1 Introduction 1
part one - LINEAR ALGEBRA IN GRAPH THEORY 7
2 The spectrum of a graph 7
3 Regular graphs and line graphs 14
4 Cycles and cuts 23
5 Spanning trees and associated structures 31
6 The tree-number 38
7 Deteminant expansions 44
8 Vertex-partitions and the spectrum 52
part two COLOURING PROBLEMS 63
9 The chromatic polynomial 63
10 Subgraph expansions 73
11 The multiplicative expansion 81
12 The induced subgraph expansion 89
13 The Tutte polynomial 97
14 Chromatic polynomials and spanning trees 106
part three - SYMMETRY AND REGULARITY 115
15 Automorphisms of graphs 115
16 Vertex-transitive graphs 122
17 Symmetric graphs 130
18 Symmetric graphs of degree three 138
19 The covering-graph construction 149
20 Distance-transitive graphs 155
21 Feasibility of intersection arrays 164
22 Imprimitivity 173
23 Minimal regular graphs with given girth 180
References 191
Index 202