A new proof of Friedman's second eigenvalue Theorem and its extension to random lifts

TL;DR

This paper provides a new proof of Friedman's theorem on the spectral gap of random regular graphs and extends the results to random lifts, improving previous bounds and understanding.

Contribution

It introduces a novel proof of Friedman's second eigenvalue theorem and extends the analysis to random graph lifts, enhancing existing spectral gap bounds.

Findings

New proof of Friedman's eigenvalue bound

Extension to random graph lifts with improved bounds

Enhanced understanding of spectral properties of random graphs

Abstract

It was conjectured by Alon and proved by Friedman that a random $d$-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most $2\sqrt{d-1} +o(1)$ with probability tending to one as the size of the graph tends to infinity. We give a new proof of this statement. We also study related questions on random $n$-lifts of graphs and improve a recent result by Friedman and Kohler.