On the average distance of the hypercube tree

Given a graph G on n vertices, the total distance of G is defined as sigma G = (1/2) Sigma(u,v is an element of V(G)) d(u, v), where d(u, v) is the number of edges in a shortest path between u and v. We define the d-dimensional hypercube tree T(d) and show that it has a minimum total distance sigma(T(d)) = 2 sigma(H(d))-(n/2) = (dn(2)/2)-(n/2) over all spanning trees of H(d), where H(d) is the d-dimensional binary hypercube. It follows that the average distance of T(d) is mu(T(d)) = 2 mu(H(d))-1 = d(1 + 1/(n-1))-1.