The Steiner $k$-Wiener index of graphs with given minimum degree

Abstract

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 $k\in \mathbb{N}$, the Steiner $k$-Wiener index $SW_k(G)$ is defined as $\sum_S d(S)$, where the sum is over all $k$-element subsets of the vertex set of $G$. The average Steiner $k$-distance $\mu_k(G)$ of $G$ is defined as $\binom{n}{k}^{-1} SW_k(G)$. 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 $\delta$. Specifically we show that $SW_k(G) \leq \frac{k-1}{k+1}\frac{3n}{\delta+1} \binom{n}{k} + O(n^{k})$, and that $\mu_k(G) \leq \frac{k-1}{k+1}\frac{3n}{\delta+1} + O(1)$. We improve this bound for triangle-free graphs to $SW_k(G) \leq \frac{k-1}{k+1}\frac{2n}{\delta} \binom{n}{k} + O(n^{k})$, and $\mu_k(G) \leq \frac{k-1}{k+1}\frac{2n}{\delta} + O(1)$. All bounds are best possible.