Steiner Wiener index of graph products

Abstract

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.