On the spectral radius and the energy of eccentricity matrix of a graph

Abstract

The eccentricity matrix $\epsilon (G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph, among the trees, in terms of invertibility of the associated eccentricity matrix. The largest eigenvalue of $\epsilon (G)$ is called the $\epsilon$-spectral radius, and the eccentricity energy (or the $\epsilon$-energy) of $G$ is the sum of the absolute values of the eigenvalues of $\epsilon (G)$. We establish some bounds for the $\epsilon$-spectral radius and characterize the extreme graphs. Two graphs are said to be $\epsilon$-equienergetic if they have the same $\epsilon$-energy. For any $n\ge 5$, we construct a pair of $\epsilon$-equienergetic graphs on $n$ vertices, which are not $\epsilon$-cospectral.

11 Pages