  We present a finer quantitative version of an observation due to Breuillard,
Green, Guralnick and Tao which tells that for finite non-bipartite Cayley
graphs, once the nontrivial eigenvalues of their normalized adjacency matrices
are uniformly bounded away from $1$, then they are also uniformly bounded away
from $-1$. Unlike previous works which depend heavily on combinatorial
arguments, we rely more on analysis of eigenfunctions. We establish a new
explicit lower bound for the gap between $-1$ and the smallest normalized
adjacency eigenvalue, which improves previous lower bounds in terms of
edge-expansion, and is comparable to the best known lower bound in terms of
vertex-expansion.
