STEINER WIENER INDEX OF GRAPH PRODUCTS

The Wiener index $W(G)$ of a connected graph $G$ is defined as $W(G)=sum_{u,vin V(G)}d_G(u,v)$ where $d_G(u,v)$ is the distance between the vertices $u$ and $v$ of $G$. For $Ssubseteq V(G)$, the Steiner distance $d(S)$ of the vertices of $S$ is the minimum size of a connected subgraph of $G$ whose vertex set is $S$. The  $k$-th Steiner Wiener index $SW_k(G)$ of $G$ is defined as $SW_k(G)=sum_{overset{Ssubseteq V(G)}{|S|=k}} d(S)$. We establish expressions for the $k$-th Steiner Wiener index on the join, corona, cluster, lexicographical product, and Cartesian product of graphs.