Spectra of graphs Andries E. Brouwer Willem H. Haemers 2 Contents 1 Graph spectrum 11 1.1 Matricesassociatedtoagraph . 11 1.2 Thespectrumofagraph ...................... 12 1.2.1 Characteristicpolynomial . 13 1.3 Thespectrumofanundirectedgraph . 13 1.3.1 Regulargraphs ........................ 13 1.3.2 Complements ......................... 14 1.3.3 Walks ............................. 14 1.3.4 Diameter ........................... 14 1.3.5 Spanningtrees ........................ 15 1.3.6 Bipartitegraphs ....................... 16 1.3.7 Connectedness ........................ 16 1.4 Spectrumofsomegraphs . 17 1.4.1 Thecompletegraph . 17 1.4.2 Thecompletebipartitegraph . 17 1.4.3 Thecycle ........................... 18 1.4.4 Thepath ........................... 18 1.4.5 Linegraphs .......................... 18 1.4.6 Cartesianproducts . 19 1.4.7 Kronecker products and bipartite double. 19 1.4.8 Strongproducts ....................... 19 1.4.9 Cayleygraphs......................... 20 1.5 Decompositions............................ 20 1.5.1 Decomposing K10 intoPetersengraphs . 20 1.5.2 Decomposing Kn into complete bipartite graphs . 20 1.6 Automorphisms ........................... 21 1.7 Algebraicconnectivity . 22 1.8 Cospectralgraphs .......................... 22 1.8.1 The4-cube .......................... 23 1.8.2 Seidelswitching. 23 1.8.3 Godsil-McKayswitching. 24 1.8.4 Reconstruction ........................ 24 1.9 Verysmallgraphs .......................... 24 1.10 Exercises ............................... 25 3 4 CONTENTS 2 Linear algebra 29 2.1 Simultaneousdiagonalization . 29 2.2 Perron-FrobeniusTheory . 29 2.3 Equitablepartitions . 32 2.3.1 Equitable and almost equitable partitions of graphs . 32 2.4 TheRayleighquotient. 33 2.5 Interlacing .............................. 34 2.6 Schur’sinequality .......................... 35 2.7 Schurcomplements.......................... 36 2.8 TheCourant-Weylinequalities . 36 2.9 Grammatrices ............................ 37 2.10 Diagonallydominantmatrices . 38 2.10.1 Gerˇsgorincircles . 38 2.11 Projections .............................. 39 2.12 Exercises ............................... 39 3 Eigenvalues and eigenvectors of graphs 41 3.1 Thelargesteigenvalue . 41 3.1.1 Graphswithlargesteigenvalueatmost2 . 42 3.1.2 Subdividinganedge . 43 3.1.3 TheKelmansoperation . 44 3.1.4 Spectral radius of a graph with a given number of edges . 44 3.2 Interlacing .............................. 45 3.3 Regulargraphs............................ 46 3.4 Bipartitegraphs ........................... 46 3.5 Cliquesandcocliques ........................ 47 3.5.1 Usingweightedadjacencymatrices . 47 3.6 Chromaticnumber.......................... 48 3.6.1 Usingweightedadjacencymatrices . 50 3.6.2 Rankandchromaticnumber . 50 3.7 Shannoncapacity .......................... 50 3.7.1 Lov´asz’ ϑ-function ...................... 51 3.7.2 The Haemers bound on the Shannon capacity . 53 3.8 Classificationofintegralcubicgraphs . 53 3.9 ThelargestLaplaceeigenvalue . 57 3.10 Laplaceeigenvaluesanddegrees . 58 3.11 TheGrone-MerrisConjecture. 60 3.11.1 Thresholdgraphs. 60 3.11.2 ProofoftheGrone-MerrisConjecture . 60 3.12 TheLaplacianforhypergraphs . 63 3.13 Applicationsofeigenvectors. 65 3.13.1 Ranking............................ 65 3.13.2 GooglePagerank. 65 3.13.3 Cutting ............................ 66 3.13.4 Graphdrawing ........................ 67 3.13.5 Clustering........................... 67 3.13.6 GraphIsomorphism . 68 3.13.7 Searchinganeigenspace . 69 3.14 Starsandstarcomplements. 69 CONTENTS 5 3.15 Exercises ............................... 70 4 The second largest eigenvalue 73 4.1 Boundsforthesecondlargesteigenvalue. 73 4.2 Largeregularsubgraphsareconnected. 74 4.3 Randomness ............................. 74 4.4 Randomwalks ............................ 75 4.5 Expansion............................... 76 4.6 ToughnessandHamiltonicity . 77 4.6.1 ThePetersengraphisnotHamiltonian . 77 4.7 Diameterbound ........................... 78 4.8 Separation .............................. 78 4.8.1 Bandwidth .......................... 80 4.8.2 Perfectmatchings . 80 4.9 Blockdesigns............................. 82 4.10 Polarities ............................... 84 4.11 Exercises ............................... 85 5 Trees 87 5.1 Characteristicpolynomialsoftrees . 87 5.2 Eigenvectorsandmultiplicities . 89 5.3 Signpatternsofeigenvectorsofgraphs. 90 5.4 Signpatternsofeigenvectorsoftrees. 91 5.5 Thespectralcenterofatree . 92 5.6 Integraltrees ............................. 93 5.7 Exercises ............................... 94 6 Groups and graphs 95 6.1 Γ(G,H,S) .............................. 95 6.2 Spectrum ............................... 95 6.3 NonabelianCayleygraphs. 96 6.4 Covers................................. 97 6.5 Cayleysumgraphs.......................... 99 6.5.1 (3,6)-fullerenes . 99 6.6 Exercises ............................... 100 7 Topology 101 7.1 Embeddings ............................. 101 7.2 Minors ................................ 101 7.3 TheColindeVerdi`ereinvariant. 102 7.4 The Van der Holst-Laurent-Schrijver invariant . 103 7.5 Spectralradiusofgraphsonasurface . 104 7.6 Exercises ............................... 104 8 Euclidean representations 105 8.1 Examples ............................... 105 8.2 Euclideanrepresentation . 105 8.3 Rootlattices ............................. 106 8.4 Cameron-Goethals-Seidel-Shult . 111 8.5 Furtherapplications . 112 6 CONTENTS 8.6 Exercises ............................... 112 9 Strongly regular graphs 113 9.1 Stronglyregulargraphs . 113 9.1.1 Simpleexamples . 113 9.1.2 ThePaleygraphs. 114 9.1.3 Adjacencymatrix . 115 9.1.4 Imprimitivegraphs. 115 9.1.5 Parameters .......................... 115 9.1.6 The half case and cyclic strongly regular graphs . 116 9.1.7 Strongly regular graphs without triangles . 116 9.1.8 Furtherparameterrestrictions . 117 9.1.9 Strongly regular graphs from permutation groups . 118 9.1.10 Strongly regular graphs from quasisymmetric designs . 119 9.1.11 Symmetric 2-designs from strongly regular graphs . 119 9.1.12 Latinsquaregraphs . 119 9.1.13 PartialGeometries . 121 9.2 Strongly regular graphs with eigenvalue 2............ 121 − 9.3 Connectivity ............................. 122 9.4 Cocliquesandcolorings . 124 9.4.1 Cliques ............................ 125 9.5 Automorphisms ........................... 126 9.6 Generalizedquadrangles. 126 9.6.1 Parameters .......................... 126 9.6.2 Constructions of generalized quadrangles . 127 9.6.3 Strongly regular graphs from generalized quadrangles . 128 9.6.4 Generalized quadrangles with lines of size 3 . 128 9.7 The(81,20,1,6)stronglyregulargraph . 129 9.7.1 Descriptions. .. 129 9.7.2 Uniqueness .......................... 130 9.7.3 Independence and chromatic numbers . 131 9.7.4 Secondsubconstituent . 132 9.7.5 Strongly regular graphs with λ = 1 and g = k ....... 133 9.8 Strongly regular graphs and 2-weight codes . 133 9.8.1 Codes,graphsandprojectivesets. 133 9.8.2 The correspondence between linear codes and subsets of a projectivespace. 133 9.8.3 The correspondence between projective two-weight codes, subsets of a projective space with two intersection num- bers, and affine strongly regular graphs . 134 9.8.4 Duality for affine strongly regular graphs . 136 9.8.5 Cyclotomy .......................... 137 9.9 Table ................................. 139 9.10 Exercises ............................... 144 CONTENTS 7 10 Regular two-graphs 147 10.1 Stronggraphs ............................ 147 10.2 Two-graphs.............................. 148 10.3 Regulartwo-graphs . 149 10.3.1 Relatedstronglyregulargraphs . 151 10.3.2 The regular two-graph on 276 points . 152 10.3.3 Coherentsubsets . 152 10.3.4 Completelyregulartwo-graphs . 152 10.4 Conferencematrices . 153 10.5 Hadamardmatrices . 154 10.5.1 Constructions. 155 10.6 Equiangularlines. 156 10.6.1 Equiangular lines in Rd and two-graphs . 156 10.6.2 Bounds on equiangular sets of lines in Rd or Cd . 157 10.6.3 Bounds on sets of lines with few angles and sets of vectors withfewdistances . 159 10.7 Exercise................................ 160 11 Association schemes 161 11.1 Definition............................... 161 11.2 TheBose-Mesneralgebra . 162 11.3 TheLinearProgrammingBound . 164 11.3.1 Equality............................ 165 11.3.2 TheCode-CliqueTheorem . 165 11.3.3 StrengthenedLPbounds . 165 11.4 TheKreinparameters . 166 11.5 Automorphisms ........................... 167 11.5.1 TheMooregraphon3250vertices . 168 11.6 P - and Q-polynomialassociationschemes . 168 11.7 Exercises ............................... 170 12 Distance-regular graphs 173 12.1 Parameters .............................. 173 12.2 Spectrum ............................... 173 12.3 Primitivity .............................. 174 12.4 Examples ............................... 174 12.4.1 Hamminggraphs . 174 12.4.2 Johnsongraphs. 175 12.4.3 Grassmanngraphs . 175 12.4.4 VanDam-Koolengraphs. 175 12.5 Bannai-Itoconjecture . 176 12.6 Connectedness ............................ 176 12.7 Growth ................................ 176 12.8 Degreeofeigenvalues . 176 12.9 Mooregraphsandgeneralizedpolygons . 177 12.10Euclideanrepresentations . 178 12.10.1Borsuk’sConjecture . 178 12.11Extremality.............................. 179 12.12Exercises ............................... 180 8 CONTENTS 13 p-ranks 183 13.1 Reduction mod p ........................... 183 13.2 Theminimalpolynomial . 184 13.3 Bounds for the p-rank........................ 184 13.4 Interesting primes p ......................... 185 13.5 Adding a multiple of J ....................... 185