Graphs determined by signless Laplacian spectra

TL;DR

This paper investigates which graphs are uniquely identified by their signless Laplacian spectra, providing new classes of such graphs and conditions under which they are determined by their spectra.

Contribution

The paper introduces new classes of graphs that are determined by their signless Laplacian spectra and establishes conditions for their spectral uniqueness.

Findings

Graphs of the form G[rK1][sK2] are DQS under certain conditions.

Identifies DQS graphs with independent edges and isolated vertices.

Provides theoretical conditions for spectral determination of specific graph classes.

Abstract

In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. A graph is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a DQS graph G, we show that G[rK1 [sK2 is DQS under certain conditions, where r, s are natural numbers and K1 and K2 denote the complete graphs on one vertex and two vertices, respectively. Applying these results, some DQS graphs with independent edges and isolated vertices are obtained