On two eccentricity-based topological indices of graphs

For a connected graph G, the eccentric connectivity index (ECI) and connective eccentricity index (CEI) of G are, respectively, defined as xi(c)(G) = Sigma(vi is an element of V(G)) deg(G)(v(i))epsilon(G)(v(i)), xi(ce)(G) = Sigma(vi is an element of V(G)) deg(G)(v(i))/epsilon(G)(v(i)) where deg(G)(v(i)) is the degree of v(i) in G and epsilon(G)(v(i)) denotes the eccentricity of vertex v(i) in G. In this paper we study on the difference of ECI and CEI of graphs G, denoted by xi(D)(G) = xi(c)(G) - xi(ce)(G). We determine the upper and lower bounds on xi(D)(T) and the corresponding extremal trees among all trees of order n. Moreover, the extremal trees with respect to xi(D) are completely characterized among all trees with given diameter d. And we also characterize some extremal general graphs with respect to xi(D). Finally we propose that some comparative relations between CEI and ECI are proposed on general graphs with given number of pendant vertices. (C) 2017 Elsevier B.V. All rights reserved.