## Expander Families and Cayley Graphs: A Beginner's GuideExpander families enjoy a wide range of applications in mathematics and computer science, and their study is a fascinating one in its own right. Expander Families and Cayley Graphs: A Beginner's Guide provides an introduction to the mathematical theory underlying these objects. The central notion in the book is that of expansion, which roughly means the quality of a graph as a communications network. Cayley graphs are certain graphs constructed from groups; they play a prominent role in the study of expander families. The isoperimetric constant, the second largest eigenvalue, the diameter, and the Kazhdan constant are four measures of the expansion quality of a Cayley graph. The book carefully develops these concepts, discussing their relationships to one another and to subgroups and quotients as well as their best-case growth rates. Topics include graph spectra (i.e., eigenvalues); a Cheeger-Buser-type inequality for regular graphs; group quotients and graph coverings; subgroups and Schreier generators; the Alon-Boppana theorem on the second largest eigenvalue of a regular graph; Ramanujan graphs; diameter estimates for Cayley graphs; the zig-zag product and its relation to semidirect products of groups; eigenvalues of Cayley graphs; Paley graphs; and Kazhdan constants. The book was written with undergraduate math majors in mind; indeed, several dozen of them field-tested it. The prerequisites are minimal: one course in linear algebra, and one course in group theory. No background in graph theory or representation theory is assumed; the book develops from scatch the required facts from these fields. The authors include not only overviews and quick capsule summaries of key concepts, but also details of potentially confusing lines of reasoning. The book contains ideas for student research projects (for capstone projects, REUs, etc.), exercises (both easy and hard), and extensive notes with references to the literature. |

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### Contents

Combinatorial Techniques | 93 |

RepresentationTheoretic Techniques | 141 |

Linear algebra | 229 |

Asymptotic analysis of functions | 244 |

247 | |

Index | 253 |

### Other editions - View all

Expander Families and Cayley Graphs: A Beginner's Guide Mike Krebs,Anthony Shaheen Limited preview - 2011 |

Expander Families and Cayley Graphs: A Beginner's Guide Mike Krebs,Anthony Shaheen No preview available - 2011 |

### Common terms and phrases

abelian groups adjacency matrix adjacency operator bijective bipartite Cay(G Cayley graph Chapter closed walks compute conjugacy classes connected construct Corollary cycle graphs d-regular graph Define Definition denote derived length diam(X dihedral group direct sum directed edge eigenvalues element of G equals Equation Example Exercise expander family Figure finite graph finite group follows function G-invariant given GL(V graph with vertex group G Hence homomorphism inequivalent irreducible representations irreps isoperimetric constant Kazhdan constant Lemma Let G linear logarithmic diameter loop matrix representations multiplicity multiset nonbipartite Nonexpansion Principle nontrivial notation Note orthogonal permutation positive integer Proof Let Prop Proposition Prove quotients Ramanujan graphs reader real numbers Recall regular representation Remark representation of G semidirect product spectral gap subgroup subspace Suppose unitary vector space vertex set vertices walks of length yield an expander zig-zag product