On graphs with large third eigenvalue

TL;DR

This paper investigates bounds on the third largest eigenvalue of graphs, proving conjectures for specific graph classes and exploring eigenvalue minimization in weighted graphs, advancing understanding of spectral graph properties.

Contribution

It generalizes constructions for the third eigenvalue, proves bounds for certain graph classes, and links eigenvalue minimization in weighted graphs to smaller unweighted graphs.

Findings

Proved the conjecture for strongly regular, line, and Cayley graphs.

Extended the problem to weighted graphs and established bounds.

Connected eigenvalue minimization to graphs on 11 vertices.

Abstract

Given a graph $G$, let $\lambda_3$ denote the third largest eigenvalue of its adjacency matrix. In this paper, we prove various results towards the conjecture that $\lambda_3(G) \le \frac{|V(G)|}{3}$, motivated by a question of Nikiforov. We generalise the known constructions that yield $\lambda_3(G) = \frac{|V(G)|}{3} - 1$ and prove the inequality holds for $G$ strongly regular, a regular line graph or a Cayley graph on an abelian group. We also consider the extended problem of minimising $\lambda_{n-1}$ on weighted graphs and reduce the existence of a minimiser with simple final eigenvalue to a vertex multiplication of a graph on 11 vertices. We prove that the minimal $\lambda_{n-1}$ over weighted graphs is at most $O(\sqrt{n})$ from the minimal $\lambda_{n-1}$ over unweighted graphs.