A lower bound on the spectrum of unimodular networks

TL;DR

This paper establishes a lower bound on the spectral radius of unimodular networks, extending classical spectral bounds to a broader stochastic graph setting and providing new insights into their spectral properties.

Contribution

It introduces a lower bound on the spectral radius of unimodular networks based on average degree, generalizing the Alon-Boppana theorem to non-regular graphs.

Findings

Lower bound on spectral radius in terms of average degree

Extension of Alon-Boppana bound to unimodular networks

Lower bound on volume growth rate of unimodular trees

Abstract

Unimodular networks are a generalization of finite graphs in a stochastic sense. We prove a lower bound to the spectral radius of the adjacency operator and of the Markov operator of an unimodular network in terms of its average degree. This allows to prove an Alon-Boppana type bound for the largest eigenvalues in absolute value of large, connected, bounded degree graphs, which generalizes the Alon-Boppana theorem for regular graphs. A key step is establishing a lower bound to the spectral radius of a unimodular tree in terms of its average degree. Similarly, we provide a lower bound on the volume growth rate of an unimodular tree in terms of its average degree.