Spectra of graphs:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
New York ; Dordrecht ; Heidelberg ; London
Springer
2012
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiii, 250 Seiten Diagramme |
| ISBN: | 9781461419389 9781489994332 |
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MARC
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Record in the Search Index
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Contents Preface. v Graph Spectrum . 1 Matrices associated to a graph. The spectrum of a graph. 1.2.1 Characteristic polynomial . The spectrum of an undirected graph. 1.3.1 Regular graphs . 1.3.2 Complements. 1.3.3 Walks. 1.3.4 Diameter . 1.3.5 Spanning trees. 1.3.6 Bipartite graphs. 1.3.7 Connectedness. Spectrum of some graphs. 1.4.1 The complete graph . 1.4.2 The complete bipartite graph. 1.4.3 The cycle.
1.4.4 The path. 1.4.5 Line graphs. 1.4.6 Cartesian products. 1.4.7 Kronecker products and bipartite double. 1.4.8 Strong products. 1.4.9 Cayley graphs. Decompositions. 1.5.1 Decomposing K1o into Petersen graphs . 1.5.2 Decomposing Kn into completebipartite graphs. Automorphisms. Algebraic connectivity. Cospectral graphs. 1.8.1 The4-cube. 1 2 3 3 4 4 4 5 5 6 7 8 8 8 8 9 9 10 10 11 11 11 12 12 12 13 14 14 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 vu
viii Contents 1.8.2 Seidel switching. 1.8.3 Godsil-McKay switching. 1.8.4 Reconstruction. Very small graphs. Exercises. 15 16 16 16 17 Linear Algebra. 21 21 22 24 25 25 26 28 28 29 29 30 31 31 32 1.9 1.10 2 2.1 2.2 2.3 Simultaneous diagonalization. Perron-Frobenius theory. Equitable partitions. 2.3.1 Equitable and almost equitable partitions of graphs. 2.4 The Rayleigh quotient. 2.5 Interlacing. 2.6 Schur’s inequality . 2.7 Schur complements. 2.8 The Courant-Weyl inequalities. 2.9 Gram matrices
. 2.10 Diagonally dominant matrices. 2.10.1 Gersgorin circles. 2.11 Projections . 2.12 Exercises. 3 Eigenvalues and Eigenvectors of Graphs . The largest eigenvalue. 3.1.1 Graphs with largest eigenvalue at most 2. 3.1.2 Subdividing an edge. 3.1.3 The Kelmans operation . 3.2 Interlacing. 3.3 Regular graphs. 3.4 Bipartite graphs. 3.5 Cliques and cocliques. 3.5.1 Using weighted adjacency matrices. 3.6 Chromatic number. 3.6.1 Using weighted adjacency
matrices. 3.6.2 Rank and chromatic number. 3.7 Shannon capacity. 3.7.1 Lovâsz’s 3-function. 3.7.2 The Haemers bound on the Shannon capacity. 3.8 Classification of integral cubic graphs. 3.8.1 A quotient of the hexagonalgrid. 3.8.2 Cubic graphs with loops. 3.8.3 The classification . 3.9 The largest Laplace eigenvalue. 3.10 Laplace eigenvalues and degrees. 3.11 The Grone-Merris conjecture. 3.1 33 33 34 35 36 37 37 38 38 39 40 42 42 42 44 45 46 47 47 47 50 51 53
ix Contents 3.11.1 Threshold graphs. 3.11.2 Proof of the Grone-Merris conjecture. The Laplacian for hypergraphs. 3.12.1 Dominance order. Applications of eigenvectors. 3.13.1 Ranking. 3.13.2 Google PageRank. 3.13.3 Cutting. 3.13.4 Graph drawing. 3.13.5 Clustering . 3.13.6 Graph isomorphism . 3.13.7 Searching an eigenspace. Stars and star complements. Exercises. 53 53 56 58 58 59 59 60 61 61 62 63 63 64 4 The Second-Largest Eigenvalue. 67 67 68 68 69 70 71 72 72 73 74 75 77 79 80 3.12 3.13 3.14 3.15 Bounds for the second-largest
eigenvalue. Large regular subgraphs are connected. Randomness. Random walks. Expansion. Toughness and Hamiltonicity. 4.6.1 The Petersen graph is not Hamiltonian. 4.7 Diameter bound. 4.8 Separation. 4.8.1 Bandwidth. 4.8.2 Perfect matchings. 4.9 Block designs. 4.10 Polarities. 4.11 Exercises. 4.1 4.2 4.3 4.4 4.5 4.6 5 Trees. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Characteristic polynomials of
trees. Eigenvectors and multiplicities. Sign patterns of eigenvectors of graphs. Sign patterns of eigenvectors of trees. The spectral center of a tree. Integral trees. Exercises. 83 83 85 86 87 88 89 90 6 Groups and Graphs. 93 6.1 6.2 6.3 6.4 r(G,H,S). Spectrum. Non-Abelian Cayley graphs. Covers. 93 93 94 95
Contents X 6.5 6.6 Cayley sum graphs. 6.5.1 (3,6)-fullerenes. Exercises. 97 97 99 7 Topology . 101 7.1 Embeddings. 101 7.2 Minors. 102 7.3 The Colin de Verdière invariant. 102 7.4 The Van der Holst-Laurent-Schrijverinvariant. 103 8 Euclidean Representations. 105 8.1 Examples. 105 8.2 Euclidean representation. 105 8.3 Root lattices. 106 8.3.1 Examples. 107 8.3.2 Root lattices.108 8.3.3
Classification. 109 8.4 The Cameron-Goethals-Seidel-Shulttheorem . Ill 8.5 Further applications. 112 8.6 Exercises. 113 9 Strongly Regular Graphs. 115 9.1 Strongly regular graphs.115 9.1.1 Simple examples.115 9.1.2 The Paley graphs. 116 9.1.3 Adjacency matrix. 117 9.1.4 Imprimitive graphs. 117 9.1.5 Parameters.118 9.1.6 The half case and cyclic strongly regular graphs. 118 9.1.7 Strongly regular graphs without triangles. 119 9.1.8 Further parameter restrictions.120 9.1.9 Strongly regular graphs from permutation groups. 121 9.1.10 Strongly regular graphs from quasisymmetric designs . 121 9.1.11 Symmetric 2-designs from strongly regular graphs. 122 9.1.12 Latin square
graphs. 122 9.1.13 Partial geometries. 124 9.2 Strongly regular graphs with eigenvalue —2. 124 9.3 Connectivity. 125 9.4 Cocliques and colorings . 127 9.5 Automorphisms.129 9.6 Generalized quadrangles.129 9.6.1 Parameters. 129 9.6.2 Constructions of generalized quadrangles.130 9.6.3 Strongly regular graphs from generalized quadrangles. 131 9.6.4 Generalized quadrangles with lines of size 3.132
Contents xi The (81,20,1,6) strongly regular graph. 132 9.7.1 Descriptions. 133 9.7.2 Uniqueness . 134 9.7.3 Independence and chromatic numbers. 135 9.7.4 Second subconstituent. 136 9.8 Strongly regular graphs and two-weight codes. 136 9.8.1 Codes, graphs, and projective sets.136 9.8.2 The correspondence between linear codes and subsets of a projective space. 137 9.8.3 The correspondence between projective two-weight codes, subsets of a projective space with two intersection numbers, and affine strongly regular graphs . 138 9.8.4 Duality for affine strongly regular graphs. 140 9.8.5 Cyclotomy. 141 9.9 Table of parameters for strongly regular graphs. 143 9.9.1 Comments. 146 9.10 Exercises. 148 9.7 10 Regular Two-graphs.151 10.1 Strong
graphs. 151 10.2 Two-graphs.152 10.3 Regular two-graphs. 154 10.3.1 Related strongly regular graphs. 155 10.3.2 The regular two-graph on 276 points. 156 10.3.3 Coherent subsets.156 10.3.4 Completely regular two-graphs.157 10.4 Conference matrices . 158 10.5 Hadamard matrices. 159 10.5.1 Constructions. 160 10.6 Equiangular lines.161 10.6.1 Equiangular lines in Rd andtwo-graphs. 161 10.6.2 Bounds on equiangular setsof lines in R^ orCd. 162 10.6.3 Bounds on sets of lines with few angles and sets of vectors with few distances. 163 11 Association Schemes.165 11.1
Definition. 165 11.2 The Bose-Mesner algebra. 166 11.3 The linear programming bound.168 11.3.1 Equality. 169 11.3.2 The code-clique theorem. 169 11.3.3 Strengthened LP bounds. 170 11.4 The Krein parameters . 170 11.5 Automorphisms. 172 11.5.1 The Moore graph on3250 vertices. 172
xii Contents 11.6 P- and Q-polynomial association schemes. 173 11.7 Exercises. 175 12 Distance-RegularGraphs. 177 12.1 Parameters . 177 12.2 Spectrum. 178 12.3 Primitivity. 178 12.4 Examples. 178 12.4.1 Hamming graphs. 178 12.4.2 Johnson graphs.179 12.4.3 Grassmann graphs. 180 12.4.4 Van Dam-Koolengraphs . 180 12.5 Bannai-Ito conjecture . 180 12.6 Connectedness. 181 12.7 Growth . 181 12.8 Degree of eigenvalues
. 181 12.9 Moore graphs and generalized polygons. 182 12.10 Euclidean representations. 183 12.11 Extremality. 183 12.12 Exercises. 185 13 p-ranks.187 13.1 Reduction mod p. 187 13.2 The minimal polynomial. 188 13.3 Bounds for the p-rank. 188 13.4 Interesting primes p. 189 13.5 Adding a multiple of J. 190 13.6 Paley graphs. 191 13.7 Strongly regular graphs. 192 13.8 Smith normal form. 194 13.8.1 Smith normal form and spectrum.
195 13.9 Exercises. 197 14 SpectralCharacterizations . 199 14.1 Generalized adjacency matrices. 199 14.2 Constructing cospectral graphs.200 14.2.1 Trees. 201 14.2.2 Partial linear spaces . 202 14.2.3 GM switching. 202 14.2.4 Sunada’s method. 204 14.3 Enumeration. 204 14.3.1 Lowerbounds. 204 14.3.2 Computer results. 205 14.4 DS graphs.206 14.4.1 Spectrum and structure. 206
Contents xiii 14.4.2 Some DS graphs. 208 14.4.3 Line graphs.210 14.5 Distance-regular graphs. 212 14.5.1 Strongly regular DS graphs. 213 14.5.2 Distance-regularity from the spectrum. 214 14.5.3 Distance-regular DS graphs. 215 14.6 The method of Wang and Xu. 217 14.7 Exercises. 219 15 Graphs with Few Eigenvalues. 221 15.1 Regular graphs with four eigenvalues. 221 15.2 Three Laplace eigenvalues. 223 15.3 Other matrices with at most threeeigenvalues. 224 15.3.1 Few Seidel eigenvalues. 224 15.3.2 Three adjacency eigenvalues. 225 15.3.3 Three signless Laplace eigenvalues. 227 15.4 Exercises. 227
References. 229 Author Index. 243 Subject Index. 247 |
| any_adam_object | 1 |
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| discipline | Mathematik |
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| isbn | 9781461419389 9781489994332 |
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| spelling | Brouwer, Andries E. 1951- (DE-588)1023667894 aut Spectra of graphs Andries E. Brouwer ; Willem H. Haemers New York ; Dordrecht ; Heidelberg ; London Springer 2012 xiii, 250 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Universitext Graphentheorie (DE-588)4113782-6 gnd rswk-swf Spektrum Mathematik (DE-588)4182180-4 gnd rswk-swf Graphentheorie (DE-588)4113782-6 s DE-604 Spektrum Mathematik (DE-588)4182180-4 s Haemers, Willem H. (DE-588)17091514X aut Erscheint auch als Online-Ausgabe 978-1-4614-1939-6 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024744217&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brouwer, Andries E. 1951- Haemers, Willem H. Spectra of graphs Graphentheorie (DE-588)4113782-6 gnd Spektrum Mathematik (DE-588)4182180-4 gnd |
| subject_GND | (DE-588)4113782-6 (DE-588)4182180-4 |
| title | Spectra of graphs |
| title_auth | Spectra of graphs |
| title_exact_search | Spectra of graphs |
| title_full | Spectra of graphs Andries E. Brouwer ; Willem H. Haemers |
| title_fullStr | Spectra of graphs Andries E. Brouwer ; Willem H. Haemers |
| title_full_unstemmed | Spectra of graphs Andries E. Brouwer ; Willem H. Haemers |
| title_short | Spectra of graphs |
| title_sort | spectra of graphs |
| topic | Graphentheorie (DE-588)4113782-6 gnd Spektrum Mathematik (DE-588)4182180-4 gnd |
| topic_facet | Graphentheorie Spektrum Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024744217&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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