On the eccentricity energy of complete mutipartite graph

TL;DR

This paper investigates the eccentricity energy of complete multipartite graphs, establishing bounds, characterizing extremal cases, and exploring non-cospectral graphs with identical eccentricity energy.

Contribution

It provides new bounds and characterizations for the eccentricity energy of complete multipartite graphs, addressing an open problem from prior research.

Findings

Derived bounds for the eccentricity energy of complete multipartite graphs.

Characterized extremal graphs with maximum or minimum eccentricity energy.

Identified classes of graphs sharing the same eccentricity energy but not being cospectral.

Abstract

The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix $\varepsilon(G),$ and the eccentricity energy (or the $\varepsilon$-energy) of $G$ is the sum of the absolute values of $\varepsilon$-eigenvalues. In this paper, we establish some bounds for the $\varepsilon$-energy of the complete multipartite graph $K_{n_1, n_2, \ldots,n_p}$ of order $n= \sum_{i=1}^p n_i $ and characterize the extreme graphs. This partially answers the problem given in Wang {\em et al.} (2019). We finish the paper showing graphs that are not $\varepsilon$-cospectral with the same $\varepsilon$-energy.