Title: A sharp lower bound on Steiner Wiener index for trees with given diameter
Abstract: 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 distance d ( 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 Steiner k -Wiener index SW k ( G ) is ∑ S ⊆ V ( G ) , | S | = k d ( 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.