  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 $C(G,S)$.
