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 ξ c ( G ) = ∑ v i ∈ V ( G ) deg G ( v i ) ε G ( v i ) , ξ c e ( G ) = ∑ v i ∈ V ( G ) deg G ( v i ) ε G ( v i ) where deg G ( v i ) is the degree of v i in G and ε 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 ξ D ( G ) = ξ c ( G ) − ξ c e ( G ) . We determine the upper and lower bounds on ξ D ( T ) and the corresponding extremal trees among all trees of order n . Moreover, the extremal trees with respect to ξ D are completely characterized among all trees with given diameter d . And we also characterize some extremal general graphs with respect to ξ D . Finally we propose that some comparative relations between CEI and ECI are proposed on general graphs with given number of pendant vertices. [ABSTRACT FROM AUTHOR]