Abstract:

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.

Key words: product graph, Steinerk-hyper Wiener index, Steiner distance