DEGREE DISTANCE AND MINIMUM DEGREE

Let G be a finite connected graph of order n, minimum degree delta and diameter d. The degree distance D'(G) of G is defined as Sigma({u,v}subset of 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'(G) <= 1/4 dn(n - d/3 (delta + 1))(2) + O(n(3)). As a corollary, we obtain the bound D'(G) <= n(4)/(9(delta + 1)) + O(n(3)) for a graph G of order n and minimum degree delta. 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].