A lower bound on the spectrum of unimodular networks

Mustazee Rahman

Published 2016-09-07Version 1

Unimodular networks are stochastic generalizations of finite graphs. We prove a lower bound on the spectral radius of both the adjacency and random walk operator of a unimodular network in terms of its average degree. Using this we prove an Alon-Boppana type bound for the largest eigenvalues in absolute value of large, connected, bounded degree graphs. We also provide a lower bound on the volume growth rate of a unimodular tree in terms of its average degree.