A Note on the Trace Method for Random Regular Graphs

TL;DR

This paper demonstrates that analyzing the non-backtracking spectrum of random regular graphs provides tighter bounds on the second largest eigenvalue than traditional methods, improving spectral analysis techniques.

Contribution

It shows that switching to non-backtracking spectrum analysis yields a better bound on the second largest eigenvalue of random regular graphs compared to previous approaches.

Findings

Bound of 2√(d-1)+2/√(d-1) on the second largest eigenvalue

Non-backtracking spectrum analysis offers advantages over ordinary spectrum

Improved spectral bounds for random d-regular graphs

Abstract

The main goal of this note is to illustrate the advantage of analyzing the non-backtracking spectrum of a regular graph rather than the ordinary spectrum. We show that by switching to non-backtracking spectrum, the method of proof used in [Puder 2015, arXiv::1212.5216] yields a bound of $2\sqrt{d-1}+\frac{2}{\sqrt{d-1}}$ instead of the original $2\sqrt{d-1}+1$ on the second largest eigenvalue of a random $d$-regular graph.