Signless Laplacian determinations of some graphs with independent edges

TL;DR

This paper investigates which graphs with independent edges are uniquely identified by their signless Laplacian spectrum, providing conditions under which certain composite graphs are determined by this spectrum.

Contribution

The paper establishes conditions for graphs with independent edges, combined with multiple copies of K2, to be uniquely determined by their signless Laplacian spectra.

Findings

Graphs of the form G⊔ rK2 are determined by their signless Laplacian spectra under specific conditions.

New classes of graphs with independent edges are identified as DQS (determined by spectrum).

Results contribute to spectral graph theory by characterizing spectral uniqueness for certain graph families.

Abstract

{Signless Laplacian determinations of some graphs with independent edges}% {Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum ({\rm DQS}, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some {\rm DQS} graphs with independent edges are obtained.