  Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized
adjacency matrix, with eigenvalues $1 = \lambda_1(A)\geq \dots \ge \lambda_n
\ge -1$. It is a classical fact that $\lambda_n = -1$ if and only if $G$ is
bipartite. Our main result provides a quantitative separation of $\lambda_n$
from $-1$ in the case of Cayley graphs, in terms of their expansion. Denoting
$h_{out}$ by the (outer boundary) vertex expansion of $G$, we show that if $G$
is a non-bipartite Cayley graph (constructed using a group and a symmetric
generating set of size $d$) then $\lambda_n \ge -1 + ch_{out}^2/d^2\,,$ for $c$
an absolute constant. We exhibit graphs for which this result is tight up to a
factor depending on $d$. This improves upon a recent result by Biswas and Saha
who showed $\lambda_n \ge -1 + h_{out}^4/(2^9d^8)\,.$ We also note that such a
result could not be true for general non-bipartite graphs.
