The Steiner [formula omitted]-Wiener index of graphs with given minimum degree.

Let G be a connected graph. The Steiner distance d (S) of a set S of vertices is the minimum size of a connected subgraph of G containing all vertices of S. For k ∈ N , the Steiner k -Wiener index S W k (G) is defined as ∑ S d (S) , where the sum is over all k -element subsets of the vertex set of G. The average Steiner k -distance μ k (G) of G is defined as n k − 1 S W k (G). In this paper we prove upper bounds on the Steiner Wiener index and the average Steiner distance of graphs with given order n and minimum degree δ. Specifically we show that S W k (G) ≤ k − 1 k + 1 3 n δ + 1 n k + O (n k) , and that μ k (G) ≤ k − 1 k + 1 3 n δ + 1 + O (1). We improve this bound for triangle-free graphs to S W k (G) ≤ k − 1 k + 1 2 n δ n k + O (n k) , and μ k (G) ≤ k − 1 k + 1 2 n δ + O (1). All bounds are best possible. [ABSTRACT FROM AUTHOR]