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Combinatorial and Spectral Properties of Graphs and Association Schemes

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Combinatorial and Spectral Properties of Graphs and Association Schemes COMBINATORIAL AND SPECTRAL PROPERTIES OF GRAPHS AND ASSOCIATION SCHEMES by Matt McGinnis A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Spring 2018 c 2018 Matt McGinnis All Rights Reserved COMBINATORIAL AND SPECTRAL PROPERTIES OF GRAPHS AND ASSOCIATION SCHEMES by Matt McGinnis Approved: Louis F. Rossi, Ph.D. Chair of the Department of Mathematical Sciences Approved: George H. Watson, Ph.D. Dean of the College of Arts and Sciences Approved: Ann L. Ardis, Ph.D. Senior Vice Provost for Graduate and Professional Education I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Sebastian M. Cioab˘a,Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Robert S. Coulter, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Nayantara Bhatnager, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: William J. Martin, Ph.D. Member of dissertation committee ACKNOWLEDGEMENTS I give my greatest thanks to my advisor, Sebi. He has helped me grow as a mathematician over the past five years. His comments on previous versions of this thesis were invaluable. I could not have done this without him. Next, I would like to thank my committee members for taking an interest in my work. I truly appreciate the time they took to look at this dissertation and the advice they had to offer. I would also like to thank all the wonderful friends I made while in graduate school. I never would have lasted this long without you all to laugh and complain with. Finally, I would like to thank my family (Mom, Dad, Molly and Brooke). With- out their love and support none of this would have been possible. iv TABLE OF CONTENTS LIST OF TABLES :::::::::::::::::::::::::::::::: vii LIST OF FIGURES ::::::::::::::::::::::::::::::: viii ABSTRACT ::::::::::::::::::::::::::::::::::: x Chapter 1 INTRODUCTION :::::::::::::::::::::::::::::: 1 1.1 Algebraic Combinatorics ::::::::::::::::::::::::: 1 1.2 Definitions, Notations and Basic Graph Theory :::::::::::: 3 1.3 Spectra of Graphs ::::::::::::::::::::::::::::: 9 1.4 Strongly Regular Graphs, Distance-Regular Graphs and Association Schemes :::::::::::::::::::::::::::::::::: 16 1.4.1 Strongly regular graphs :::::::::::::::::::::: 17 1.4.2 Association schemes ::::::::::::::::::::::: 18 1.4.3 Distance-regular graphs ::::::::::::::::::::: 22 2 THE SMALLEST EIGENVALUES OF HAMMING GRAPHS AND JOHNSON GRAPHS :::::::::::::::::::::::: 25 2.1 The Hamming Association Scheme ::::::::::::::::::: 26 2.1.1 The graphs in the Hamming scheme ::::::::::::::: 26 2.1.2 Krawtchouk polynomials ::::::::::::::::::::: 27 2.2 The Johnson Association Scheme :::::::::::::::::::: 33 2.2.1 The graphs in the Johnson scheme ::::::::::::::: 33 2.2.2 Eberlein polynomials ::::::::::::::::::::::: 34 2.3 The Max-Cut Problem :::::::::::::::::::::::::: 41 2.4 Distance-j Hamming Graphs ::::::::::::::::::::::: 46 v 2.5 Distance-j Johnson Graphs ::::::::::::::::::::::: 53 2.6 Open Problems :::::::::::::::::::::::::::::: 61 3 COSPECTRAL MATES FOR GRAPHS IN THE JOHNSON ASSOCIATION SCHEME ::::::::::::::::::::::::: 63 3.1 Godsil-McKay Switching and Cospectral Graphs :::::::::::: 64 3.2 Spectral Characterization of Graphs in the Johnson Scheme ::::: 68 3.3 J(n; 2k + 1;S) where S = f0; 1; : : : ; kg ::::::::::::::::: 72 3.4 J(3m − 2k − 1; m; S) where S = f0; : : : ; kg ::::::::::::::: 75 3.5 Relation to Previous Work :::::::::::::::::::::::: 78 3.5.1 Relation to Kneser graphs and Johnson graphs ::::::::: 78 3.5.2 Vertices containing [m − 2] as a possible switching set ::::: 79 3.6 Computational Results :::::::::::::::::::::::::: 80 3.7 Open Problems :::::::::::::::::::::::::::::: 83 4 CONNECTIVITY OF SOME GRAPHS IN ASSOCIATION SCHEMES :::::::::::::::::::::::::::::::::: 84 4.1 Some Conjectures of Brouwer and Godsil :::::::::::::::: 84 4.2 Connectivity of Graphs in Association Schemes :::::::::::: 85 4.3 Disconnecting Distance-Regular Graphs into Nonsingleton Components by Removing Edges :::::::::::::::::::: 87 4.4 The Grassmann Graphs and Twisted Grassmann Graphs ::::::: 97 4.5 Edge-Connectivity of Distance-j Twisted Grassmann Graphs ::::: 100 4.6 Open Problems :::::::::::::::::::::::::::::: 102 4.6.1 Conjecture 4.1.3 and Conjecture 4.1.4 :::::::::::::: 102 4.6.2 Vertex-connectivity of graphs in association schemes :::::: 102 4.6.3 Disconnecting distance-regular graphs into nonsingleton components by removing vertices :::::::::::::::: 103 BIBLIOGRAPHY :::::::::::::::::::::::::::::::: 107 vi LIST OF TABLES 3.1 Switching sets checked for in J(n; m; S) with m = 3 ::::::::: 82 3.2 Switching sets checked for in J(n; m; S) with m = 4 ::::::::: 82 3.3 Switching sets checked for in J(n; m; S) with m = 5 ::::::::: 82 4.1 Cubic distance-regular graphs of diameter at least three :::::: 90 vii LIST OF FIGURES 1.1 The graphs P3 and C6 ::::::::::::::::::::::::: 4 1.2 The complements of P3 and C6 :::::::::::::::::::: 6 1.3 The graphs K4 and K3;3 :::::::::::::::::::::::: 7 1.4 Graph isomorphism example :::::::::::::::::::::: 7 1.5 Cartesian product K2 K2 :::::::::::::::::::::: 8 1.6 Two nonisomorphic graphs with spectrum f2(1); 0(3); (−2)(1)g :::: 16 1.7 The graphs C5 and J(5; 2) ::::::::::::::::::::::: 17 2.1 Hamming scheme H(3; 2) ::::::::::::::::::::::: 27 2.2 H(4; 3; 1) with the first eigenmatrix of H(4; 3) :::::::::::: 33 2.3 Johnson scheme J (5; 2) :::::::::::::::::::::::: 34 2.4 J(7; 3; 1) with the first eigenmatrix of J (7; 3) :::::::::::: 40 2.5 Max-Cut visualization ::::::::::::::::::::::::: 41 3.1 GM-switching produces a nonisomorphic graph ::::::::::: 66 3.2 GM-switching produces an isomorphic graph ::::::::::::: 67 3.3 GM-switching on a regular graph ::::::::::::::::::: 67 3.4 Switching sets for the Chang graphs from J(8; 2) :::::::::: 68 3.5 Example of GM-switching on J(7; 3) ::::::::::::::::: 70 3.6 Example of GM-switching on K(8; 3) ::::::::::::::::: 71 viii 3.7 Example of GM-switching on J(11; 5; f0; 1; 2g) :::::::::::: 73 3.8 Example of GM-switching on J(9; 4; f0; 1g) :::::::::::::: 76 4.1 Taylor graph with local graph Sp(2r; 2) :::::::::::::::: 105 ix ABSTRACT The main topics of this dissertation are related to spectral graph theory, a subtopic of algebraic combinatorics. Algebraic combinatorics is the area of mathemat- ics that implements techniques from linear and abstract algebra to solve problems in combinatorics as well as techniques from combinatorics to study algebraic structures. Spectral graph theory focuses on the study of the eigenvalues associated with various matrices for a graph and how the eigenvalues relate to structural properties of the graph. Properties such as connectedness, diameter, independence number, chromatic number and regularity, among others, are all related to the spectrum of a graph. In this dissertation we will study the spectra of various graphs and incorporate well known techniques in spectral graph theory to gain a better understanding of the structure of these graphs. We focus on three topics (Chapters2,3 and4). The variation of these topics reinforces how diverse and useful spectral techniques in graph theory can be. In Chapter1 we cover notation and basic definitions used throughout this disser- tation. We also introduce some well known and powerful results relating the structural properties of a graph to its spectrum. This includes a discussion of the properties of graphs that can be established from the spectrum as well as which graphs are deter- mined by their spectrum. Finally, we give definitions and basic results for association schemes and distance-regular graphs since the results of this dissertation are related to these structures. In Chapter2 we study the smallest eigenvalues for distance- j graphs in both the Hamming and Johnson association schemes. Our results for distance-j Hamming graphs settle a conjecture proposed by Van Dam and Sotirov in [29]. In fact, we reach a stronger conclusion than the one proposed by Van Dam and Sotirov. Our results for x distance-j Johnson graphs settle a conjecture proposed by Karloff in [46]. Again, we are able to obtain a stronger conslusion than what is presented in Karloff's conjecture. In Chapter3 we use the technique of Godsil-McKay switching to construct cospectral mates for graphs formed by taking the union of relations in the Johnson association scheme. Our results offer insight into which graphs in this scheme are not determined by their spectrum. Our work also unifies the switching sets previously found for Johnson graphs in [26] and Kneser graphs in [43]. We also present some open problems related to our work, including a switching set that we would like to see generalized in order to obtain a new infinite family of graphs in the Johnson scheme that are not determined by their spectrum. In Chapter4 we examine connectivity properties of distance-regular
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