On the average distance of the hypercube tree

Abstract

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 T d and show that it has a minimum total distance σ (T d ) = 2σ (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 μ(T d ) = 2μ(H d ) − 1 = d(1 + 1/(n − 1)) − 1.