A proof of Alon's second eigenvalue conjecture and related problems

TL;DR

This paper proves Noga Alon's conjecture that the probability of a random d-regular graph having a second eigenvalue exceeding 2√(d-1)+ε approaches zero as the number of vertices grows, confirming spectral properties of such graphs.

Contribution

The paper provides a proof of Alon's second eigenvalue conjecture for various models of random d-regular graphs, including cases with odd degree d.

Findings

The probability of large second eigenvalues tends to zero as n increases.

In many models, this probability decays polynomially with n.

The conjecture is proven for multiple notions of random d-regular graphs.

Abstract

In this paper we show the following conjecture of Noga Alon. Fix a positive integer d>2 and real epsilon > 0; consider the probability that a random d-regular graph on n vertices has the second eigenvalue of its adjacency matrix greater than 2 sqrt(d-1) + epsilon; then this probability goes to zero as n tends to infinity. We prove the conjecture for a number of notions of random d-regular graph, including models for d odd. We also estimate the aforementioned probability more precisely, showing in many cases and models (but not all) that it decays like a polynomial in 1/n.