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An Introduction to Expander Graphs AN INTRODUCTION TO EXPANDER GRAPHS A REPORT submitted in partial fulfillment of the requirements for the award of the dual degree of Bachelor of Science-Master of Science in MATHEMATICS by ASHWIN K (12026) DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH BHOPAL BHOPAL - 462066 April 2017 i CERTIFICATE This is to certify that Ashwin K, BS-MS (Mathematics), has worked on the project entitled `An Introduction to Expander graphs' under my supervision and guidance. The content of this report is original and has not been submitted elsewhere for the award of any academic or professional degree. April 2017 Dr. Kashyap Rajeevsarathy IISER Bhopal Committee Member Signature Date ii ACADEMIC INTEGRITY AND COPYRIGHT DISCLAIMER I hereby declare that this project is my own work and, to the best of my knowledge, it contains no materials previously published or written by another person, or substantial proportions of material which have been ac- cepted for the award of any other degree or diploma at IISER Bhopal or any other educational institution, except where due acknowledgement is made in the document. I certify that all copyrighted material incorporated into this document is in compliance with the Indian Copyright Act (1957) and that I have received written permission from the copyright owners for my use of their work, which is beyond the scope of the law. I agree to indemnify and save harmless IISER Bhopal from any and all claims that may be asserted or that may arise from any copyright violation. April 2017 Ashwin K IISER Bhopal iii ACKNOWLEDGEMENT I thank my supervisor for his wonderful guidance, his useful explanations, both motivational and technical, and for being with me for long hours, when- ever I needed his help. I also thank all my professors from the department of mathematics at IISER Bhopal. I want to thank Mr.Neeraj Dhanwani(Ph.D Scholar IISER Bhopal) for helping me in doing this project by giving the right ideas whenever I needed. I am very grateful to my mother and father for their immense support whenever I was struggling in my life. Lastly, I want to thank the IISER Bhopal community for providing the best days of my life. Ashwin K iv ABSTRACT We will begin by discussing some of the basic facts about graph theory in view towards understanding this thesis. We then will derive bounds on the spectrum of a k-regular graph, and understand the relation between the spec- trum, and the bipartiteness and connectivity of such a graph. Following a brief introduction to group actions on graphs, we will establish that all Cay- ley graphs are vertex-transitive.We will then introduce Ramanujan Graphs, and establish that the undirected n-cycle is Ramanujan. After describing about Ramanujan graphs we then discuss some of the basic properties of ex- pander families of graphs,begin by defining some operators associated with the graph such as adjacency operator,laplacian operators.Then we will define the Isoperimetric constant associated with the graph and from that we will describe the definition of expander families of graphs.We will also understand the proof of Rayleigh-Ritz theorem which will help us in understanding the relationship between spectral gap and isoperimetric constant.We will see on what bounds of the diameter's of families of graphs will make those families to expander.Then we will see why Abelian groups do not yield expander families of graph.Finally, we will state Alon-Boppana theorem and will understand the proof,and will show why Ramanujan graphs with regularity greater than three turns out to be expander graphs. v LIST OF SYMBOLS OR ABBREVIATIONS • Diam(X) represents the diameter of graph X. • λ(X) denotes the eigenvalue of graph X. • kgkΓ represents the word norm of g wrt subset Γ. • dist(x; y) denotes the distance between vertices x and y. • We denote the group of integers modulo n under addition by Zn: • We denote the sets of integers, real numbers, and complex numbers, respectively, by Z; R; and C. • Let x be a real number. We write bxc for the greatest integer less than or equal to x. For example, b2:55c = 2. • Let A and B are two sets. We write A n B for the set difference; that is A n B = fx 2 Ajx2 = Bg. CONTENTS Certificate ::::::::::::::::::::::::::::::::: i Academic Integrity and Copyright Disclaimer :::::::::: ii Acknowledgement :::::::::::::::::::::::::::: iii Abstract :::::::::::::::::::::::::::::::::: iv List of Symbols or Abbreviations :::::::::::::::::: v 1. Introduction :::::::::::::::::::::::::::::: 2 1.1 Background . 2 2. Preliminaries ::::::::::::::::::::::::::::: 4 2.1 Graphs . 4 2.2 Degree and Regularity . 9 2.3 Connected Graph and Path . 10 2.4 Adjacency Matrix . 13 3. Spectrum and Cayley graph :::::::::::::::::::: 16 3.1 Graph Isomorphism . 16 3.2 Spectra of Graphs . 19 3.3 Bipartite Graph . 21 3.4 Graph Automorphism and Vertex Transitivity . 22 3.5 Cayley graph . 23 4. Expander Graphs :::::::::::::::::::::::::: 29 4.1 Adjacency Operator . 29 Contents 1 4.2 Isoperimetric Constant . 33 4.3 Expander families of graphs . 34 4.4 Spectral gap and the isoperimetric constant . 37 4.5 Properties of Expander graphs . 43 4.6 Diameter of Cayley graph . 48 4.7 Alon-Boppana Theorem . 50 Appendices :::::::::::::::::::::::::::::::: 57 I Basic Definitions . I II Additional Theorems . I Bibliography ::::::::::::::::::::::::::::::: III 1. INTRODUCTION 1.1 Background Algebraic graph theory is a branch of Mathematics in which algebraic meth- ods, particularly those employed in group theory and linear algebra, are use to solve graph-theoretic problems. An important subbranch of algebraic graph theory is spectral graph theory, which involves the study of the spectra of ma- trices associated with the graph such as its adjacency matrix, and its relation to the properties of the graph. Expander graph are graphs with the special property that any set of vertices S (unless very large) has a number of outgo- ing edges proportional to jSj. Expansion can be defined both with respect to the number of the edges or vertices on the boundary of S. We will stick with edge expansion, which is more directly related to eigenvalues The spectral gap of a graph is the difference in magnitude of the two largest eigenvalues of its adjacency matrix. Graphs which have large spectral gaps are of great in- terest from the viewpoint of communication networks, as they exhibit strong connectivity properties. As Ramanujan Graphs are known to maximize the spectral gap, they have been widely studied from an application perspective. However, Ramanujan graphs have also fascinated pure mathematicians alike, as they lie in the interface of Number Theory, Representation Theory, and Al- gebraic Geometry [6]. In 1988, a beautiful paper by A. Lubotsky, R. Phillips, and P. Sarnak [5], described the explicit construction of a Ramanujan Graph for every pair of distinct primes congruent to 1 modulo 4, thereby establish- ing the existence of an infinite family of such graphs. Expander graphs have found extensive applications in computer science, in designing algorithms, er- ror correcting codes, extractors, pseudorandom generators, sorting networks 1. Introduction 3 (Ajtai, Komls and Szemerdi (1983)) and robust computer networks. They have also been used in proofs of many important results in computational complexity theory, such as SL = L (Reingold (2008)) and the PCP theorem (Dinur (2007)). In cryptography, expander graphs are used to construct hash functions. The primary goal of this thesis is to study some basic concepts in algebraic graph theory, with a view towards understanding expander graphs and their properties. We will mostly follow [3, 4], and we will use [2, 6] and [1] as additional references. 2. PRELIMINARIES In this introductory chapter we provide the background to the material that we present more formally in later chapters. We will begin our discussions by describing some basic notations and definitions in graph theory. 2.1 Graphs Graphs are mathematical structures used to model pairwise relations between objects. We will now formally define an undirected graph. Definition 2.1. An undirected graph X is defined to be a pair (V (X);E(X)), where (i) V (X) is a set called the set of vertices, and (ii) E(X) is a set of unordered pairs of vertices (i.e E(X) ⊂ ffx; yg j x; y 2 V (X)g called the set of edges. For a graph X, jV (X)j is called the order of graph, which is denoted as jXj. We say a graph X is finite if jXj < 1, otherwise, we say a graph is infinite. Example 2.2 (Undirected Graph). Let X = (V (X);E(X)) be a graph with V (X) = f1; 2; 3; 4; 6g and E(X) = ff1; 2g; f2; 3g; f3; 1g; f4; 6g; f4; 2g; f1; 4gg: Figure 2.1 gives the pictorial representation of an undirected graph X = (V (X);E(X)) . 2. Preliminaries 5 Fig. 2.1: An undirected graph Since f2; 1g; f2; 6g and f2; 5g 2 E(X) we say that 1; 6 and 5 are adja- cent to 2. Also, note that since 4 2 V (X) has no adjacent vertices we will call it a isolated vertex. Remark 2.3. If the graph is directed then in E(X) instead of taking un- ordered pairs of vertices we will take ordered pairs of vertices, that is, E(X) = f(x; y)j x; y 2 V (X)g: Definition 2.4. Consider the graph X = (V (X);E(X)) .Then (i) For x; y 2 V (X), we say x is adjacent to y if fx; yg 2 E(X). (ii) An edge of the form e = (x; x) or e = fxg for x 2 V (X) is called a loop. (iii) If E(X) is a multiset then X = (V (X);E(X)) is called multigraph.
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