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Isospectral Graphs and the Expander Coefficient Western Michigan University ScholarWorks at WMU Dissertations Graduate College 8-1994 Isospectral Graphs and the Expander Coefficient Ian Campbell Walters Jr. Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Applied Mathematics Commons Recommended Citation Walters, Ian Campbell Jr., "Isospectral Graphs and the Expander Coefficient" (1994). Dissertations. 1849. https://scholarworks.wmich.edu/dissertations/1849 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. ISOSPECTRAL GRAPHS AND THE EXPANDER COEFFICIENT by Ian Campbell Walters Jr. A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Department of Mathematics and Statistics Western Michigan University Kalamazoo, Michigan August 1994 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ISOSPECTRAL GRAPHS AND THE EXPANDER COEFFICIENT Ian Campbell Walters Jr., Ph.D. Western Michigan University, 1994 The expander coefficient of a graph is a parameter that is utilized to quantify the rate at which information is spread throughout a graph. The eigenvalues of the Lapladan of a graph provide a bound for the expander coefficient of the graph. In this dissertation, we construct many pairs of isospectral graphs with different expander coefficients. In Chapter I, we define the problem and present some preliminary definitions. We then introduce two constructions that are related to graph composition and that may be employed to produce cospectral and isospectral graphs. In Chapter II, we investigate the connectivity of and distance in the graphs formed by the constructions of Chapter I. In Chapter HI, we construct four pairs of infinite sequences of isospectral graphs. For each pair of infinite sequences, it is demonstrated that the expander coefficients of the corresponding graphs of the sequences are different. We determine the limit of each sequence of each pair, and see that these limits need not be equal. In Chapter IV, the Folkman graph is used to construct a pair of isospectral graphs that more accurately model networks. The expander coefficients for the graphs are then shown to be unequal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI MICROFILMED 1994 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The q u alityof this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. 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Order Number 9502504 Isospectral graphs and the expander coefficient Walters, Ian Campbell, Jr., PhJD. Western Michigan University, 1994 UMI 300 N. ZeebRd. Ann Arbor. MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I wish to thank Professor Allen Schwenk for all of his guidance and support throughout the research conducted for this dissertation, for all of the personal insight he has shared with me, and the patience he has shown me. It has been a pleasure to work with him, and a privilege to have Professor Schwenk as my advisor. A special thank you to Professor Morteza Shafii-Mousavi, for encouraging me to pursue a graduate degree in Mathematics while I was an undergraduate at Indiana University at South Bend. Another special thank you to Professor Yousef Alavi, for bringing me to Western Michigan University, imparting his inexhaustible supply of enthusiasm for the beauty of the mathematics, and for his insistence that I see the whole picture. I wish to thank Professors Allen Schwenk, Gary Chartrand, Arthur T. White and Joseph McCanna, for their insightful and entertaining lectures in graph theory, and for sharing their knowledge. Many thanks to Professors Arthur T. White, Yousef Alavi, Joseph Buckley, and Morteza Shafii-Mousavi for serving on my committee. I want to thank my mother and sisters for their patience, love and support over the past six years. And a special thank you to Donna Szarowicz, for all of her love, support and caring. Ian Campbell Walters Jr. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGMENTS............................................................................. ii CHAPTER I. INTRODUCTION......................................................................... 1 1.1 Definitions .................................................................. 1 1.2 Preliminary Results .................................................. 6 H. SOME PARAMETERS OF 2-SET COMPOSITION AND BIPARTITE COMPOSITION GRAPHS........................ 14 2.1 Connectivity of 2-set Composition and Bipartite Composition Graphs ................................ 14 2.2 Distance in 2-set Composition and Bipartite Composition Graphs ................................ 22 m. 2-SET AND BIPARTITE COMPOSITION GRAPHS AND THE EXPANDER COEFFICIENT................................... 30 3.1 Preliminary Results ................................................. 30 3.2 2-set Composition and the Expander Coefficient ................................................ 31 3.3 Bipartite Composition and the Expander Coefficient................................................. 42 IV. THE FOLKMAN GRAPH, 2-SET COMPOSITION AND THE EXPANDER COEFFICIENT................................... 55 REFERENCES.......................................!...................................................... 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I INTRODUCTION 1.1 Definitions Graphs have been used effectively to model discrete situations, such as people and friendship, intersections and roads, or information networks. When modeling a network, one might be curious as to how to quantify the rate at which information is spread throughout the network. One such measure is the expander coefficient. In some lecture notes, F.R.K. Chung gave one definition of the expander coefficient, and provided an eigenvalue bound for the value of the expander coefficient ([3]). In his paper in the Notices of the American Mathematical Society [1], Frederick Bien gave another definition of the expander coefficient, and also presented some results relating the spectrum of the Lapladan of a graph to the expander coefficient of the graph. He then suggested that there was no reason to believe the spectrum of the Lapladan of a graph should uniquely determine the expander coeffident of the graph, but had no examples to demonstrate that it did not. Actually, there are two versions of this question. Must cospectral graphs have the same expander coeffident? Bien gave an example of two graphs that are cospectral and have unequal expander coeffidents. Must isospectral graphs have the same expander coeffident? He did not answer this question. We answer both of these questions negatively for the definition of the expander 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. coefficient given by Chung. We present two constructions that yield cospectral graphs, discuss some of the properties of the constructed graphs, and then employ the constructions to form isospectral expander graphs with unequal expander coefficients. The spectrum of a graph G is the collection of eigenvalues (including multiplicities) of the adjacency matrix of the graph. Two graphs G and H are said to be cospectral if the spectra of the two graphs are identical.
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