the notation into line with current practice. In the course of revision it was a pleasant surprise to find that the original text remained a fairly good introduction to the subject, both in outline and in detail. For this reason I have resisted the temptation to reorganise the material in order to make the book rather more like a standard textbook.

• Many Additional Results are now included at the end of each chapter. These replace the rather patchy selection in the old version, and they are intended to cover most of the major advances in the last twenty years. It is hoped that the combination of the revised text and the additional results will render the book of service to a wide range of readers.

I am grateful to all those people who have helped by commenting upon the old version and the draft of the new one. Particular thanks are due to Peter Rowlinson, Tony Gardiner, Ian Anderson, Robin Wilson, and Graham Brightwell. On the practical side, I thank Alison Adcock, who prepared a TEX version of the old book, and David Tranah of Cambridge University Press, who has been constant in his support.

About the book

This book is concerned with the use of algebraic techniques in the study of graphs. The aim is to translate properties of graphs into algebraic properties and then, using the results and methods of algebra, to deduce theorems about graphs.

It is fortunate that the basic terminology of graph theory has now become part of the vocabulary of most people who have a serious interest in studying mathematics at this level. A few basic definitions are gathered together at the end of this chapter for the sake of convenience and standardization. Brief explanations of other graph-theoretical terms are included as they are needed. A small number of concepts from matrix theory, permutation-group theory, and other areas of mathematics, are used, and these are also accompanied by a brief explanation.

The literature of algebraic graph theory itself has grown enormously since 1974, when the original version of this book was published. Literally thousands of research papers have appeared, and the most relevant ones are cited here, both in the main text and in the Additional Results at the end of each chapter. But no attempt has been made to provide a complete bibliography, partly because there are now several books dealing with aspects of this subject. In particular there are two books which contain massive quantities of information, and on which it is convenient to rely for ‘amplification and exemplification’ of the main results discussed here.

The condition of vertex-transitivity is not a very powerful one, as is demonstrated by the fact that we can construct at least one vertex-transitive graph from each finite group, by means of the Cayley graph construction. A vertex-transitive graph is symmetric if and only if each vertex-stabilizer Gv acts transitively on the set of vertices adjacent to v. For example, there are just two distinct 3-regular graphs with 6 vertices; one is K3,3 and the other is the ladder L3. Both these graphs are vertex-transitive, and K3,3 is symmetric, but L3 is not because there are two ‘kinds’ of edges at each vertex.

Although the property of being symmetric is apparently only slightly stronger than vertex-transitivity, symmetric graphs do have distinctive properties which are not shared by all vertex-transitive graphs. This was first demonstrated by Tutte (1947a) in the case of 3-regular graphs. More recently his results have been extended to graphs of higher degree, and it has become apparent that the results are closely related to fundamental classification theorems in group theory. (See 17a, 17f, 17g.)

We begin by defining a t-arc [α] in a graph Г to be a sequence (α0, α1,…, αt) of t + 1 vertices of Г, with the properties that {αi−1, αi} is in EГ for 1 ≤ i ≤ t, and αi−1 ≠ αi+1 for 1 ≤ i ≤ t − 1. A t-arc is not quite the same thing as the sequence of vertices underlying a path of length t, because it is convenient to allow repeated vertices.

In this chapter we investigate the relationship between primitivity and distance-transitivity. We shall prove that the automorphism group of a distance-transitive graph can act imprimitively in only two ways, both of which have simple characterizations in terms of the structure of the graph.

We begin by summarizing some terminology. If G is a group of permutations of a set X, a block B is a subset of X such that B and g(B) are either disjoint or identical, for each g in G. If G is transitive on X, then we say that the permutation group (X, G) is primitive if the only blocks are the trivial blocks, that is, those with cardinality 0, 1 or |X|. If B is a non-trivial block and G is transitive on X, then each g(B) is a block, and the distinct blocks g(B) form a partition of X which we refer to as a block system. Further, G acts transitively on these blocks.

A graph Г is said to be primitive or imprimitive according as the group G = Aut(Г) acting on VГ has the corresponding property. For example, the ladder graph L3 is imprimitive: there is a block system with two blocks, the vertices of the triangles in L3.

Proposition 22.1Let Г be a connected graph for which the group of automorphisms acts imprimitively and symmetrically (in the sense of Definition 15.5). Then a block system for the action of Aut(Г) on VГ must be a colour-partition of Г.