On the spectral gap of Cayley graphs

Let $\Gamma$ 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 $\Gamma$ is undirected and non-bipartite. Let $\mu$ (resp. $\mu_2$) denote the smallest (resp. the second largest) eigenvalue of the normalized adjacency operator of $\Gamma$, and $d$ denote the degree of $\Gamma$. We show that $1+ \mu = \Omega((1-\mu_2)/d)$ holds.