THe largest eigenvalue of sparse random graphs

TL;DR

This paper establishes a precise asymptotic formula for the largest eigenvalue of sparse random graphs G(n,p), relating it to the maximum degree and average degree, with high probability as the graph size grows.

Contribution

It provides a rigorous proof connecting the largest eigenvalue to the maximum degree and average degree for all edge probabilities p(n), extending understanding of spectral properties of sparse graphs.

Findings

Largest eigenvalue asymptotically equals max{√Δ, np}

Almost sure convergence as graph size increases

Applicable for all edge probability regimes

Abstract

We prove that for all values of the edge probability p(n) the largest eigenvalue of a random graph G(n,p) satisfies almost surely: \lambda_1(G)=(1+o(1))max{\sqrt{\Delta},np}, where \Delta is a maximal degree of G, and the o(1) term tends to zero as max{\sqrt{\Delta},np} tends to infinity.