# A Cheeger type inequality in finite Cayley sum graphs

## TL;DR

This paper establishes a Cheeger-type inequality for finite Cayley sum graphs, providing explicit spectral bounds based on the Cheeger constant, and improves existing bounds on the spectrum of related Cayley graphs.

## Contribution

It introduces a Cheeger-type inequality for Cayley sum graphs, linking spectral bounds to the Cheeger constant, and refines previous spectral bounds for Cayley graphs.

## Key findings

- Spectral bounds depend on the Cheeger constant and degree.
- Non-bipartite Cayley sum graphs have spectra bounded away from -1.
- Improved bounds on the spectrum of non-bipartite Cayley graphs.

## Abstract

Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}{\eta}, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_{\Sigma}(G,S)$ and $\eta = 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.