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

Authors :
Mahato, Iswar
Gurusamy, R.
Kannan, M. Rajesh
Arockiaraj, S.
Publication Year :
2019

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.<br />Comment: 11 Pages

Subjects

Subjects :
Mathematics - Combinatorics

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.1909.05609
Document Type :
Working Paper