On the eccentric connectivity index and Wiener index of a graph

Abstract

Let G be a finite connected graph of order n and minimum degree δ. The eccentric connectivity index ξc (G) of G is defined as ξc (G) = Σv∊V (G) ecG (v)degG (v), where ecG (x) and degG (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.