Multiplicative version of eccentric connectivity index

Abstract

The eccentric connectivity index is defined for a connected graph G as the summation of the terms ɛG(u)+ɛG(v) over all edges uv of G, in which ɛG(u) denotes the eccentricity of the vertex u in G. In this paper, we study a multiplicative version of this graph invariant under the name multiplicative eccentric connectivity index. We present a sharp lower bound on the multiplicative eccentric connectivity index of n-vertex trees with fixed diameter and a sharp upper bound on the multiplicative eccentric connectivity index of n-vertex trees with given number of pendent vertices. Using these results, the trees on n vertices with the first three minimum and maximum values of the multiplicative eccentric connectivity index are characterized. Moreover, we present sharp bounds on the multiplicative eccentric connectivity index of connected graphs in terms of certain graph parameters including the order, size, radius, diameter, and number of universal vertices, and study the relationship between this invariant and several other vertex-eccentricity-based invariants. In particular, the minimal graphs with respect to the multiplicative eccentric connectivity index in the set of all connected graphs of given order, connected graphs of given size, and unicyclic and bicyclic graphs of given order are characterized and a Nordhaus–Gaddum-type result for this invariant is given.