Comparison Between Two Eccentricity-based Topological Indices of Graphs

For a connected graph G, the eccentric connectivity index (ECI) and the first Zagreb eccentricity index of G are defined as ( ) ( ) deg ( ) ( ) i c G i G i v V G ξ G v ε v   and ( 2 1 ) ( ) ( ) i G i v G V E G ε v   , respectively, where deg ( ) G i v is the degree of i v in G and ( ) G i ε v denotes the eccentricity of vertex i v in G. In this paper we compare the eccentric connectivity index and the first Zagreb eccentricity index of graphs. It is proved that  1( ) ( ) c E T ξ T for any tree T. This improves a result by Das[25] for the chemical trees. Moreover, we also show that there are infinite number of chemical graphs G with  1( ) ( ) c E G ξ G . We also present an example in which infinite graphs G are constructed with  1( ) ( ) c E G ξ G and give some results on the graphs G with  1( ) ( ) c E G ξ G . Finally, an effective construction is proposed for generating infinite graphs with each comparative inequality possibility between these two topological indices.