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

Abstract: The eccentricity matrix of a simple connected graph is derived from its distance matrix by preserving the largest non-zero distance in each row and column, while the other entries are set to zero. This article examines the $\epsilon$-spectrum, $\epsilon$-energy, $\epsilon$-inertia and irreducibility of the central graph (respectively complement of the central graph) of a triangle-free regular graph(respectively regular graph). Also look into the $\epsilon-$spectrum and the irreducibility of different central graph operations, such as central vertex join, central edge join, and central vertex-edge join. We also examine the $\epsilon-$ energy of some specific graphs. These findings allow us to construct new families of $\epsilon$-cospectral graphs and non $\epsilon$-cospectral $\epsilon-$equienergetic graphs. Additionally, we investigate certain upper and lower bounds for the eccentricity Wiener index of graphs. Also, provide an upper bound for the eccentricity energy of a self-centered graph.