On a Cheeger Type Inequality in Cayley Graphs of Finite Groups

Abstract

[nicht gut kopierbar]Let G be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph C(G,S) is an expander graph and is non-bipartite then the spectrum of the adjacency operator T is bounded away from −1. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval [−1+h(G)⁴/γ,1−h(G)²/2d²], where h(G) denotes the (vertex) Cheeger constant of the d regular graph C(G,S) with respect to a symmetric set S of generators and γ=2⁹d⁶(d+1)².