On the Main Signless Laplacian Eigenvalues of a Graph

TL;DR

This paper characterizes graphs with one or two main signless Laplacian eigenvalues, providing necessary and sufficient conditions and classifying trees and unicyclic graphs with exactly two such eigenvalues.

Contribution

It offers new characterizations of graphs based on their main signless Laplacian eigenvalues, including specific classifications for trees and unicyclic graphs.

Findings

Characterization of graphs with one main signless Laplacian eigenvalue

Conditions for graphs with two main signless Laplacian eigenvalues

Classification of trees and unicyclic graphs with exactly two main eigenvalues

Abstract

A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, we first give the necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues, and then characterize the trees and unicyclic graphs with exactly two main signless Laplacian eigenvalues, respectively.