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)= sum_{v_i in V(G)}deg_G(v_i)varepsilon_G(v_i)) and (E_1(G)=sum_{v_iin V(G)}varepsilon_{G}(v_i)^{2}), respectively, where (deg_G(v_i)) is the degree of (v_i) in (G) and (varepsilon_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)