On the eccentric connectivity index of a graph

If GG is a connected graph with vertex set VV, then the eccentric connectivity index of GG, ξC(G)ξC(G), is defined as ∑v∈Vdeg(v)ec(v) where deg(v) is the degree of a vertex vv and ec(v) is its eccentricity. We obtain an exact lower bound on ξC(G)ξC(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.