Arbitrary Spectral Edge of Regular Graphs

TL;DR

This paper characterizes the possible limit points of the first k eigenvalues of sequences of d-regular graphs, confirming a conjecture and extending previous results using an infinite random graph model.

Contribution

It proves the set of limit points for the first k eigenvalues of d-regular graphs, generalizing prior results and confirming a conjecture by Alon and Wei.

Findings

Characterizes limit points of eigenvalues for regular graphs.

Extends Friedman's theorem to a new infinite graph model.

Provides bounds on the trace of the non-backtracking operator.

Abstract

We prove that for each $d\geq 3$ and $k\geq 2$, the set of limit points of the first $k$ eigenvalues of sequences of $d$-regular graphs is \[ \{(\mu_1,\dots,\mu_k): d=\mu_1\geq \dots\geq \mu_{k}\geq2\sqrt{d-1}\}. \] The result for $k=2$ was obtained by Alon and Wei, and our result confirms a conjecture of theirs. Our proof uses an infinite random graph sampled from a distribution that generalizes the random regular graph distribution. To control the spectral behavior of this infinite object, we show that Huang and Yau's proof of Friedman's theorem bounding the second eigenvalue of a random regular graph generalizes to this model. We also bound the trace of the non-backtracking operator, as was done in Bordenave's separate proof of Friedman's theorem.