Abstract
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.
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