The Average Steiner Distance of a Graph

The average distance μ(G) of a graph G is the average among the distances between all pairs of vertices in G. For n ≥ 2, the average Steiner n-distance μ_n(G) of a connected graph G is the average Steiner distance over all sets of n vertices in G. It is shown that for a connected weighted graph G, μ_n(G) ≤ μ_k(G) + μ_n+1-k(G) where 2 ≤ k ≤ n - 1. The range for the average Steiner n-distance of a connected graph G in terms of n and |V(G)| is established. Moreover, for a tree T and integer k, 2 ≤ k ≤ n - 1, it is shown that μ_n(T) ≤ (n/k)μ_k(T) and the range for μ_n(T) in terms of n and |V(T)| is established. Two efficient algorithms for finding the average Steiner n-distance of a tree are outlined.