Geometry and Combinatorics (eBook)

Front Cover 1
Geometry and Combinatorics: Selected Works of J. J. Seidel 4
Copyright Page 5
Table of Contents 6
Preface 10
Acknowledgments 12
List of Publications of J. J. Seidel 14
Part I: Graphs and Designs 22
Chapter 1. EQUILATERAL POINT SETS IN ELLIPTIC GEOMETRY 24
1. Introduction on geometry 24
2. Introduction on matrices 25
3. Re-wording of the problem 26
4. Tables 27
5. C-matrices 30
6. Results on n(r) 31
7. Equilateral point sets in Er-1 34
REFERENCES 36
Chapter 2. STRONGLY REGULAR GRAPHS OF L2-TYPE AND OF TRIANGULAR TYPE 38
1. INTRODUCTION 38
2. DEFINITIONS 39
3. EIGENVALUES 40
4. GRAPHS WITH TWO EIGENVALUES, Q1 = 28 + 1, Q2 = – 3 42
REFERENCES 45
Chapter 3. Strongly Regular Graphs with (—1,1,0) Adjacency Matrix Having Eigenvalue 3 47
1. INTRODUCTION 47
2. STRONG GRAPHS 48
3. CLASSIFICATION AND EXAMPLES 51
4. COMPLETE BIPARTITE INDUCED SUBGRAPHS 53
5. THE STANDARD ADJACENCY MATRIX FOR .1 = 3, .0 . 3 56
6. STRONGLY REGULAR GRAPHS WITH .1 = 3 59
REFERENCES 63
Chapter 4. STRONGLY REGULAR GRAPHS DERIVED FROM COMBINATORIAL DESIGNS 65
1. Introduction 65
2. A construction method for graphs 66
3. Quasi-symmetric block designs 70
4. Symmetric Hadamard matrices with constant diagonal 72
5. Tactical configurations 78
6. The extended Golay code 80
REFERENCES 82
Chapter 5. A Strongly Regular Graph Derived from the Perfect Ternary Golay Code 83
1. Introduction 83
2. Strongly regular graphs with P211—P111 = 1 84
3. The perfect ternary Golay code 86
4. Constitution of the 243-graph 87
References 88
Chapter 6. SPHERICAL CODES AND DESIGNS 89
1. INTRODUCTION 89
2. GEGENBAUER POLYNOMIALS 90
3. HARMONIC POLYNOMIALS 92
4. SPHERICAL CODES 94
5. SPHERICAL DESIGNS 97
6. SPHERICAL (d, n, s, t)- CONFIGURATIONS 102
7. DISTANCE INVARIANCE AND ASSOCIATION SCHEMES 104
8. EXAMPLES FROM SETS OF LINES AND DERIVED CONFIGURATIONS 106
9. EXAMPLES FROM ASSOCIATION SCHEMES 111
BIBLIOGRAPHY 113
Chapter 7. GRAPHS RELATED TO EXCEPTIONAL ROOT SYSTEMS 115
ANNOUNCEMENT OF RESULTS 115
REFERENCES 120
Chapter 8. Strongly Regular Graphs Having Strongly Regular Subconstituents 122
1. INTRODUCTION 122
2. STRONGLY REGULAR GRAPHS 123
3. KREIN PARAMETERS AND TENSORS 126
4. SPHERICAL DESIGNS 128
5. THE SUBCONSTITUENTS 130
6. SMITH GRAPHS 131
7. PSEUDO-GEOMETRIC GRAPHS 137
8. GENERALIZED QUADRANGLES WITH PARAMETERS (q, q2) 139
9. UNIQUENESS PROOFS 142
REFERENCES 144
Part II: Lines with Few Angles 146
Chapter 9. Equiangular Lines 148
1. INTRODUCTION 148
2. DEFINITIONS AND EXAMPLES 149
3. BOUNDS FOR .(r) 151
4. PILLARS 155
5. DETERMINATION OF .1/5(r) 160
REFERENCES 166
Chapter 10. A SURVEY OF TWO–GRAPHS 167
1. INTRODUCTION 167
2. INTRODUCTORY EXAMPLES 168
3. SWITCHING OF GRAPHS 170
4. TWO-GRAPHS 171
5. EQUIANGULAR LINES 174
6. STRONG GRAPHS 175
7. REGULAR TWO-GRAPHS 177
8. RANK 3 GRAPHS 181
9. SYMPLECTIC AND ORTHOGONAL TWO-GRAPHS 184
10. UNITARY TWO-GRAPHS 186
11. SPORADIC TWO-GRAPHS 187
12. HADAMARD MATRICES 189
13. CONFERENCE MATRICES 191
REFERENCES 196
Chapter 11. THE REGULAR TWO-GRAPH ON 276 VERTICES 198
1. Introduction 198
2. Regular two-graphs 199
3. The case n = 276 203
4. Ternary codes 205
5. The 276-two-graph 207
References 212
Chapter 12. BOUNDS FOR SYSTEMS OF LINES AND JACOBI POLYNOMIALS 214
Abstract 214
1. Introduction 214
2. Jacobi polynomials 215
3. Addition formulae 217
4. Characteristic matrices 218
5. Special bounds for A-sets 221
6. Absolute bounds for A-sets 223
7. Properties of extremal A-sets 226
REFERENCES 228
Chapter 13. Line Graphs, Root Systems, and Elliptic Geometry 229
1. INTRODUCTION 229
2. LINES AT 60° AND 90° 230
3. ROOT SYSTEMS 234
4. GRAPHS WITH LEAST EIGENVALUES —2 239
5. SPECTRAL CHARACTERIZATION OF CERTAIN GRAPHS 245
6. AN APPLICATION TO HADAMARD MATRICES 249
REFERENCES 251
Chapter 14. TWO-GRAPHS, A SECOND SURVEY 252
1. INTRODUCTION 252
2. DEFINITION AND ENUMERATION 253
3. EQUIANGULAR LINES 256
4. AUTOMORPHISMS 258
5. ENUMERATION OF REGULAR TWO-GRAPHS 265
6. CONFERENCE TWO-GRAPHS OF ORDER pq2 + 1 267
7. MÖBIUS AND MINKOWSKI TWO-GRAPHS 271
REFERENCES 274
Part III: Matrices and Forms 276
Chapter 15. ORTHOGONAL MATRICES WITH ZERO DIAGONAL 278
1. Introduction 278
2. Paley matrices 279
3. Symmetric C-matrices 281
4. C-matrices and Hadamard matrices 284
REFERENCES 286
Chapter 16. QUASIREGULAR TWO-DISTANCE SETS 288
1. Introduction 288
2. Two-distance sets in Rn 288
3. The case of even dimensions 291
4. The case of odd dimensions 291
REFERENCES 294
Chapter 17. A SKEW HADAMARD MATRIX OF ORDER 36 295
References 296
Chapter 18. SYMMETRIC HADAMARD MATRICES OF ORDER 36 297
1. Introduction 297
2. Latin square graphs and Steiner graphs 298
3. Equivalence under switching 299
4. Regular Steiner graphs 301
5. The lines of PG(3,2) 304
6. Rank 3 graphs of order 36 308
Referencs 309
Chapter 19. QUADRATIC FORMS OVER GF(2) 311
§ 1. Introduction 311
§ 2. Quadratic and bilinear forms 311
§ 3. The rational vectors and their Gramian matrix 313
§ 4. Configurations, designs, and codes 315
REFERENCES 318
Chapter 20. On two-graphs, and Shult's characterization of symplectic and orthogonal geometries over GF(2) 319
1. Introduction 319
2. Regular two-graphs 320
3. Symplectic and orthogonal geometries over GF(2) 324
4. Characterization of the symplectic and orthogonal graphs 330
5. A problem by Hamelink 340
References 343
Chapter 21. The Krein condition, spherical designs, Norton algebras and permutation groups 344
1. INTRODUCTION 344
2. THE KREIN PARAMETERS 345
3. IMPRIMITIVE ASSOCIATION SCHEMES 346
4. SPHERICAL DESIGNS 347
5. NORTON ALGEBRAS 349
6. PERMUTATION GROUPS 350
7. A FINAL REMARK 353
REFERENCES 353
Part IV: Non-Euclidean Geometry 356
Chapter 22. The congruence order of the elliptic plane 358
Chapter 23. EQUI-ISOCLINIC SUBSPACES OF EUCLIDEAN SPACES 362
1. Introduction 362
2. Two n-subspaces 363
3. Equi-isoclinic subspaces 365
4. Block matrices 367
5. Determination of ..(n, 2n) 369
REFERENCES 371
Chapter 24. METRIC PROBLEMS IN ELLIPTIC GEOMETRY 372
1. Introduction 372
2. Congruence Order 373
3. Elliptic Space En-1 373
4. Pillars 374
5. Further Results 378
6. Finite Groups, Error-Correcting Codes, and Two-Graphs 380
REFERENCES 383
Chapter 25. THE FOOTBALL 384
2. THE ICOSAHEDRON 384
3. ORBITS 385
4. APPROXIMATION OF STRENGTH t 387
5. INVARIANTS 388
6. THE DIHEDRAL GROUP OF ORDER 10 389
7. THE ICOSAHEDRAL GROUP 390
REFERENCES 391
Chapter 26. DISCRETE HYPERBOLIC GEOMETRY 393
1. Introduction 393
2. Lorentz space 395
3. Lifting 396
4. Graphs with .2 = 1 398
5. Reflexive graphs 400
6. Unimodular Euclidean lattices 404
7. Unimodular lattices of signature (p, 1) 407
References 410
Chapter 27. FEW - DISTANCE SETS IN Rp, q 412
1. Introduction 412
2. The generalized addition formula 413
3. Bounds for s-distance sets 416
4. Equiangular lines in Rp,1 418
5. Two-distance sets 420
6. Two-angle sets of lines 421
7. Final remarks 423
REFERENCES 424
Chapter 28. Remark on Wielandt's Visibility Theorem 426
ABSTRACT 426
1. INTRODUCTION 426
2. THE THEOREM 426
3. THE PROOF 427
REFERENCES 427
Chapter 29. Complete List of Permissions 428
I. GRAPHS AND DESIGNS 428
II. LINES WITH FEW ANGLES 429
III. MATRICES AND FORMS 429
IV. NON-EUCLIDEAN GEOMETRY 430