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 xi(c)(G)=Sigma(vi epsilon V(G))deg(G)(V-i)epsilon(G)(V-i) and E-1(G) = Sigma(vi epsilon V(G))epsilon(G)(V-i)(2), respectively, 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 compare the eccentric connectivity index and the first Zagreb eccentricity index of graphs. It is proved that E-1(T) > xi(c)(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 E-1(G) > xi(c)(G). We also present an example in which infinite graphs G are constructed with E-1(G) > xi(c)(G) and give some results on the graphs G with E-1(G) > xi(c)(G). Finally, an effective construction is proposed for generating infinite graphs with each comparative inequality possibility between these two topological indices.