On the eccentricity energy of complete mutipartite graph

Abstract

The eccentricity (anti-adjacency) matrix $\epsilon (G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. The $\epsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix $\epsilon (G),$ and the eccentricity energy (or the $\epsilon$-energy) of $G$ is the sum of the absolute values of $\epsilon$-eigenvalues. In this paper, we establish some bounds for the $\epsilon$-energy of the complete multipartite graph $K_{n_1,n_2,\dots ,n_p}$ 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 $\epsilon$-cospectral with the same $\epsilon$-energy.

11 pages