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Interactions Between Coherent Configurations and Some Classes Interactions between Coherent Configurations and Some Classes of Objects in Extremal Combinatorics Thesis submitted in partial fulfillment of the requirements for the degree of \DOCTOR OF PHILOSOPHY" by Matan Ziv-Av Submitted to the Senate of Ben-Gurion University of the Negev November 9, 2013 Beer-Sheva Interactions between Coherent Configurations and Some Classes of Objects in Extremal Combinatorics Thesis submitted in partial fulfillment of the requirements for the degree of \DOCTOR OF PHILOSOPHY" by Matan Ziv-Av Submitted to the Senate of Ben-Gurion University of the Negev Approved by the Advisor Approved by the Dean of the Kreitman School of Advanced Graduate Studies November 9, 2013 Beer-Sheva This work was carried out under the supervision of Mikhail Klin In the Department of Mathematics Faculty of Natural Sciences Research-Student's Affidavit when Submitting the Doctoral Thesis for Judgment I, Matan Ziv-Av, whose signature appears below, hereby declare that: x I have written this Thesis by myself, except for the help and guidance offered by my Thesis Advisors. x The scientific materials included in this Thesis are products of my own research, culled from the period during which I was a research student. This Thesis incorporates research materials produced in cooperation with others, excluding the technical help commonly received during ex- perimental work. Therefore, I am attaching another affidavit stating the contributions made by myself and the other participants in this research, which has been approved by them and submitted with their approval. Date: November 9, 2013 Student's name: Matan Ziv-Av Signature: Acknowledgements First, I'd like to thank my advisor, Misha Klin, for being my advisor. While this work is mine, it is much easier for me to imagine another student producing a similar thesis under Prof. Klin's guidance, than to imagine myself producing a similar thesis with another advisor. I would like to thank Mati Rubin, who was very helpful in my life in BGU, both mathematical and personal, ever since my first class here 23 years ago. I'd like to thank Yoav Segev, my advisor in my previous, failed, attempt as a graduate student. I hope that this thesis indicates to him that I did not completely waste his time. Work on this thesis was facilitated by cooperation with many of Prof. Klin's colleagues, mostly through him. Misha Muzychuk's ideas and advice are used in almost every chapter of the thesis. M. Maˇcaj, J. Lauri, A. Woldar, and G. Jones were also very kind with their assistance. Two of my advisor's previous students, Christian Pech and Sven Re- ichard, allowed me to use their work in progress COCO-II package for GAP. Discussion with both of them allowed me to improve my mathematical pro- gramming skills. Many discussions with students of Klin, Danny Kalmanovich and Nim- rod Kriger were also very useful to me. I thank the entire staff of the Department of Mathematics in BGU for the support from the department and its people, even though I was a somewhat problematic student. While working on this thesis, I discussed its results in seminars in JKU (Linz, Austria), in Wroclaw University (Poland), TU/e (Eindhoven, Nether- lands) and The University of West Bohemia (Plzen, Czech Republic)). I thank Pech, Andrzej Kisielewicz, Aart Blokhuis, and Zdenek Ryjacek for inviting me, as well as Hapoel Tel-Aviv FC for making those visits possible. Last, but certainly not least, I thank my family (parents, grandparents, brothers, sisters, aunts, uncles, cousins, in-laws) for being very supportive and pushing me just enough to make sure this project actually reached its destination. Contents 1 Introduction 1 1.1 Motivation . 1 1.1.1 Repeatability of results . 2 1.2 Outline of the thesis . 3 1.2.1 Preliminaries . 3 1.2.2 Summary of results . 4 2 Preliminaries 9 2.1 Coherent configurations and association schemes . 9 2.1.1 Axioms and basic definitions . 9 2.1.2 Isomorphism and automorphism types . 12 2.1.3 Coherent closure . 13 2.1.4 Mergings . 14 2.2 Triangle-free strongly regular graphs . 15 2.2.1 Strongly regular graphs . 15 2.2.2 Triangle free strongly regular graphs . 16 2.2.3 The 7 known tfSRGs . 17 2.3 Equitable partitions . 20 2.3.1 Definitions and basic properties . 20 2.3.2 Global vs. local approaches . 22 2.4 Schur rings . 22 2.5 Semisymmetric graphs . 24 2.5.1 Definitions and basic facts . 24 2.5.2 Nikolaev graph N .................... 26 i ii CONTENTS 2.5.3 Ljubljana graph L ................... 27 2.6 Cages . 28 2.7 Computer tools . 30 2.7.1 COCO . 30 2.7.2 WL-stabilization . 31 2.7.3 GAP . 31 2.7.4 COCO v.2 . 32 2.7.5 Ad-hoc tools . 32 3 Triangle-free strongly regular graphs 33 3.1 Embeddings of tfSRGs inside tfSRGs . 33 3.1.1 Computer results . 34 3.1.2 Theoretical view of some embeddings . 34 3.1.3 Imprimitive tfSRGs inside primitive tfSRGs . 39 3.2 Equitable partitions of tfSRGs . 41 3.2.1 Pentagon . 43 3.2.2 Petersen graph . 43 3.2.3 Clebsch graph . 44 3.2.4 Hoffman-Singleton graph . 44 3.2.5 Gewirtz graph . 55 3.2.6 Mesner graph . 55 3.2.7 NL2(10) . 56 3.3 Understanding some EPs . 56 3.3.1 Some models of the Hoffman-Singleton graph . 56 3.3.2 Some models of the Gewirtz graph . 58 3.3.3 Quadrangle EP of NL2(10) . 61 4 Schur rings over A5, AGL1(8) and Z11oZ5 64 4.1 S-rings over A5 ......................... 65 4.1.1 Computer results . 65 4.1.2 Rational S-rings . 68 4.1.3 General Outline of Non-Schurian S-rings . 70 4.1.4 The Exceptional non-Schurian S-ring #115 . 73 CONTENTS iii 4.2 S-rings over AGL1(8) . 74 4.3 Symmetric S-rings over Z11oZ5 . 75 5 Other results 78 5.1 Links between two semisymmetric graphs on 112 vertices . 78 5.1.1 A master association scheme on 56 points . 78 5.1.2 Embeddings of L into N . 82 5.1.3 The Ljubljana configuration . 83 5.2 Coherent cages . 85 5.2.1 Coherency of small cages . 85 5.2.2 Non-Schurian association scheme on 90 points . 85 5.3 Non-Schurian association scheme . 86 5.3.1 General case . 86 5.3.2 Non-Schurian association scheme on 125 points . 86 5.4 Miscellanea . 87 5.4.1 Modified Weisfeiler-Leman stabilization . 87 5.4.2 Pseudo S-rings . 92 6 Concluding remarks 93 A Data 96 A.1 Equitable partitions of Clebsch graph . 96 A.2 Mergings of master association scheme M . 97 A.3 Basic sets of pseudo S-rings . 99 A.4 Computer readable data . 100 Bibliography 102 List of Figures 1 Petersen graph . 17 2 Clebsch graph . 18 3 Hoffman-Singleton graph, Robertson model . 19 4 Mesner's model of NL2(10) . 20 2 5 Mesner decomposition of NLg(g + 3g) . 37 6 A split of the non-edge decomposition of NL2(10) . 62 7 Paley tournament P (7) with isolated vertex . 83 List of Tables 1 Mesner's table of feasible parameter sets of tfSRGs . 17 2 Number of tfSRGs inside tfSRGs . 37 3 Number of orbits of tfSRGs inside tfSRGs . 38 4 Orders of stabilizers of tfSRGs inside tfSRGs . 38 5 Structure of stabilizers of tfSRGs inside tfSRGs . 38 6 Number of imprimitive tfSRGs inside tfSRGs . 40 7 Number of orbits of imprimitive tfSRGs inside tfSRGs . 40 iv List of Tables v 8 Number of orbits of equitable partitions and of automorphic equitable partitions for known tfSRGs. 41 9 Number of orbits of EPs and automorphic EPs of the pentagon by size of partition . 41 10 Number of orbits of EPs and automorphic EPs of the Petersen graph by size of partition . 42 11 Number of orbits of EPs and automorphic EPs of the Clebsch graph by size of partition . 42 12 Number of orbits of EPs and automorphic EPs of the Hoffman- Singleton graph by size of partition . 42 13 Number of orbits of automorphic and non-rigid EPs of the Gewirtz graph by size of partition . 42 14 Number of orbits of automorphic EPs of the Mesner graph by size of partition . 42 15 Number of orbits of automorphic EPs of NL2(10) by size of partition . 43 16 Equitable partitions of a Petersen graph . 43 17 Summary of equitable partitions of a Hoffman-Singleton graph . 45 18 Distribution of S-rings over A5 with respect to rank . 66 19 List of orders of small groups over A5 . 67 20 Rational S-rings over A5 having a small automorphism group . 69 21 Information about non-Schurian S-rings over A5 . 71 22 Distribution of S-rings over AGL1(8) with respect to rank . 74 23 List of orders of small groups over AGL1(8) . 75 24 Orbits of symmetric S-rings over Z11oZ5 . 76 25 Orbits of non-symmetric S-rings over Z11oZ5 . 77 26 2-orbits of M and their covers . 81 Interactions between Coherent Configurations and Some Classes of Objects in Extremal Combinatorics Submitted by: Matan Ziv-Av Advisor: Mikhail Klin Abstract A strongly regular graph with parameters (v; k; λ, µ) is a regular graph on v vertices with valency k, such that two neighbors have λ common neighbors and two non-neighbors have µ common neighbors. An equitable partition of a graph is a partition of the vertex set such that the number of neighbors a vertex from cell X has in cell Y depends only on the selection of the cells, not the selection of vertex.
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