Tree walks and the spectrum of random graphs

TL;DR

This paper investigates the spectral distribution of sparse random graphs G(n, c/n), extending combinatorial methods to approximate spectral moments through walk counts on trees, advancing understanding of their spectral properties.

Contribution

It combines and extends previous combinatorial approaches to analyze the spectral measure of sparse random graphs G(n, c/n).

Findings

Approximate moments of the spectral measure using walk counts.

Extended combinatorial methods for spectral analysis.

Improved understanding of spectral distribution in sparse graphs.

Abstract

It is a classic result in spectral theory that the limit distribution of the spectral measure of random graphs G(n, p) converges to the semicircle law in case np tends to infinity with n. The spectral measure for random graphs G(n, c/n) however is less understood. In this work, we combine and extend two combinatorial approaches by Bauer and Golinelli (2001) and Enriquez and Menard (2016) and approximate the moments of the spectral measure by counting walks that span trees.