Partial characterization of graphs having a single large Laplacian eigenvalue

TL;DR

This paper investigates graphs with exactly one Laplacian eigenvalue above the average degree, proposing a conjecture that such graphs are stars possibly with isolated vertices, supported by partial characterizations and class-specific results.

Contribution

It introduces a conjecture linking graphs with a single large Laplacian eigenvalue to star graphs plus isolated vertices and provides partial characterizations and related results.

Findings

Conjecture that graphs with one large Laplacian eigenvalue are stars plus isolated vertices.

Established a connection between the parameter σ(G) and the number of anticomponents.

Provided results supporting the conjecture within specific graph classes.

Abstract

The parameter $\sigma(G)$ of a graph $G$ stands for the number of Laplacian eigenvalues greater than or equal to the average degree of $G$. In this work, we address the problem of characterizing those graphs $G$ having $\sigma(G)=1$. Our conjecture is that these graphs are stars plus a (possible empty) set of isolated vertices. We establish a link between $\sigma(G)$ and the number of anticomponents of $G$. As a by-product, we present some results which support the conjecture, by restricting our analysis to some classes of graphs.