Gap sets for the spectra of regular graphs with minimum spectral gap

TL;DR

This paper investigates spectral gaps in large regular graphs, identifying specific gap intervals for certain symmetric graph families and conjecturing their maximality, with implications for eigenvalue distributions.

Contribution

It establishes explicit spectral gap intervals for symmetric cubic and quartic graphs and conjectures their maximality, advancing understanding of eigenvalue distributions in regular graphs.

Findings

Identified gap interval (1,√5] for Δ_n graphs.

Identified gap interval [(−1+√17)/2,3] for Γ_n graphs.

Eigenvalues outside these gaps tend to specific limits as n increases.

Abstract

Following recent work by Koll\'{a}r and Sarnak, we study gaps in the spectra of large connected cubic and quartic graphs with minimum spectral gap. We focus on two sequences of graphs, denoted $\Delta_n$ and $\Gamma_n$ which are more `symmetric' compared to the other graphs in these two families, respectively. We prove that $(1,\sqrt{5}]$ is a gap interval for $\Delta_n$, and $[(-1+\sqrt{17})/2,3]$ is a gap interval for $\Gamma_n$. We conjecture that these two are indeed maximal gap intervals. As a by-product, we show that the eigenvalues of $\Delta_n$ lying in the interval $[-3,-\sqrt{5}]$ (in particular, its minimum eigenvalue) converge to $(1-\sqrt{33})/2$ and the eigenvalues of $\Gamma_n$ lying in the interval $[-4,-(1+\sqrt{17})/2]$ (and in particular, its minimum eigenvalue) converge to $1-\sqrt{13}$ as $n$ tends to infinity. The proofs of the above results heavily depend on the following property which can be of independent interest: with few exceptions, all the eigenvalues of connected cubic and quartic graphs with minimum spectral gap are simple.