Let G be a finite connected graph of order n, minimum degree δ and diameter d. The degree distance D′(G) of G is defined as ∑ {u,v}⊆V (G)(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. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that
\[ D^\prime (G)\le \frac {1}{4}\,dn\biggl (n-\frac {d}{3}(\delta +1)\biggr )^2+O(n^3). \]
As a corollary, we obtain the bound
D′ (
G)≤
n4 /(9(
δ+1) )+
O(
n3) for a graph
G of order
n and minimum degree
δ. This result, apart from improving on a result of Dankelmann
et al. [‘On the degree distance of a graph’,
Discrete Appl. Math.157 (2009), 2773–2777] for graphs of given order and minimum degree, completely settles a conjecture of Tomescu [‘Some extremal properties of the degree distance of a graph’,
Discrete Appl. Math.98(1999), 159–163].