Two Kinds of Laplacian Spectra and Degree Kirchhoff Index of the Weighted Corona Networks

Haiqin Liu,Yanling Shao,Azhar Hussain

doi:10.1155/2022/6884839

Abstract

Recently, the study related to network has aroused wide attention of the scientific community. Many problems can be usefully represented by corona graphs or networks. Meanwhile, the weight is a vital factor in characterizing some properties of real networks. In this paper, we give complete information about the signless Laplacian spectra of the weighted corona of a graph G 1 and a regular graph G 2 and the complete information about the normalized Laplacian spectra of the weighted corona of two regular graphs. The corresponding linearly independent eigenvectors of all these eigenvalues are also obtained. The spanning trees’ total number and the degree Kirchhoff index of the weighted corona graph are computed.

Highlights

In recent years, the network research has attracted more and more attention from scientific communities [1,2,3,4,5], for example, chemical networks, social networks, disease transmission networks, and product networks
The fact is that the weighted networks which are the extension of the binary networks are more in conformity to the networks in real world. rough the weighted networks, we can model the real world much better
Dai et al [22] deduced the complete conclusion of generalized adjacency eigenvalues and generalized Laplacian eigenvalues of the weighted corona graph with different structures

Summary

Introduction

The network research has attracted more and more attention from scientific communities [1,2,3,4,5], for example, chemical networks, social networks, disease transmission networks, and product networks. Dai et al [22] deduced the complete conclusion of generalized adjacency eigenvalues and generalized Laplacian eigenvalues of the weighted corona graph with different structures. Liu et al [23] determined the generalized adjacency, Laplacian, and signless Laplacian spectra of the weighted edge corona networks. We calculate the spanning trees’ number and the degree Kirchhoff index of the weighted corona graph as the application of the normalized Laplacian spectrum. Let intensity matrix of G. en, the signless Laplacian matrix of the weighted graph G is defined by Q(G) D􏽥 (G) + W(G). En, the signless Laplacian matrix of the weighted corona graph G1􏽥°G2 is as follows: Let G1 be a graph with n1 vertices and G2 be a k-regular graph with n2 vertices. en, the signless Laplacian matrix of the weighted corona graph G1􏽥°G2 is as follows:

Figure 2:

Application

Conclusion