Eccentricity energy change of complete multipartite graphs due to edge deletion

TL;DR

This paper investigates how the eccentricity energy of complete multipartite graphs changes when edges are deleted, revealing cases where it increases or decreases, and providing specific proofs for complete k-partite graphs.

Contribution

It introduces the study of eccentricity energy change due to edge deletion and proves its increase in complete k-partite graphs, highlighting differences from distance energy behavior.

Findings

Eccentricity energy can increase or decrease after edge deletion.

Examples show eccentricity and distance energies can change in opposite directions.

Eccentricity energy of complete k-partite graphs increases with edge deletion.

Abstract

The eccentricity matrix $\varepsilon(G)$ 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 $\varepsilon(G)$. 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 $K_{n_1,\hdots,n_k}$ with $k\geq 2$ and $ n_i\geq 2$, increases due to an edge deletion.