Steiner Wiener Index of Complete m-Ary Trees

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 $2 le k le n - 1$, the $k$-th Steiner Wiener index of a graph $G$ is defined as $SW_k(G) = sum_{substack{Ssubseteq V(G)\ |S|=k}}d(S)$. In this paper, we present exact values of the $k$-th Steiner Wiener index of complete $m$-ary trees by using inclusion-excluision principle for various values of $k$.