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On the spectrum of closed neighborhood corona product of graph and its application

Bishal Sonar, Ravi Srivastava

TL;DR

This work analyzes the closed neighbourhood corona product of graphs by deriving explicit expressions for the adjacency, Laplacian, and signless Laplacian spectra in terms of the factor graphs' spectra and coronals. It establishes cospectrality transfer criteria, provides formulas for Kirchhoff index and the number of spanning trees, and demonstrates how to construct non-cospectral equienergetic families. It also gives necessary and sufficient conditions for the product to be an integral graph. Overall, the results deepen the algebraic and combinatorial understanding of this graph product and support spectral graph theory applications related to graph invariants and graph construction.

Abstract

In this paper, we investigate the spectral properties of the closed neighborhood corona product of graphs, which was introduced by Harishchandra S. Ramane et al.~\cite{ramane2021polynomials} (cf. Polynomials Associated with Closed Neighborhood Corona and Neighborhood Complement Corona of Graphs). Based on their results, such as characteristic polynomials of the adjacency, Laplacian, and signless Laplacian matrices, we further investigate the spectral characteristics of this product graph. Specifically, we investigate conditions under which cospectrality occurs for this operation. Further, we determine the Kirchhoff index and count spanning trees and identify sequences of non-cospectral equienergetic product graphs. Finally, we develop criteria for when the product graph is integral and thereby contribute to a deeper understanding of the algebraic and combinatorial structure of the product graph.

On the spectrum of closed neighborhood corona product of graph and its application

TL;DR

This work analyzes the closed neighbourhood corona product of graphs by deriving explicit expressions for the adjacency, Laplacian, and signless Laplacian spectra in terms of the factor graphs' spectra and coronals. It establishes cospectrality transfer criteria, provides formulas for Kirchhoff index and the number of spanning trees, and demonstrates how to construct non-cospectral equienergetic families. It also gives necessary and sufficient conditions for the product to be an integral graph. Overall, the results deepen the algebraic and combinatorial understanding of this graph product and support spectral graph theory applications related to graph invariants and graph construction.

Abstract

In this paper, we investigate the spectral properties of the closed neighborhood corona product of graphs, which was introduced by Harishchandra S. Ramane et al.~\cite{ramane2021polynomials} (cf. Polynomials Associated with Closed Neighborhood Corona and Neighborhood Complement Corona of Graphs). Based on their results, such as characteristic polynomials of the adjacency, Laplacian, and signless Laplacian matrices, we further investigate the spectral characteristics of this product graph. Specifically, we investigate conditions under which cospectrality occurs for this operation. Further, we determine the Kirchhoff index and count spanning trees and identify sequences of non-cospectral equienergetic product graphs. Finally, we develop criteria for when the product graph is integral and thereby contribute to a deeper understanding of the algebraic and combinatorial structure of the product graph.
Paper Structure (13 sections, 12 theorems, 42 equations, 1 figure)

This paper contains 13 sections, 12 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition
  4. Notation
  5. Spectral properties
  6. Adjacency spectrum
  7. Laplacian spectrum
  8. Signless Laplacian spectrum
  9. Application
  10. Kirchhoff Index
  11. Spanning Tree
  12. Equienergetic Graph
  13. Integral Graph

Key Result

Lemma 2.1

bapat2010graphs Let $\mathcal{A}_1,\mathcal{A}_2,\mathcal{A}_3,$ and $\mathcal{A}_4$ be matrix of order $a_1\times a_1,~ a_1\times a_2, ~a_2\times a_1, ~a_2\times a_2$ respectively. If $\mathcal{A}_1$ and $\mathcal{A}_4$ are invertible, then

Figures (1)

  • Figure 1: Closed neighbourhood corona product of $C_4$ and $C_3$.

Theorems & Definitions (24)

  • Lemma 2.1
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Corollary 3.1.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.2.1
  • proof
  • ...and 14 more