  For a non-bipartite finite Cayley graph, we show the non-trivial eigenvalues
of its normalized adjacency matrix lie in the interval
$$\left[-1+\frac{ch_{out}^2}{d},1-\frac{Ch_{out}^2}{d}\right],$$ for some
absolute constant $c$ and $C$, where $h_{out}$ stands for the outer vertex
boundary isoperimetric constant. This improves upon recent obtained estimates
aiming at a quantitative version of an observation due to Breuillard, Green,
Guralnick and Tao stating that if a non-bipartite finite Cayley graph is an
expander then the non-trivial eigenvalues of its normalized adjacency matrix is
not only bounded away from $1$ but also bounded away from $-1$. We achieve this
by extending the work of Bobkov, Houdr\'e and Tetali on vertex isoperimetry to
the setting of signed graphs. We further extend our interval estimate to the
settings of vertex transitive graphs and Cayley sum graphs, the latter of which
need not be vertex transitive. Our approach answers positively recent open
questions proposed by Moorman, Ralli and Tetali.
