Resolvent of Large Random Graphs

TL;DR

This paper investigates how the spectral properties of large random graphs converge to those of an infinite limit graph, providing new formulas and applications to various graph models.

Contribution

It introduces a new formula for the spectral measure's Stieltjes transform for graphs converging locally to trees, extending spectral analysis methods.

Findings

Spectral convergence of large random graphs to infinite graph limits

New formula for the Stieltjes transform of spectral measures

Applications to various graph models like Erdős-Rényi and preferential attachment

Abstract

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and preferential attachment graphs. We sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices.