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Integral Cayley Graphs INTEGRAL CAYLEY GRAPHS by Azhvan Ahmady M.Sc., Sharif University of Technology, 2006 B.Sc., Teacher Training University, 2004 a Thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics Faculty of Science c Azhvan Ahmady 2013 SIMON FRASER UNIVERSITY Summer 2013 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for \Fair Dealing." Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Azhvan Ahmady Degree: Doctor of Philosophy Title of Thesis: Integral Cayley Graphs Examining Committee: Dr. Karen Yeats, Assistant Professor Chair Dr. Bojan Mohar, Professor Senior Supervisor Dr. Jason Bell, Professor Supervisor Dr. Ladislav Stacho, Associate Professor SFU Examiner Dr. Joel Friedman, Professor of Mathematics, University of British Columbia, External Examiner Date Approved: August 22nd, 2013 ii Partial Copyright Licence iii Abstract A graph X is said to be integral if all eigenvalues of the adjacency matrix of X are integers. This property was first defined by Harary and Schwenk who suggested the problem of classifying integral graphs. Since the general problem of classifying integral graphs seemed too difficult, graph theorists started to investigate special classes of graphs which included trees, graphs of bounded degree, regular graphs and Cayley graphs. What proves so interesting about this problem is that no one can yet identify what the integral trees are or which 5-regular graphs are integral. In this thesis, integral Cayley graphs are studied. Several topics on the integral Cayley graphs are presented. First, a classification of integral Cayley graphs over abelian groups in terms of the associated Boolean algebra of the subgroups is presented. Secondly, the notions of character and representation integrality are introduced. It has been shown that character integrality is a weaker notion than representation integrality. An internal classification of character integral subsets is proved. General results about representation integral subsets are presented and in an attempt to generalize the results from abelian to non-abelian case, Hamiltonian and dihedral groups are studied. Thirdly, two open problems about integrality of Cayley graphs are solved. Simple eigenvalues in Cayley graphs are studied, and some observations lead to two interesting results in this topic. Finally, the classification of cubic and 4-regular integral Cayley graphs are presented. A general approach to characterize all integral Cayley graphs over abelian groups is presented. Furthermore, a sharp upper bound over the size of the group in terms of the graph degree has been suggested and proved. The thesis concludes with a section devoted to open problems and conjectures in this area. iv To Sweet Penguin v Acknowledgments Firstly, I would like to thank my advisor, Bojan Mohar, for all his help and guidance that he has given me throughout my graduate studies. I would like to show my sincere gratitude to Jason Bell, my supervisor from whom I have learned a great deal and from his enthusiasm for mathematics I have greatly benefited. I want to thank my great friends Azad Shokri, for being a good chiropractor and cracking my back many times, and Sophie Burrill who has been very supportive and has made my studies in SFU enjoyable. My family has always been very supportive of my academic career. Their constant love and encouragement have always been with me, for which I am very grateful. Special thanks goes to my love Skyler M. Warren for all of her support during the writing of this thesis. vi Contents Approval ii Partial Copyright License iii Abstract iv Dedication v Acknowledgments vi Contents vii List of Figures x 1 Preliminaries 1 1.1 Introduction . .1 1.1.1 Graphs . .2 1.1.2 Linear Algebra . .5 1.1.3 Group theory and Algebra . .7 1.2 Overview of the thesis . 12 2 Integral Cayley graphs over abelian groups 13 2.1 Introduction . 14 2.2 Integral Cayley graphs . 18 2.3 B-integrality . 21 2.4 Integral Cayley graphs over abelian groups . 24 vii 3 Integral Cayley graphs over non-abelian groups 27 3.1 Introduction . 27 3.2 Character and representation integrality . 30 3.3 Hamiltonian groups . 37 3.4 Some special cases . 40 3.4.1 Simple Cayley graphs of Q8 × Cp, where p 6=3 ............. 40 3.4.2 Q8 × C3 ................................... 43 3.4.3 Cayley multigraphs of Q8 × Cp ...................... 43 d 3.4.4 Simple Cayley graphs of Q8 × Cp ..................... 45 3.5 Dihedral groups . 47 3.5.1 Irreducible representations of Dn ..................... 47 3.5.2 The case Dp where p isaprimenumber ................. 50 3.5.3 The general case . 52 4 CIS and Cayley integral groups 54 4.1 Introduction . 54 4.2 Cayley integral simple groups . 56 4.3 Cayley Integral Groups . 62 5 Integral Cayley graphs of small degree 78 5.1 Abelian groups admitting integral Cayley graphs of small degree . 79 5.2 Non-abelian groups admitting cubic integral Cayley graphs . 86 5.3 Non-abelian groups admitting 4-regular integral Cayley graphs . 89 6 Miscellaneous results 94 6.1 Cayley graphs with small number of distinct eigenvalues . 94 6.2 Simple eigenvalues in Cayley graphs . 96 6.3 Open problems and conjectures . 100 Appendix A Representation theory in GAP 102 A.1 Representation theory of non-abelian groups of order 12 . 103 A.2 Representation theory of non-abelian groups of order 18 . 106 A.3 Representation theory of non-abelian groups of order 24 . 111 viii Appendix B Cayley graphs programming 133 B.1 Spectrum computation with GAP . 133 B.2 Cayley graph construction with GRAPE . 137 B.3 (Z6 × Z2) o Z2 admits no connected cubic integral Cayley graph . 139 Bibliography 143 ix List of Figures 2.1 Non-cubic connected integral graphs with ∆ ≤ 3 ................ 16 2.2 Tutte 8-cage, smallest cubic graph of girth 8 . 17 2.3 Complete Gray Code in hypercube Q4 ...................... 18 5.1 Connected cubic bipartite integral graphs. 92 5.2 Connected cubic non-bipartite integral graphs. 93 x Chapter 1 Preliminaries 1.1 Introduction A graph essentially is a discrete mathematical model of a network of objects. For example, such objects can be cities, computers, atoms, or those that are more abstract. Graphs are applied in numerous fields, like chemistry, social science, electrical engineering, architecture, computer science and many others. Roughly speaking, a graph is a set of vertices representing the nodes of the network (cities, computers or atoms), and a set of edges between vertices. These edges can represent roads between cities, links between computers, or bonds between atoms. These edges may have weights, representing distances, capacities, forces, and they can be directed (one-way roads). There is a large variety of problems in graph theory. One classic example is the famous traveling salesman problem: given a list of cities and the distances between each pair of cities, what is the shortest possible route to visit each city exactly once and return to the origin city? For small graphs this problem may seem easy, but as the number of vertices increases, the problem becomes very difficult. Such problem can be applied to quite different areas, for example in planning, logistics, and the manufacture of microchips or DNA sequencing, the process of determining the precise order of nucleotides within a DNA molecule. One way to cope with the difficulty of working with large graphs is to use computers. In order to read and store graphs into computers we represent graphs with matrices. Depending on the specific problem and personal preference, graph theorists use different kinds of matrices to represent a graph, the most popular ones being the (0; 1)-adjacency matrix and the Laplace matrix. Often, the algebraic properties of the matrix are used as a 1 CHAPTER 1. PRELIMINARIES 2 bridge between different kinds of structural properties of the graph. The relation between the structural (combinatorial, topological) properties of the graph and the algebraic ones of the corresponding matrix is therefore a very interesting one. For example, using the spectrum of light, scientists have had great success in being able to indirectly determine the compounds of chemicals that could not be directly measured. For instance, by examining the light given off by stars we can determine their chemical composition, even though we could never directly gather any material from those stars. Another interesting example is from theoretical chemistry, where chemists associate a graph with hydrocarbon molecule. The eigenvalues of the matrix of this graph are used to predict stability of the molecule. In an analogous way we can use the spectra of various matrices (i.e., the eigenvalues of the matrices) to get information about a graph that would otherwise be difficult to obtain. In this chapter, we will provide some introductory comments about connections between the eigenvalues of matrices and the properties of graphs in a general setting. The study of the relations between these two objects is spectral graph theory. Thus to work in spectral graph theory, one not only need to be familiar with graph theory but also must understand the basic tools from algebra. Eigenvalues, eigenvectors, determinants, Courant-Fischer formula, Perron-Frobenius and others are the tools of the trade. In this thesis we study special classes of graphs which have a lot of structure. In the eye of the mathematical beholder, graphs with significant structure and symmetry are the most beautiful graphs. Cayley graphs are the main graphs we study and they are used in many different areas. In computer science Cayley graphs are used for the design and analysis of network architectures for parallel computers.
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