Eccentricity spectrum of join of central graphs and eccentricity Wiener index of graphs

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.