令图G是一个连通图。当$ 2\leqslant k\leqslant n-1$时, 图G的Steiner k-hyper Wiener指标定义为$ {\rm SWW}_{k}(G)=\frac{1}{2}\sum_{S\subseteq V (G), |S|=k}d_{G}(S)+\frac{1}{2}\sum_{S\subseteq V (G), |S|=k}d_{G}(S)^{2}$, 其中$ d_{G}(S)$表示图G中S的Steiner距离, 即连通图G中包含点集S的最小连通子图的边数。本文中我们确定了连图和字典积图的Steiner k-hyper Wiener指标的表达式, 给出了笛卡尔积图, 聚类图和冠状图的Steiner k-hyper Wiener指标的下限。

Let G be a connected graph. For $ 2\leqslant k\leqslant n-1$, the Steiner k-hyper Wiener index $ {\rm SWW}_{k}(G)$ is defined as ${\rm SWW}_{k}(G)=\frac{1}{2}\sum_{S\subseteq V(G), |S|=k}d_{G}(S)+\frac{1}{2}\sum_{S\subseteq V(G), |S|=k}d_{G}(S)^{2} $, where $d_{G}(S) $ is the Steiner distance of S, means the minimum size of a connected subgraph which vertex set contains S. In this paper, we establish expressions for the Steiner k-hyper Wiener index on the join and lexicographical product of graphs and give lower bounds for the Steiner $k$-hyper Wiener index on cartesian, cluster and corona product of graphs.