Degree distance and vertex-connectivity

Let GG be a finite connected graph of order nn, vertex-connectivity κκ, and diameter dd. The degree distance D′(G)D′(G) of GG is defined as ∑{u,v}⊆V(G)(degu+degv)dG(u,v), where degw is the degree of vertex ww and dG(u,v)dG(u,v) denotes the distance between uu and vv in GG. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, vertex-connectivity, and diameter. In particular, we prove that D′(G)≤{14dn(n−κd)2+O(n3)ifd