On the Local Geometry of Graphs in Terms of Their Spectra

TL;DR

This paper explores the relationship between the spectrum of large graphs and their local structure, showing that graphs with spectra close to their universal cover are asymptotically locally tree-like.

Contribution

It establishes that sequences of graphs with spectra concentrated in the universal cover’s support are asymptotically locally tree-like, extending previous results to more general graph classes.

Findings

Graphs with spectra near their universal cover are asymptotically locally tree-like.

Extends spectral and local geometric relations to infinite sofic graphs and unimodular networks.

Generalizes results from regular graphs and Cayley graphs to broader classes.

Abstract

In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose $G_1, G_2, G_3, \ldots$ is a sequence of finite and connected graphs that share a common universal cover $T$ and such that the proportion of eigenvalues of $G_n$ that lie within the support of the spectrum of $T$ tends to 1 in the large $n$ limit. This is a weak notion of being Ramanujan. We prove such a sequence of graphs is asymptotically locally tree-like. This is deduced by way of an analogous theorem proved for certain infinite sofic graphs and unimodular networks, which extends results for regular graphs and certain infinite Cayley graphs.