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 $ε$-spectrum, $ε$-energy, $ε$-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 $ε-$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 $ε-$ energy of some specific graphs. These findings allow us to construct new families of $ε$-cospectral graphs and non $ε$-cospectral $ε-$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.