Distance regular graphs:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
Berlin <<[u.a.]>>
Springer
1989
|
| Schriftenreihe: | Ergebnisse der Mathematik und ihrer Grenzgebiete
Folge 3 ; 18 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 495 S. Ill. |
| ISBN: | 3540506195 0387506195 |
Internformat
MARC
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| 100 | 1 | |a Brouwer, Andries E. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Distance regular graphs |c A. E. Brouwer ; A. M. Cohen ; A. Neumaier |
| 264 | 1 | |a Berlin <<[u.a.]>> |b Springer |c 1989 | |
| 300 | |a XVII, 495 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 |v 18 | |
| 700 | 1 | |a Cohen, Arjeh M. |d 1949- |e Verfasser |0 (DE-588)121202119 |4 aut | |
| 700 | 1 | |a Neumaier, Arnold |d 1954- |e Verfasser |0 (DE-588)108567214 |4 aut | |
| 830 | 0 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete |v Folge 3 ; 18 |w (DE-604)BV000899194 |9 18 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-018045227 | ||
Datensatz im Suchindex
| _version_ | 1804140047814885376 |
|---|---|
| adam_text | Contents
Preface vii
Contents xi
1. SPECIAL REGULAR GRAPHS
1.1 Edge regular and co-edge-regular graphs 3
1.2 Line graphs 6
1.3 Strongly regular graphs 8
Conference matrices and Paley graphs 10
The Hoffman bound 10
1.4 Strongly regular graphs as extremal graphs 11
1.5 Taylor graphs and regular two-graphs 13
1.6 Square 2-designs 16
1.7 Partial X-geometries 17
A connection with affine resolvable designs 19
1.8 Hadamard matrices 19
1.9 Hadamard graphs as extremal graphs 20
1.10 Square divisible designs 22
1.11 The bipartite double of a graph 24
The extended bipartite double of a graph 26
1.12 Direct products and Hamming graphs 26
1.13 /-cubes as extremal graphs 27
1.14 Gamma spaces and singular lines 28
1.15 Generalized quadrangles with line size three 29
1.16 Regular graphs without quadrangles 33
1.17 Geodetic graphs of diameter two 38
2. ASSOCIATION SCHEMES
2.1 Association schemes and coherent configurations 43
2.2 The Bose-Mesner algebra 44
The Frame quotient 46
Pseudocyclic association schemes 48
2.3 The Krein parameters 48
2.4 Imprimitivity 51
Dual imprimitivity 52
2.5 Subsets in association schemes 54
2.6 Characterization of the Bose-Mesner algebra 57
2.7 Metric and cometric schemes 58
The Frame quotient in a metric scheme 59
2.8 Subsets of cometric schemes; the Assmus-Mattson theorem 60
2.9 Distribution diagrams and the group case 62
2.10 Translation association schemes 65
Multiplier theorems and cyclotomic schemes 66
Duality 68
Additive codes 71
2.11 Representation diagrams, Krein modules and spherical designs 73
xii Contents
3. REPRESENTATION THEORY
3.1 Nonnegative matrices 80
3.2 Adjacency matrices and eigenvalues of graphs 82
3.3 Interlacing 85
3.4 Gram matrices 86
3.5 Graph representations 87
3.6 The absolute bound 90
3.7 Representations of subgraphs 91
3.8 Graph switching, equiangular lines, and representations of two-graphs 94
3.9 Lattices and integral representations 97
3.10 Root systems and root lattices 98
Fundamental systems and classification 99
The irreducible root lattices 100
Another proof of the classification 102
3.11 Graphs represented by roots of £g 103
3.12 Graphs with smallest eigenvalue at least —2 106
3.13 Equiangular lines Ill
3.14 Root graphs 113
Examples 114
3.15 Classification of amply regular root graphs 116
Amply regular root graphs in E% 120
Amply regular root graphs with /i = 2 123
4. THEORY OF DISTANCE-REGULAR GRAPHS
4.1 Distance-regular graphs 126
Parameters 126
Eigenvalues 128
Eigenspaces 131
Feasible parameter sets 133
Imprimitivity and the Q-polynomial property 135
Distance transitivity 136
Distance-biregular graphs 138
Weakenings of distance-regularity 139
4.2 Imprimitivity; new graphs from old 139
Imprimitivity 140
Parameters of halved graphs, folded graphs, and covers 141
Structural conditions for the existence of covers 143
Generalized Odd graphs; several P-polynomial structures 145
Distance-regular line graphs 148
Merging classes in distance-regular graphs 149
4.3 Substructures 151
Lines 151
Cubes 152
Moore geometries and Petersen graphs 155
7-point biplanes 156
4.4 Representations of distance-regular graphs 157
Contents xiii
5. PARAMETER RESTRICTIONS FOR DISTANCE-REGULAR GRAPHS
5.1 Unimodality of the sequence (/c(), 167
5.2 Diameter bounds by Terwilliger 169
5.3 Godsil s diameter bound. Graphs with Z», = 1 171
5.4 Restrictions for n 1 173
5.5 Further restrictions from counting arguments 174
5.6 Graphs with small kd 179
5.7 The case p2M = 0 181
5.8 A lower bound for p%2 183
5.9 Ivanov-Ivanov Theory 184
5.10 Circuit chasing 191
6. CLASSIFICATION OF THE KNOWN DISTANCE-REGULAR GRAPHS
6.1 Graphs with classical parameters 193
6.2 Computation of classical parameters 195
6.3 Imprimitive graphs with classical parameters; partition graphs 197
6.4 Regular near polygons 198
6.5 Generalized polygons 200
6.6 Other regular near polygons 205
6.7 Moore graphs 206
6.8 Moore geometries 207
6.9 Cages 209
6.10 The remaining primitive graphs 210
6.11 Bipartite distance-regular graphs; imprimitive regular near polygons 211
6.12 Antipodal distance-regular graphs 212
7. DISTANCE-TRANSITIVE GRAPHS
7.1 Some elementary group theory 214
7.2 The Thompson-Wielandt Theorem 216
7.3 A diameter bound for distance-transitive graphs 218
7.4 The case of large girth 221
7.5 Graphs with small valency 221
7.6 Imprimitive distance-transitive graphs 225
2-transitive square designs 226
2-transitive Hadamard matrices 227
2-transitive regular two-graphs 228
7.7 Towards the classification of all distance-transitive graphs 229
7.8 Further transitivity in graphs 231
Distance-transitive digraphs 232
Infinite distance-transitive graphs 232
8. Q-POLYNOMIAL DISTANCE-REGULAR GRAPHS
8.1 Leonard s characterization of g-polynomial graphs 235
Recurrence relations for g-sequences 237
Reduction of parameters 240
xiv Contents
8.2 Imprimitive Q-polynomial distance-regular graphs 241
8.3 Further results on g-polynomial graphs 244
Q-polynomial distance-regular graphs as extremal graphs 244
Explicit formulae for eigenmatrices, eigenvalues, and multiplicities 245
Integrality of eigenvalues 247
Bounds for girth and diameter 248
8.4 Graphs with classical parameters 249
A characterization of graphs with classical parameters 249
8.5 The known Q-polynomial distance-regular graphs 252
9. THE FAMILIES OF GRAPHS WITH CLASSICAL PARAMETERS
9.1 Johnson graphs 255
Characterizations by structure 256
Characterization by parameters 258
Folded Johnson graphs 259
Odd graphs and doubled Odd graphs 259
9.2 Hamming graphs 261
Geometric characterization 262
Characterization by parameters - Pseudo Hamming graphs 262
Characterization by spectrum 263
Halved and folded cubes 264
Covers of cubes and folded cubes - the Wells graph 266
9.3 Grassmann graphs 268
Characterization by structure 270
Characterization by parameters 271
Graphs related to Grassmann graphs 272
9.4 Dual polar graphs 274
Geometric characterization 276
Characterization by parameters 277
Related graphs 278
9.5 Sesquilinear forms graphs 280
Bilinear forms graphs 280
Alternating forms graphs 282
Hermitean forms graphs 285
Symmetric bilinear forms graphs 285
Affine subspaces of dual polar spaces 286
Antipodal covers 288
9.6 The quadratic forms graphs 290
10. GRAPHS OF COXETER AND LIE TYPE
10.1 Coxeter systems 294
The Coxeter group as a reflection group 296
The length function; reduced expressions 296
The word problem in Coxeter groups 297
Bruhat order 298
10.2 Coxeter graphs 299
The neighbourhood of a point 299
Contents xv
The 2-neighbourhood of a point 301
Subgraphs from subdiagrams 303
Objects and their shadows 304
Association scheme and double coset diagram 306
Product expressions for k and v 308
Incidence graphs 309
10.3 The finite Coxeter graphs; root systems and presentations 310
Root systems 310
10.4 Global properties 315
Finiteness 315
Diameter and permutation rank 316
Amply regular Coxeter graphs 318
Distance-regular Coxeter graphs 319
Multiplicity-free representations 320
10.5 Tits Systems 323
The association scheme of a Tits system 325
Nonexistence results 326
10.6 Graphs of Lie Type 326
Subgraphs from subdiagrams 327
Objects 327
Lines 327
Singular lines 328
Transitivity properties 329
Relation between a graph of Lie type and the associated Coxeter graph 329
Incidence graphs 332
10.7 Chevalley Groups 332
Graphs of Lie Type from Chevalley groups 334
Parameters 336
Computation of the parameters of E71(q) - geometric approach 339
10.8 The affine E6 graph 340
10.9 Distance-transitive representations of Chevalley groups 341
11. GRAPHS RELATED TO CODES
11.1 Completely regular codes 345
Codes in distance-regular graphs 345
Completely regular partitions and distance-regular quotient graphs 350
Distance-regular graphs with regular abelian automorphism groups 353
Completely regular codes in the Hamming scheme 354
Completely regular codes in other distance-regular graphs 357
11.2 Graphs from the Kasami codes 358
11.3 Graphs from the Golay codes 359
The coset graph of the extended ternary Golay code 359
The coset graph of the ternary Golay code 360
The coset graph of the truncated ternary Golay code 360
The coset graph of the extended binary Golay code 360
The coset graph of the binary Golay code 361
The coset graph of the truncated binary Golay code 362
xvi Contents
The coset graph of the doubly truncated binary Golay code and the graph
of the unitals in PG(2,4) 363
Variations on the theme of coset graph - some antipodal covers 364
11.4 Graphs related to the Witt designs 366
The Witt graph associated to M24 366
The truncated Witt graph associated to Af23 367
The doubly truncated Witt graph associated to M12 368
The Ivanov-Ivanov-Faradjev graph 369
Higman s symmetric design 370
The Leonard graph - Af12.2 over PGL(2, l) 371
The Hadamard association scheme 371
Antipodal 2-covers of the Gewirtz graph 372
The regular two-graph on 276 vertices and the McLaughlin graph 372
11.5 The van Lint-Schrijver partial geometry 373
12. GRAPHS RELATED TO CLASSICAL GEOMETRIES
12.1 The even orthogonal case; the Doro graph 374
12.2 The odd orthogonal case 380
12.3 The Coxeter graph for PSL(2,7) 382
12.4 The unitary case 383
12.5 Antipodal covers of complete graphs 385
12.6 Thin near octagons from Denniston s complete arcs 387
12.7 Antipodal covers of complete graphs from pseudocyclic association schemes 388
Cyclotomic schemes 389
The Mathon and Hollmann schemes on 28 points, and conies in PG(2,q) 390
The Hollmann scheme on 496 points 390
13. SPORADIC GRAPHS
13.1 Graphs related to the Hoffman - Singleton graph 391
Sylvester s double six graph 394
13.2 Commuting involutions graphs and Fischer spaces 395
Five antipodal 3-covers 397
The Foster graph for 3-Sym(6).2 and the hexacode 397
The Conway-Smith graph for 3Sym(7) 399
The locally Gg(4,2) graph on 3X126 points 399
The 3.O7(3) graph on 3X378 points 400
13.3 The Perkel graph for L(2,19) 401
13.4 The Biggs-Smith graph for L(2,17) 403
13.5 The Livingstone graph for Jx 406
13.6 The near octagon associated with the Hall-Janko group 408
13.7 The Patterson graph for Suz 410
Contents xvii
14. TABLES OF PARAMETERS FOR DISTANCE REGULAR GRAPHS 413
A. APPENDIX
A.I Graphs 433
A.2 Permutation groups 435
A.3 Automorphisms 435
A.4 Regular partitions, distribution diagrams and double coset graphs 436
A.5 Primitivity 437
A.6 Designs 438
A.7 Codes 440
A.8 Singular subspaces 440
A.9 Geometries 441
A.10 Miscellaneous notation 442
References 444
Symbols and notation 477
Intersection arrays 480
Author index 484
Subject index 489
|
| any_adam_object | 1 |
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| author_role | aut aut aut |
| author_sort | Brouwer, Andries E. |
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| building | Verbundindex |
| bvnumber | BV024632896 |
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| id | DE-604.BV024632896 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:53:46Z |
| institution | BVB |
| isbn | 3540506195 0387506195 |
| language | Undetermined |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018045227 |
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| physical | XVII, 495 S. Ill. |
| publishDate | 1989 |
| publishDateSearch | 1989 |
| publishDateSort | 1989 |
| publisher | Springer |
| record_format | marc |
| series | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 |
| spelling | Brouwer, Andries E. Verfasser aut Distance regular graphs A. E. Brouwer ; A. M. Cohen ; A. Neumaier Berlin <<[u.a.]>> Springer 1989 XVII, 495 S. Ill. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 18 Cohen, Arjeh M. 1949- Verfasser (DE-588)121202119 aut Neumaier, Arnold 1954- Verfasser (DE-588)108567214 aut Ergebnisse der Mathematik und ihrer Grenzgebiete Folge 3 ; 18 (DE-604)BV000899194 18 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018045227&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brouwer, Andries E. Cohen, Arjeh M. 1949- Neumaier, Arnold 1954- Distance regular graphs Ergebnisse der Mathematik und ihrer Grenzgebiete |
| title | Distance regular graphs |
| title_auth | Distance regular graphs |
| title_exact_search | Distance regular graphs |
| title_full | Distance regular graphs A. E. Brouwer ; A. M. Cohen ; A. Neumaier |
| title_fullStr | Distance regular graphs A. E. Brouwer ; A. M. Cohen ; A. Neumaier |
| title_full_unstemmed | Distance regular graphs A. E. Brouwer ; A. M. Cohen ; A. Neumaier |
| title_short | Distance regular graphs |
| title_sort | distance regular graphs |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018045227&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000899194 |
| work_keys_str_mv | AT brouwerandriese distanceregulargraphs AT cohenarjehm distanceregulargraphs AT neumaierarnold distanceregulargraphs |