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

TL;DR

This paper investigates the spectral properties and energy of the eccentricity matrix of graphs, characterizes star graphs among trees, and constructs pairs of non-cospectral graphs with equal eccentricity energy.

Contribution

It provides a characterization of star graphs via eccentricity matrix invertibility and constructs non-cospectral graphs with identical eccentricity energy.

Findings

Characterization of star graphs among trees based on eccentricity matrix invertibility

Bounds established for the eccentricity spectral radius

Construction of non-cospectral graphs with equal eccentricity energy

Abstract

The eccentricity matrix $\varepsilon(G)$ 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 $\varepsilon(G)$ is called the $\varepsilon$-spectral radius, and the eccentricity energy (or the $\varepsilon$-energy) of $G$ is the sum of the absolute values of the eigenvalues of $\varepsilon(G)$. We establish some bounds for the $\varepsilon$-spectral radius and characterize the extreme graphs. Two graphs are said to be $\varepsilon$-equienergetic if they have the same $\varepsilon$-energy. For any $n \geq 5$, we construct a pair of $\varepsilon$-equienergetic graphs on $n$ vertices, which are not $\varepsilon$-cospectral.