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 $S{W}_k(G)$ 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 ${\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^{-1}S{W}_k(G)$. 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 $S{W}_k(G)\le \frac{k-1}{k+1}\frac{3n}{\delta +1}\left(\genfrac{}{}{0ex}{}{n}{k}\right)+O(n^k)$, and that ${\mu }_k(G)\le \frac{k-1}{k+1}\frac{3n}{\delta +1}+O(1)$. We improve this bound for triangle-free graphs to $S{W}_k(G)\le \frac{k-1}{k+1}\frac{2n}{\delta }\left(\genfrac{}{}{0ex}{}{n}{k}\right)+O(n^k)$, and ${\mu }_k(G)\le \frac{k-1}{k+1}\frac{2n}{\delta }+O(1)$. All bounds are best possible.

References in corpus (1)

• The Steiner diameter of a graph