On the degree distance of a graph

If G is a connected graph with vertex set V , then the degree distance of G, D′(G), is defined as ∑ {u,v}⊆V (deg u + deg v) d(u, v), where degw is the degree of vertex w, and d(u, v) denotes the distance between u and v. We prove the asymptotically sharp upper bound D′(G) ≤ 14 nd(n− d) 2 +O(n7/2) for graphs of order n and diameter d. As a corollary we obtain the bound D′(G) ≤ 1 27 n 4 + O(n7/2) for graphs of order n. This essentially proves a conjecture by Tomescu [20].