SPECTRAL AND COMBINATORIAL PROPERTIES OF FRIENDSHIP GRAPHS, SIMPLICIAL ROOK GRAPHS, AND EXTREMAL EXPANDERS by Jason R. Vermette 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 2015 c 2015 Jason R. Vermette All Rights Reserved ProQuest Number: 3718380 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. ProQuest 3718380 Published by ProQuest LLC (2015). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 SPECTRAL AND COMBINATORIAL PROPERTIES OF FRIENDSHIP GRAPHS, SIMPLICIAL ROOK GRAPHS, AND EXTREMAL EXPANDERS by Jason R. Vermette 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 & Sciences Approved: James G. Richards, Ph.D. 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: Felix G. Lazebnik, 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: Steven K. Butler, Ph.D. Member of dissertation committee ACKNOWLEDGEMENTS I would like to thank Andrew Lang for sparking my interest in the theoretical side of mathematics during my time as an undergraduate. Thanks to my dissertation committee for taking the time to carefully read this thesis and offer advice and suggestions to improve it. Thanks to my adviser, Sebi Cioab˘a,for his patience and instruction for the past several years. His guidance was integral to my success as a graduate student. I would also like to thank him for his extremely useful comments on earlier versions of this thesis. Thanks most of all to my wife, Sharayah, for her endless support throughout my graduate career. I would certainly not have made it this far without her. iv TABLE OF CONTENTS LIST OF TABLES :::::::::::::::::::::::::::::::: vii LIST OF FIGURES ::::::::::::::::::::::::::::::: viii ABSTRACT ::::::::::::::::::::::::::::::::::: xi Chapter 1 INTRODUCTION :::::::::::::::::::::::::::::: 1 1.1 Algebraic Combinatorics ::::::::::::::::::::::::: 1 1.2 Definitions, Notation, and Basics of Graph Theory ::::::::::: 3 1.3 Spectra of Graphs ::::::::::::::::::::::::::::: 14 1.4 Vertex Partitions, Quotient Matrices, and Eigenvalue Interlacing ::: 19 1.5 Which Graphs are Determined by Their Spectra? ::::::::::: 24 2 THE FRIENDSHIP GRAPH AND GRAPHS WITH 4 DISTINCT EIGENVALUES ::::::::::::::::::::::::::::::: 28 2.1 The Friendship Graph :::::::::::::::::::::::::: 28 2.2 Connected Graphs Cospectral to the Friendship Graph :::::::: 31 2.3 The Graphs With All But Two Eigenvalues Equal to 1 or −1 ::::: 37 2.4 Characterization of the Graphs in G. :::::::::::::::::: 38 2.5 The Proof of the Characterization of G ::::::::::::::::: 42 3 SIMPLICIAL ROOK GRAPHS ::::::::::::::::::::: 56 3.1 Rook Graphs and Simplicial Rook Graphs ::::::::::::::: 56 3.2 Independence Number :::::::::::::::::::::::::: 60 3.3 The Smallest Eigenvalue ::::::::::::::::::::::::: 64 3.4 Partial Spectrum ::::::::::::::::::::::::::::: 67 3.5 Spectrum of SR(m; n) for m ≤ 3 or n ≤ 3 ::::::::::::::: 71 3.6 Integrality of the Spectrum :::::::::::::::::::::::: 76 3.7 Diameter :::::::::::::::::::::::::::::::::: 79 3.8 Clique Number :::::::::::::::::::::::::::::: 82 v 3.9 Cospectral Mates ::::::::::::::::::::::::::::: 84 3.10 Multiplicity of the Smallest Eigenvalue ::::::::::::::::: 88 3.11 Spectrum of SR(m; n) for m; n ≥ 4 ::::::::::::::::::: 91 4 LARGE REGULAR GRAPHS WITH FIXED VALENCY AND SECOND EIGENVALUE ::::::::::::::::::::::::: 98 4.1 The Second Eigenvalue of Regular Graphs ::::::::::::::: 98 4.2 Interlacing Bound ::::::::::::::::::::::::::::: 100 4.3 Results of the Interlacing Bound ::::::::::::::::::::: 102 4.4 Linear Programming Bound ::::::::::::::::::::::: 115 5 OPEN PROBLEMS AND FUTURE WORK ::::::::::::: 121 5.1 The Graphs With All But Two Eigenvalues Equal to 0 or −2 ::::: 121 5.2 Partial Permutohedra ::::::::::::::::::::::::::: 126 5.3 Problems and Questions on the Second Eigenvalue of Regular Graphs 136 BIBLIOGRAPHY :::::::::::::::::::::::::::::::: 139 vi LIST OF TABLES 3.1 Spectrum of SR(3; n). ::::::::::::::::::::::::: 71 3.2 Spectrum of SR(m; 3). ::::::::::::::::::::::::: 72 3.3 Spectrum of SR(m; 4). ::::::::::::::::::::::::: 93 3.4 Spectrum of SR(m; 5). ::::::::::::::::::::::::: 94 3.5 Spectra of SR(4; n). :::::::::::::::::::::::::: 96 3.6 Spectra of SR(5; n). :::::::::::::::::::::::::: 97 4.1 Families of graphs meeting the bound M(k; t; c). ::::::::::: 118 4.2 Sporadic of graphs meeting the bound M(k; t; c). :::::::::: 118 4.3 Summary of our results. :::::::::::::::::::::::: 120 vii LIST OF FIGURES 1.1 The Graphs P3 and C5. :::::::::::::::::::::::: 3 1.2 The complements of P3 and C5. :::::::::::::::::::: 6 ∼ 1.3 The graphs K5, K3;2, Ci10(1; 4), and K2;2;2 = CP (3). :::::::: 7 1.4 The graphs P3, C3, and P3 [ C3. :::::::::::::::::::: 8 1.5 The graphs P3, C3, P3 + C3, and P3 r C3. :::::::::::::: 8 1.6 The graphs P3, C3, and P3C3. :::::::::::::::::::: 8 1.7 The graphs Q0, Q1, Q2, Q3, and Q4. ::::::::::::::::: 9 1.8 Block and standard representation of graphs. :::::::::::: 9 1.9 The Petersen graph. :::::::::::::::::::::::::: 11 1.10 The Rook graph R(2; 3). :::::::::::::::::::::::: 11 1.11 The Heawood graph. :::::::::::::::::::::::::: 12 1.12 The Tutte{Coxeter graph. ::::::::::::::::::::::: 12 1.13 Two nonisomorphic graphs with spectrum f21; 03; −21g. :::::: 24 1.14 A pair of nonisomorphic cospectral graphs on 9 vertices obtained by GM-switching. ::::::::::::::::::::::::::::: 27 1.15 A pair of nonisomorphic cospectral graphs on 7 vertices obtained by GM-switching. ::::::::::::::::::::::::::::: 27 2.1 The Friendship graph Fk for several values of k. ::::::::::: 28 2.2 Graphs with λ2 > 1 or λmin −1 < −1. ::::::::::::::::: 33 viii 2.3 The graphs in G. :::::::::::::::::::::::::::: 40 2.4 The graph F16 and its cospectral mate B6(3; 5) + 10K2. ::::::: 41 2.5 A subgraph induced by G if a vertex in Z0 has two neighbors in Z0. 51 3.1 The integer lattice points in the n-th dilate of the standard simplex in R3, n = 1; 2. ::::::::::::::::::::::::::::::: 57 3.2 Lattice points in the n-th dilate of the standard simplex in R3 viewed as 3-tuples summing to n, n = 2; 3. :::::::::::::::::: 58 3.3 The Simplicial Rook graphs SR(3; 4) and SR(4; 3). ::::::::: 59 3.4 Maximal independent sets in SR(3; 8), SR(3; 9), and SR(3; 10). :: 61 3.5 Maximal independent sets in SR(3; 3) and SR(4; 3). ::::::::: 65 3.6 Maximal cliques of size m = 4 and n + 1 = 4 in SR(4; 3). :::::: 82 3.7 The set V1 as a GM-switching set in SR(4; 3). :::::::::::: 85 3.8 The set f3e1; 2e1 + e2; e1 + 2e2; 3e2g as a GM-switching set in SR(4; 3). ::::::::::::::::::::::::::::::::: 87 4.1 Two small graphs with spectral radius greater than 2. :::::::: 100 4.2 The Tutte{Coxeter graph. ::::::::::::::::::::::: 103 4.3 The Odd graph O4. ::::::::::::::::::::::::::: 105 4.4 The Heawood graph. :::::::::::::::::::::::::: 107 4.5 A 3-regular graph on 8 vertices with girth 4. ::::::::::::: 108 p 4.6 The 4-regular graphs λ2 = 5 − 1. :::::::::::::::::: 110 4.7 The 3-regular graphs on 18 vertices with λ2 < 1:9. :::::::::: 113 4.8 A graph with spectral radius 2. :::::::::::::::::::: 113 5.1 Graphs inducing line graphs with two positive eigenvalues. ::::: 125 ix 5.2 The π-admissible permutations for π = (3; 2; 1; 4). :::::::::: 128 5.3 The subgraph SR(4; 3)jParp(π) for π = (3; 2; 1; 4). ::::::::::: 129 5.4 The subgraph SR(4; 3)jParp(π) for π = (3; 2; 1; 4) within SR(4; 3). :: 129 5.5 The unique 3-regular graph with largest least eigenvalue less than −2. 137 x ABSTRACT Algebraic combinatorics is the area of mathematics that uses the theories and methods of abstract and linear algebra to solve combinatorial problems, or conversely applies combinatorial techniques to solve problems in algebra. In particular, spectral graph theory applies the techniques of linear algebra to study graph theory. Spec- tral graph theory is the study of the eigenvalues of various matrices associated with graphs and how they relate to the properties of those graphs. The graph properties of diameter, independence number, chromatic number, connectedness, toughness, hamil- tonicity, and expansion, among others, are all related to the spectra of graphs. In this thesis we study the spectra of various families of graphs, how their spectra relate to their properties, and when graphs are determined by their spectra. We focus on three topics (Chapters 2{4) in spectral graph theory. The wide range of these topics showcases the power and versatility of the eigenvalue techniques such as interlacing, the common thread that ties these topics together. In Chapter1, we review the basic definitions, notations, and results in graph theory and spectral graph theory. We also introduce powerful tools for determining the structure of a graph and its subgraphs using eigenvalue interlacing.