A graph product and its Application

TL;DR

This paper investigates the spectral properties of a specific graph product, deriving formulas for Laplacian spectra, and applies these results to compute indices and structural properties like spanning trees.

Contribution

It introduces a new graph product, analyzes its Laplacian spectra, and provides conditions for integrality, enabling computation of indices and structural graph properties.

Findings

Derived Laplacian and signless Laplacian spectra for the graph product.

Computed Kirchhoff and Wiener indices for the product.

Determined conditions for Laplacian and signless Laplacian integrality.

Abstract

The spectrum of Laplacian and signless Laplacian matrix for a graph product is obtained, where both underlying graphs are regular. As an application of this, we have been able to generate the Kirchhoff Index and Wiener Index and determine the number of spanning trees. Additionally, we derived the conditions necessary for obtaining a Laplacian and signless Laplacian integral product graph.