A sharp lower bound on Steiner Wiener index for trees with given diameter

Let G be a connected graph with vertex set V(G) and edge set E(G). For a subset S of V(G), the Steiner distanced(S) of S is the minimum size of a connected subgraph whose vertex set contains S. For an integer k with 2≤k≤n−1, the Steinerk-Wiener indexSWk(G) is ∑S⊆V(G),|S|=kd(S). In this paper, we introduce some transformations for trees that do not increase their Steiner k-Wiener index for 2≤k≤n−1. Using these transformations, we get a sharp lower bound on Steiner k-Wiener index for trees with given diameter, and obtain the corresponding extremal graph as well.