ALGEBRAIC GRAPH THEORY

Second Edition

NORMAN BIGGS London School of Economics

CAMBRIDGE UNKVERSITY PRESS Contents

Preface 1 Introduction

PART ONE -- LINEAR ALGEBRA IN GRAPH THEORY 2 The spectrum of a graph 3 Regular graphs and line graphs 4 Cycles and cuts 5 Spanning trees and associated structures 6 The tree-number 7 Deterninant expansions 8 Vertex-partitions and the spectrum

PART TWO - COLOURING PROBLEMS 9 The chromatic polynomial 10 Subgraph expansions 11 The multiplicative expansion 12 The induced subgraph expansion 13 The Tutte polynomial 14 Chromatic polynomials and spanning trees

PART THREE - SYMMETRY AND REGULARITY 15 Automorphisms of graphs 16 vertex-transitive graphs 17 Symmetric graphs vi Contents

18 Symmetric graphs of degree three 19 The covering-graph construction 20 Distance-transitive graphs 21 Feasibility of intersection arrays 22 Imprimitivity 23 Minimal regular graphs with given girth References Index Index

acyclic orientation 70 complete graph 8 adjacent 7 complete matching 50 adjacency algebra 9 complete multipartite graph 41 adjacency matrix 7 conductance 34 almost-complete 43 cone 66 alternating knot 105 confluence 72 angles 51 conforms 30 antipodal 177 conjugate Bell polynomials 72 antipodal r-fold covering 178 connected 10 augmentation 29 contracting 64 automorphic 178 Conway's presentations 145 automorphism 115 co-rank 25, 97 automorphism group 115 coset graph 128 cospectral graphs 12, 49 bicentroid 119 cover 50 bigrading 97 covering graph 149 bipartite 11 cube 43, 69, 140, 157, 161, 169 biplane 189 cubic graph 138 block 81 current 34 block system 173 cut 26 broken cycle 77 cut-orientation 26 Brooks's theorem 55 cut-subspace 26 buckminsterfullerene 127 cut-vertex 67 cage 181, 188, 189 cycle 25 Cayley graph 123 cycle graph 17, 65 centroid 119 , cycle-orientation 25 characteristic polynomial 8 cycle-subspace 26 chromatically unique 69 chromatic invariant 107 degree 4 chromatic number' 52 deletion-contraction 65, 72 chromatic polynomial 63 density 94 chromatic root 71 derived graph 178 circulant graph 16, 126 Desargues graph 148, 153 circulant matrix 16 diameter 10 closed walk 12 dihedral group 126 coboundary mapping 28 distance 10 cocktail-party graph 17, 68 distance matrices 13, 159 colour-class 52 distance-regular 13, 159 colour-partition 52 distance-transitive 118, 155 compatible 150 dodecahedron 69, 178 complete bipartite graph 21 double pyramid 68 Index double-transitivity 118 incidence mapping 24, 29 dual 29,43 incidence matrix 24 independent 98 edge 3 indicator function 74 edge space 23 induced subgraph 4 edge-transitive 115. 118, 120 interaction model 80 effective resistance 3G internal activity 99 eigenvalue 8 internally active 99 clectrical network 34 intersection array 157, 159 elrrnentary 44 intersection matrix 165 ends 4 intersection numbers 156 equipartition 58 Ising model 80 even subgraph 110 isoperimetric number 28, 58 excess 28, 189 isthmus 30 expansion 147 external activity 99 join 66 externally active 99 Jones polynomial 105 feasible array 168 K-chain 149 flow 29 Kelly's lemma 50 flow polynomial 110 Kirchhoff !s laws 34 forest 47 Kocay's lemma 50 Foster's census 147 Krein parameters 170 friendship theorern 171 labelled tree 104 generalized d-gon 187 ladder 69. 126 generalized line graph 21 Laplacian matrix 27 generalized polygon graph 181 Laplacian spectrum 29. 40 general graph 3 line graph 17, 120 girth 28, 76. 131. 180 logarithmic transform 82 g~aph4 loop 3 graphical regular representation medial graph 104 124. 128 minimal support 29 graph types 87 Mobius ladder 20: 42. 69. 110 Harniltonian cycle 50 modified rank polynomial 101 Hamming graph 169 modular flour 30 Hcawood graph 148. 154. 163 Moore graph 181 Hoffman-Singleton graph 189 Motzkin-Straus formula 59 homeomorphic 79> 108 negative end 24 homogeneous 120 nowhere-zero 30 homological covering 154 Hopf algebra 88 octahedron 43 hyperoctahedral graph 17 odd graphs 20: 58, 137. 161: 170 orbit 115 icosahedral group 127 orientation 24 icosahedron 69, 1.78 ilrlprimitiw 177 Paley graph 129 Index

Pappus graph 148, 154 spectruni 8 partial geometry 162 sporadic groups 172 partition function 80 square lattice 96 path graph 11 stabilizer 122 perfect code 22, 171 stabilizer sequence 133, 137, 147 permutation character 172 standard bases 24 permutation matrix 116 star graph 49 Petersen graph 20, 95, 103, 133 star types 87 planar 29 strict graph 4 positive end 24 strongly regular graph 16, 20, 159, potential 36 171 Potts model 80 subdividing 79 power 36 subgraph 4 primitive 30, 173 successor 132 principal minors 8 support 29 projective plane 163 suspension 66 proper 90 symmetric 118, 126 pyramid 68 symmetric cycle 137 symmetric design 163 quasi-separable 67 symmetric group 118, 148 quasi-separation 67 t-arc 130 rank 25 tetrahedral group 127 rank matrix 73 thermodynamic limit 94 rank polynomial 73 theta graph 86 Rayleigh quotient 54 Thomson's principle 36 Rayleigh's monotonicity law 37 topological invariant 79 reconstructible 50, 91 totally unimodular 34 reconstruction conjecture 50 tree 47, 49, 65, 119 recursive family 70, 103 tree-number 38 regular graph 14 triangle graph 19, 169 regular action 122 tridiagonal 165 resonant model 80 t-transitive 131 rewriting rules 72 Turan's Theorem 59 root systems 22 Tutte polynomial 97, 100 r-ply transitive 162 umbra1 chromatic polynomial 72 semi-direct product 150 unimodal conjecture 108 separable 67 separation 67 vertex 3 series-parallel 109 vertex-colouring 52 sextet graph 145 vertex space 23 Shannon capacity 51 vertex-stabilizer 122, 127 sides 149 vertex-transitive 115, 120, 125 simple eigenvalues 116, 125 V-function 79 spanning elementary subgraph 44 voltage 34 spanning tree 31 spectral decomposition 13 walk 9 Index 205

walk-generating function 13 weakly homogeneous 120 walk-generating matrix 12 wheel 68