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 deg w 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) ≤ frac(1, 4) n d (n - d)² + O (n^{7 / 2}) for graphs of order n and diameter d. As a corollary we obtain the bound D^′ (G) ≤ frac(1, 27) n⁴ + O (n^{7 / 2}) for graphs of order n. 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].