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

TL;DR

This paper explores the spectral properties of the closed neighborhood corona product of graphs, analyzing conditions for cospectrality, calculating invariants like Kirchhoff index, and identifying when the product graph is integral.

Contribution

It provides new spectral characterizations, criteria for integrality, and insights into the algebraic and combinatorial structure of the closed neighborhood corona product of graphs.

Findings

Conditions for cospectrality are established.

Kirchhoff index and spanning trees are determined.

Criteria for graph integrality are developed.

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.