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

Abstract

Let $G$ be a connected graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. For a subset $S$ of $V\left(G\right)$, the Steiner distance $d\left(S\right)$ of $S$ is the minimum size of a connected subgraph whose vertex set contains $S$. For an integer $k$ with $2\le k\le n-1$, the Steiner $k$-Wiener index ${\text{SW}}_k\left(G\right)$ is ${\sum }_{S\subseteq V\left(G\right),\left|S\right|=k}d\left(S\right)$. In this paper, we introduce some transformations for trees that do not increase their Steiner $k$-Wiener index for $2\le k\le n-1$. 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.