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A graph product and its Application

Bishal Sonar, Ravi Srivastava

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.

A graph product and its Application

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.

Paper Structure

This paper contains 13 sections, 12 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kronecker product
  4. Definition
  5. Spectral properties
  6. Definition
  7. Laplacian Spectrum for Regular graphs
  8. Signless Laplacian Spectrum for Regular graphs
  9. Applications
  10. Kirchhoff Index
  11. Spanning Tree
  12. Wiener Index
  13. Integral Graph

Key Result

Lemma 2.1

bapat2010graphs Let $\mathcal{S}_1,\mathcal{S}_2,\mathcal{S}_3,$ and $\mathcal{S}_4$ be matrix of order $q_1\times q_1,~ q_1\times q_2, ~q_2\times q_1, ~q_2\times q_2$ respectively with $\mathcal{S}_1$ and $\mathcal{S}_4$ are invertible. Then

Figures (1)

  • Figure 1: $A=K_2\circledast K_3$ and $B=K_3\circledast K_2.$

Theorems & Definitions (21)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.1.1
  • proof
  • Corollary 3.1.2
  • Theorem 3.2
  • proof
  • Corollary 3.2.1
  • proof
  • ...and 11 more