On a cheeger type inequality in Cayley graphs of finite groups

TL;DR

This paper establishes explicit spectral bounds for Cayley graphs of finite groups, linking the spectrum to the Cheeger constant, thus extending Cheeger-type inequalities to these algebraic structures.

Contribution

It provides explicit bounds on the spectrum of Cayley graphs of finite groups in terms of the Cheeger constant, advancing understanding of their spectral properties.

Findings

Spectrum lies in a specific interval depending on Cheeger constant

Non-trivial spectrum is bounded away from -1 in expander graphs

Results apply to non-bipartite Cayley graphs with explicit constants

Abstract

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 $\left[-1+\frac{h(\mathbb{G})^{4}}{\gamma}, 1-\frac{h(\mathbb{G})^{2}}{2d^{2}}\right]$, where $h(\mathbb{G})$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma = 2^{9}d^{6}(d+1)^{2}$.