On the spectral gap of Cayley graphs

Let $\mathrm{\Gamma }$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Suppose $\mathrm{\Gamma }$ is undirected and non-bipartite. Let $\mu$ (resp. ${\mu }_2$) denote the smallest (resp. the second largest) eigenvalue of the normalized adjacency operator of $\mathrm{\Gamma }$, and $d$ denote the degree of $\mathrm{\Gamma }$. We show that $1+\mu =\mathrm{\Omega }((1-{\mu }_2)\mathrm{/}d)$ holds.