On the nontrivial extremal eigenvalues of graphs

Abstract

We present improved bounds on a quantitative version of an observation originally due to Breuillard, Green, Guralnick and Tao which says 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.