On the eccentric connectivity index of a graph

If G is a connected graph with vertex set V , then the eccentric connectivity index of G , ξ C ( G ) , is defined as ∑ v ∈ V deg ( v ) ec ( v ) where deg ( v ) is the degree of a vertex v and ec ( v ) is its eccentricity. We obtain an exact lower bound on ξ C ( G ) in terms of order, and show that this bound is sharp. An asymptotically sharp upper bound is also derived. In addition, for trees of given order, when the diameter is also prescribed, precise upper and lower bounds are provided.