Seidel / Corneil / Mathon Geometry and Combinatorics

1. Auflage 2014
ISBN: 978-1-4832-6800-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 430 Seiten, Web PDF

Geometry and Combinatorics: Selected Works of J. J. Seidel brings together some of the works of J. J. Seidel in geometry and combinatorics. Seidel's selected papers are divided into four areas: graphs and designs; lines with few angles; matrices and forms; and non-Euclidean geometry. A list of all of Seidel's publications is included. Comprised of 29 chapters, this book begins with a discussion on equilateral point sets in elliptic geometry, followed by an analysis of strongly regular graphs of L2-type and of triangular type. The reader is then introduced to strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3; graphs related to exceptional root systems; and equiangular lines. Subsequent chapters deal with the regular two-graph on 276 vertices; the congruence order of the elliptic plane; equi-isoclinic subspaces of Euclidean spaces; and Wielandt's visibility theorem. This monograph will be of interest to students and practitioners in the field of mathematics.

Autoren/Hrsg.

Seidel, J. J.

Corneil, D. G.

Mathon, R.

Inhaltsverzeichnis

1;Front Cover;1
2;Geometry and Combinatorics: Selected Works of J. J. Seidel;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Acknowledgments;12
7;List of Publications of J. J. Seidel;14
8;Part I: Graphs and Designs;22
8.1;Chapter 1. EQUILATERAL POINT SETS IN ELLIPTIC GEOMETRY;24
8.1.1;1. Introduction on geometry;24
8.1.2;2. Introduction on matrices;25
8.1.3;3. Re-wording of the problem;26
8.1.4;4. Tables;27
8.1.5;5. C-matrices;30
8.1.6;6. Results on n(r);31
8.1.7;7. Equilateral point sets in Er-1;34
8.1.8;REFERENCES;36
8.2;Chapter 2. STRONGLY REGULAR GRAPHS OF L2-TYPE AND OF TRIANGULAR TYPE;38
8.2.1;1. INTRODUCTION;38
8.2.2;2. DEFINITIONS;39
8.2.3;3. EIGENVALUES;40
8.2.4;4. GRAPHS WITH TWO EIGENVALUES, Q1 = 28 + 1, Q2 = – 3;42
8.2.5;REFERENCES;45
8.3;Chapter 3. Strongly Regular Graphs with (—1,1,0) Adjacency Matrix Having Eigenvalue 3;47
8.3.1;1. INTRODUCTION;47
8.3.2;2. STRONG GRAPHS;48
8.3.3;3. CLASSIFICATION AND EXAMPLES;51
8.3.4;4. COMPLETE BIPARTITE INDUCED SUBGRAPHS;53
8.3.5;5. THE STANDARD ADJACENCY MATRIX FOR .1 = 3, .0 . 3;56
8.3.6;6. STRONGLY REGULAR GRAPHS WITH .1 = 3;59
8.3.7;REFERENCES;63
8.4;Chapter 4. STRONGLY REGULAR GRAPHS DERIVED FROM COMBINATORIAL DESIGNS;65
8.4.1;1. Introduction;65
8.4.2;2. A construction method for graphs;66
8.4.3;3. Quasi-symmetric block designs;70
8.4.4;4. Symmetric Hadamard matrices with constant diagonal;72
8.4.5;5. Tactical configurations;78
8.4.6;6. The extended Golay code;80
8.4.7;REFERENCES;82
8.5;Chapter 5. A Strongly Regular Graph Derived from the Perfect Ternary Golay Code;83
8.5.1;1. Introduction;83
8.5.2;2. Strongly regular graphs with P211—P111 = 1;84
8.5.3;3. The perfect ternary Golay code;86
8.5.4;4. Constitution of the 243-graph;87
8.5.5;References;88
8.6;Chapter 6. SPHERICAL CODES AND DESIGNS;89
8.6.1;1. INTRODUCTION;89
8.6.2;2. GEGENBAUER POLYNOMIALS;90
8.6.3;3. HARMONIC POLYNOMIALS;92
8.6.4;4. SPHERICAL CODES;94
8.6.5;5. SPHERICAL DESIGNS;97
8.6.6;6. SPHERICAL (d, n, s, t)- CONFIGURATIONS;102
8.6.7;7. DISTANCE INVARIANCE AND ASSOCIATION SCHEMES;104
8.6.8;8. EXAMPLES FROM SETS OF LINES AND DERIVED CONFIGURATIONS;106
8.6.9;9. EXAMPLES FROM ASSOCIATION SCHEMES;111
8.6.10;BIBLIOGRAPHY;113
8.7;Chapter 7. GRAPHS RELATED TO EXCEPTIONAL ROOT SYSTEMS;115
8.7.1;ANNOUNCEMENT OF RESULTS;115
8.7.2;REFERENCES;120
8.8;Chapter 8. Strongly Regular Graphs Having Strongly Regular Subconstituents;122
8.8.1;1. INTRODUCTION;122
8.8.2;2. STRONGLY REGULAR GRAPHS;123
8.8.3;3. KREIN PARAMETERS AND TENSORS;126
8.8.4;4. SPHERICAL DESIGNS;128
8.8.5;5. THE SUBCONSTITUENTS;130
8.8.6;6. SMITH GRAPHS;131
8.8.7;7. PSEUDO-GEOMETRIC GRAPHS;137
8.8.8;8. GENERALIZED QUADRANGLES WITH PARAMETERS (q, q2);139
8.8.9;9. UNIQUENESS PROOFS;142
8.8.10;REFERENCES;144
9;Part II: Lines with Few Angles;146
9.1;Chapter 9. Equiangular Lines;148
9.1.1;1. INTRODUCTION;148
9.1.2;2. DEFINITIONS AND EXAMPLES;149
9.1.3;3. BOUNDS FOR .(r);151
9.1.4;4. PILLARS;155
9.1.5;5. DETERMINATION OF .1/5(r);160
9.1.6;REFERENCES;166
9.2;Chapter 10. A SURVEY OF TWO–GRAPHS;167
9.2.1;1. INTRODUCTION;167
9.2.2;2. INTRODUCTORY EXAMPLES;168
9.2.3;3. SWITCHING OF GRAPHS;170
9.2.4;4. TWO-GRAPHS;171
9.2.5;5. EQUIANGULAR LINES;174
9.2.6;6. STRONG GRAPHS;175
9.2.7;7. REGULAR TWO-GRAPHS;177
9.2.8;8. RANK 3 GRAPHS;181
9.2.9;9. SYMPLECTIC AND ORTHOGONAL TWO-GRAPHS;184
9.2.10;10. UNITARY TWO-GRAPHS;186
9.2.11;11. SPORADIC TWO-GRAPHS;187
9.2.12;12. HADAMARD MATRICES;189
9.2.13;13. CONFERENCE MATRICES;191
9.2.14;REFERENCES;196
9.3;Chapter 11. THE REGULAR TWO-GRAPH ON 276 VERTICES;198
9.3.1;1. Introduction;198
9.3.2;2. Regular two-graphs;199
9.3.3;3. The case n = 276;203
9.3.4;4. Ternary codes;205
9.3.5;5. The 276-two-graph;207
9.3.6;References;212
9.4;Chapter 12. BOUNDS FOR SYSTEMS OF LINES AND JACOBI POLYNOMIALS;214
9.4.1;Abstract;214
9.4.2;1. Introduction;214
9.4.3;2. Jacobi polynomials;215
9.4.4;3. Addition formulae;217
9.4.5;4. Characteristic matrices;218
9.4.6;5. Special bounds for A-sets;221
9.4.7;6. Absolute bounds for A-sets;223
9.4.8;7. Properties of extremal A-sets;226
9.4.9;REFERENCES;228
9.5;Chapter 13. Line Graphs, Root Systems, and Elliptic Geometry;229
9.5.1;1. INTRODUCTION;229
9.5.2;2. LINES AT 60° AND 90°;230
9.5.3;3. ROOT SYSTEMS;234
9.5.4;4. GRAPHS WITH LEAST EIGENVALUES —2;239
9.5.5;5. SPECTRAL CHARACTERIZATION OF CERTAIN GRAPHS;245
9.5.6;6. AN APPLICATION TO HADAMARD MATRICES;249
9.5.7;REFERENCES;251
9.6;Chapter 14. TWO-GRAPHS, A SECOND SURVEY;252
9.6.1;1. INTRODUCTION;252
9.6.2;2. DEFINITION AND ENUMERATION;253
9.6.3;3. EQUIANGULAR LINES;256
9.6.4;4. AUTOMORPHISMS;258
9.6.5;5. ENUMERATION OF REGULAR TWO-GRAPHS;265
9.6.6;6. CONFERENCE TWO-GRAPHS OF ORDER pq2 + 1;267
9.6.7;7. MÖBIUS AND MINKOWSKI TWO-GRAPHS;271
9.6.8;REFERENCES;274
10;Part III: Matrices and Forms;276
10.1;Chapter 15. ORTHOGONAL MATRICES WITH ZERO DIAGONAL;278
10.1.1;1. Introduction;278
10.1.2;2. Paley matrices;279
10.1.3;3. Symmetric C-matrices;281
10.1.4;4. C-matrices and Hadamard matrices;284
10.1.5;REFERENCES;286
10.2;Chapter 16. QUASIREGULAR TWO-DISTANCE SETS;288
10.2.1;1. Introduction;288
10.2.2;2. Two-distance sets in Rn;288
10.2.3;3. The case of even dimensions;291
10.2.4;4. The case of odd dimensions;291
10.2.5;REFERENCES;294
10.3;Chapter 17. A SKEW HADAMARD MATRIX OF ORDER 36;295
10.3.1;References;296
10.4;Chapter 18. SYMMETRIC HADAMARD MATRICES OF ORDER 36;297
10.4.1;1. Introduction;297
10.4.2;2. Latin square graphs and Steiner graphs;298
10.4.3;3. Equivalence under switching;299
10.4.4;4. Regular Steiner graphs;301
10.4.5;5. The lines of PG(3,2);304
10.4.6;6. Rank 3 graphs of order 36;308
10.4.7;Referencs;309
10.5;Chapter 19. QUADRATIC FORMS OVER GF(2);311
10.5.1;§ 1. Introduction;311
10.5.2;§ 2. Quadratic and bilinear forms;311
10.5.3;§ 3. The rational vectors and their Gramian matrix;313
10.5.4;§ 4. Configurations, designs, and codes;315
10.5.5;REFERENCES;318
10.6;Chapter 20. On two-graphs, and Shult's characterization of symplectic and orthogonal geometries over GF(2);319
10.6.1;1. Introduction;319
10.6.2;2. Regular two-graphs;320
10.6.3;3. Symplectic and orthogonal geometries over GF(2);324
10.6.4;4. Characterization of the symplectic and orthogonal graphs;330
10.6.5;5. A problem by Hamelink;340
10.6.6;References;343
10.7;Chapter 21. The Krein condition, spherical designs, Norton algebras and permutation groups;344
10.7.1;1. INTRODUCTION;344
10.7.2;2. THE KREIN PARAMETERS;345
10.7.3;3. IMPRIMITIVE ASSOCIATION SCHEMES;346
10.7.4;4. SPHERICAL DESIGNS;347
10.7.5;5. NORTON ALGEBRAS;349
10.7.6;6. PERMUTATION GROUPS;350
10.7.7;7. A FINAL REMARK;353
10.7.8;REFERENCES;353
11;Part IV: Non-Euclidean Geometry;356
11.1;Chapter 22. The congruence order of the elliptic plane;358
11.2;Chapter 23. EQUI-ISOCLINIC SUBSPACES OF EUCLIDEAN SPACES;362
11.2.1;1. Introduction;362
11.2.2;2. Two n-subspaces;363
11.2.3;3. Equi-isoclinic subspaces;365
11.2.4;4. Block matrices;367
11.2.5;5. Determination of ..(n, 2n);369
11.2.6;REFERENCES;371
11.3;Chapter 24. METRIC PROBLEMS IN ELLIPTIC GEOMETRY;372
11.3.1;1. Introduction;372
11.3.2;2. Congruence Order;373
11.3.3;3. Elliptic Space En-1;373
11.3.4;4. Pillars;374
11.3.5;5. Further Results;378
11.3.6;6. Finite Groups, Error-Correcting Codes, and Two-Graphs;380
11.3.7;REFERENCES;383
11.4;Chapter 25. THE FOOTBALL;384
11.4.1;2. THE ICOSAHEDRON;384
11.4.2;3. ORBITS;385
11.4.3;4. APPROXIMATION OF STRENGTH t;387
11.4.4;5. INVARIANTS;388
11.4.5;6. THE DIHEDRAL GROUP OF ORDER 10;389
11.4.6;7. THE ICOSAHEDRAL GROUP;390
11.4.7;REFERENCES;391
11.5;Chapter 26. DISCRETE HYPERBOLIC GEOMETRY;393
11.5.1;1. Introduction;393
11.5.2;2. Lorentz space;395
11.5.3;3. Lifting;396
11.5.4;4. Graphs with .2 = 1;398
11.5.5;5. Reflexive graphs;400
11.5.6;6. Unimodular Euclidean lattices;404
11.5.7;7. Unimodular lattices of signature (p, 1);407
11.5.8;References;410
11.6;Chapter 27. FEW - DISTANCE SETS IN Rp, q;412
11.6.1;1. Introduction;412
11.6.2;2. The generalized addition formula;413
11.6.3;3. Bounds for s-distance sets;416
11.6.4;4. Equiangular lines in Rp,1;418
11.6.5;5. Two-distance sets;420
11.6.6;6. Two-angle sets of lines;421
11.6.7;7. Final remarks;423
11.6.8;REFERENCES;424
11.7;Chapter 28. Remark on Wielandt's Visibility Theorem;426
11.7.1;ABSTRACT;426
11.7.2;1. INTRODUCTION;426
11.7.3;2. THE THEOREM;426
11.7.4;3. THE PROOF;427
11.7.5;REFERENCES;427
11.8;Chapter 29. Complete List of Permissions;428
11.8.1;I. GRAPHS AND DESIGNS;428
11.8.2;II. LINES WITH FEW ANGLES;429
11.8.3;III. MATRICES AND FORMS;429
11.8.4;IV. NON-EUCLIDEAN GEOMETRY;430