Signless Laplacian spectral characterization of some graphs

TL;DR

This paper explores the spectral properties of graphs with star components, proving specific spectral characterizations and identifying unique cospectral mates for certain join graphs.

Contribution

It introduces new spectral characterization results for graphs with star components and identifies unique cospectral mates for a class of join graphs.

Findings

Proves that certain join graphs are DQS for specified parameters.

Identifies the unique Q-cospectral mate for a specific join graph when alpha=3.

Provides spectral characterization results for graphs with star components.

Abstract

In this paper, we investigate (signless) Laplacian spectral characterization of graphs with star components. Also, we prove that the join graph ${\small K_{n-\alpha}-e\vee \alpha K_{1}}$ is $DQS$ for $n-\alpha >3$ and $\alpha\neq3$, and for $\alpha=3$, we show that its only $Q$-cospectral mate is $\bar{K_{1,3}\cup K_{2}\cup (n-6)K_{1}}$.