Cubic graphs with no eigenvalues in the interval (-1,1)

TL;DR

This paper fully characterizes cubic graphs that lack eigenvalues in the interval (-1,1), identifying infinite families and sporadic cases, and establishes that (-1,1) is the maximal spectral gap for such graphs.

Contribution

It provides a complete classification of cubic graphs with no eigenvalues in (-1,1), combining known families and new sporadic examples, and proves the maximality of this spectral gap.

Findings

Identified two infinite families of such graphs.

Discovered 14 sporadic graphs with at most 32 vertices.

Proved (-1,1) is the maximal spectral gap for cubic graphs.

Abstract

We give a complete characterisation of the cubic graphs with no eigenvalues in the open interval $(-1,1)$. There are two infinite families, one due to Guo and Mohar [Linear Algebra Appl. 449:68--75] the other due to Koll\'ar and Sarnak [Communications of the AMS. 1,1--38], and $14$ "sporadic" graphs on at most $32$ vertices. This allows us to show that $(-1,1)$ is a maximal spectral gap set for cubic graphs. Our techniques including examination of various substructure and an application of the classification of generalized line graphs.