On the average distance of the hypercube tree

Given a graph G on n vertices, the total distance of G is defined as σ(G) = 1 2 ∑ u,v∈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 Td and show that it has a minimum total distance σ(Td) = 2σ(Hd) − ( n 2 ) = dn 2 2 − ( n 2 ) over all spanning trees of Hd, where Hd is the d-dimensional binary hypercube. It follows that the average distance of Td is μ(Td) = 2μ(Hd)− 1 = d ( 1 + 1 n−1 ) − 1.