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(degu+degv)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)@?14nd(n-d)^2+O(n^7^/^2) for graphs of order n and diameter d. As a corollary we obtain the bound D^'(G)@?127n^4+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].

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