Graph energy based on the eccentricity matrix.

The eccentricity matrix E (G) of a graph G is derived from the distance matrix by keeping for each row and each column only the eccentricities. The E -eigenvalues of a graph G are those of its eccentricity matrix E (G) , and the eccentricity energy (or the E -energy) of G is the sum of the absolute values of E -eigenvalues. A graph is called self-centered graph if its diameter and radius are equal. In this paper, we investigate the relation between the E -energy and the ordinary energy, and we determine the exact values of E -energies of paths, cycles and double stars. Moreover, when G is an r -antipodal graph, we show that the E -energy of strong product of graphs G and H only depends on the structure of G. We finally provide upper and lower bounds for the E -energy whose extreme graphs are kinds of self-centered graphs, and we propose some potential topics for further study. [ABSTRACT FROM AUTHOR]