The upper bounds on the Steiner k-Wiener index in terms of minimum and maximum degrees

Abstract

For $k \in {\mathbb {N}},$ Ali et al. (Discrete Appl Math 160:1845-1850, 2012) introduce the Steiner k-Wiener index $SW_{k}(G)=\sum _{S\in V(G)} d(S),$ where d(S) is the minimum size of a connected subgraph of G containing the vertices of S. The average Steiner k-distance $\mu _{k}(G)$ of G is defined as $\genfrac(){0.0pt}1{n}{k}^{-1} SW_{k}(G)$. In this paper, we give some upper bounds on $SW_{k}(G)$ and $\mu _{k}(G)$ in terms of minimum degree, maximum degree and girth in a triangle-free or a $C_{4}$-free graph.