On the eccentric connectivity index and Wiener index of a graph

Let G be a finite connected graph of order n and minimum degree δ. The eccentric connectivity index ξ^c (G) of G is defined as ξ^c (G) = Σ_{v{small element of}V (G)} ec_G (v)deg_G (v), where ec_G (x) and deg_G (x) denote the eccentricity and degree of vertex x in G, respectively. We prove that the eccentric connectivity index of G satisfies, and construct graphs which asymptotically attain the bound. Our bound implies some known results by Došlić, Saheli & Vukičević [4], Morgan, Mukwembi & Swart [11], and Zhou & Du [16]. Further, we also determine upper bounds on the well-studied Wiener index in terms of the eccentric connectivity index.