On the degree distance of a graph

If GG is a connected graph with vertex set VV, then the degree distance of GG, D′(G)D′(G), is defined as ∑{u,v}⊆V(degu+degv)d(u,v), where degwdegw is the degree of vertex ww, and d(u,v)d(u,v) denotes the distance between uu and vv. We prove the asymptotically sharp upper bound D′(G)≤14nd(n−d)2+O(n7/2) for graphs of order nn and diameter dd. As a corollary we obtain the bound D′(G)≤127n4+O(n7/2) for graphs of order nn. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159–163].