ECCENTRICITY SPECTRUM OF JOIN OF CENTRAL GRAPHS AND ECCENTRICITY WIENER INDEX OF GRAPHS

ECCENTRICITY SPECTRUM OF JOIN OF CENTRAL GRAPHS AND ECCENTRICITY WIENER INDEX OF GRAPHS

A. Ashokan, A. V. Chithra

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Abstract

The eccentricity matrix of a simple connected graph is obtained from its distance matrix by preserving the largest non-zero distance in each row and column, while setting all other entries to zero. This article examines the ϵ-spectrum and ϵ-spectral radius of central graphs of triangle-free regular graphs. The study further explores the irreducibility of the eccentricity matrix of central graphs. Moreover, we investigate the ϵ-spectrum and the irreducibility of various central graph operations, such as the central vertex join, central edge join, and central vertex-edge join. Also, we compute the ϵ-energy of specific graphs. These findings allow us to construct new families of ϵ-cospectral graphs and non ϵ-cospectral ϵ-equienergetic graphs. Additionally, we estimate certain upper and lower bounds for the eccentricity Wiener index of graphs and an upper bound for the eccentricity energy of self-centered graphs.

Keywords

eccentricity matrix, ϵ-spectrum, ϵ- energy, central graph, ϵ-Wiener index.