Laplacian spectral characterization of some graph products

TL;DR

This paper investigates the Laplacian spectral properties of certain graph products, identifying conditions under which these products are uniquely determined by their spectra, and characterizing specific cospectral graphs.

Contribution

It characterizes when graph products with certain connected graphs are Laplacian spectral determined, extending spectral characterization to new classes of graph products.

Findings

Most graphs in class $\\mathscr{G}$ are $L$-DS, except for $C_6$ and $\Theta_{3,2,5}$.

The paper identifies all $L$-cospectral graphs for $C_6 \times K_m$ and $\Theta_{3,2,5} \times K_m$.

Product graphs with these exceptions are not $L$-DS, and their cospectral counterparts are characterized.

Abstract

This paper studies the Laplacian spectral characterization of some graph products. We consider a class of connected graphs: $\mathscr{G}={G : |EG|\leq|VG|+1}$, and characterize all graphs $G\in\mathscr{G}$ such that the products $G\times K_m$ are $L$-DS graphs. The main result of this paper states that, if $G\in\mathscr{G}$, except for $C_{6}$ and $\Theta_{3,2,5}$, is $L$-DS graph, so is the product $G\times K_{m}$. In addition, the $L$-cospectral graphs with $C_{6}\times K_{m}$ and $\Theta_{3,2,5}\times K_{m}$ have been found.