Steiner Wiener index of graph products

The Wiener index W(G) of a connected graph G‎ ‎is defined as W(G)=∑u,v∈V(G)dG(u,v) ‎ ‎where dG(u,v) is the distance between the vertices uu and vv of‎ ‎ G‎. ‎For S⊆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 ‎SWk(G) of G is defined as‎ ‎SWk(G)=∑|S|=kS⊆V(G)d(S) ‎. ‎We establish‎ ‎expressions for the k-th Steiner Wiener index on the join‎, ‎corona‎, ‎cluster‎, ‎lexicographical product‎, ‎and Cartesian product of graphs‎.