# 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.

## Key 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 $\gamma = 2^{9}d^{6}(d+1)^{2}$.