On a Cheeger type inequality in Cayley graphs of finite groups

Let $G$ be a finite group. It was remarked in Breuillard et al. (2015) 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 $\left.?1+\frac{h{\left(\mathbb{G}\right)}^4}{\gamma },1?\frac{h{\left(\mathbb{G}\right)}^2}{2d^2}\right]$, where $h\left(\mathbb{G}\right)$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma =2^9{d}^6{(d+1)}^2$.