Eccentricity energy change of complete multipartite graphs due to edge deletion

Iswar Mahato,M Rajesh Kannan

doi:10.1515/spma-2021-0156

Abstract

Abstract The eccentricity matrix ɛ(G) of a graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of G is sum of the absolute values of the eigenvalues of ɛ(G). Although the eccentricity matrices of graphs are closely related to the distance matrices of graphs, a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The change in eccentricity energy of a graph due to an edge deletion is one such property. In this article, we give examples of graphs for which the eccentricity energy increase (resp., decrease) but the distance energy decrease (resp., increase) due to an edge deletion. Also, we prove that the eccentricity energy of the complete k-partite graph Kn 1, ... , nk with k ≥ 2 and ni ≥ 2, increases due to an edge deletion.

Highlights

The eccentricity matrix ε(G) of a graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones
We prove that the eccentricity energy of the complete k-partite graph Kn,...,nk with k ≥ and ni ≥, increases due to an edge deletion
The distance matrix of a connected graph G on n vertices, denoted by D(G), is an n × n matrix whose rows and columns are indexed by the vertices of G and the entries are de ned by D(G)uv = dG(u, v)

Summary

Introduction

Throughout this paper, we consider nite simple connected graphs. For a graph G, let V(G) and E(G) denote the vertex set and the edge set of G, respectively. The change in eccentricity energy of a graph due to an edge deletion is one such property. We prove that the eccentricity energy of the complete k-partite graph Kn ,...,nk with k ≥ and ni ≥ , increases due to an edge deletion.

Results

Conclusion