2023/10/26 by Li, Wenbo, Liu, Shiping
#05C48 #05C50 #20F65 #Combinatorics (math.CO) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2310.17520
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.