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 ξc(G)=∑vi∈V(G)degG(vi)εG(vi), ξce(G)=∑vi∈V(G)degG(vi)εG(vi) where degG(vi) is the degree of vi in G and εG(vi) denotes the eccentricity of vertex vi in G. In this paper we study on the difference of ECI and CEI of graphs G, denoted by ξD(G)=ξc(G)−ξce(G). We determine the upper and lower bounds on ξD(T) and the corresponding extremal trees among all trees of order n. Moreover, the extremal trees with respect to ξD are completely characterized among all trees with given diameter d. And we also characterize some extremal general graphs with respect to ξD. Finally we propose that some comparative relations between CEI and ECI are proposed on general graphs with given number of pendant vertices.