Steiner Wiener index of graph products

The Wiener index W(G) of a connected graph G‎ ‎is defined as W(G)=\sum_{u,v\in V(G)}d_G(u,v)‎ ‎where d_G(u,v) is the distance between the vertices u and v of‎ ‎G‎. ‎For S\subseteq 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{S\subseteq 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.

Keywords:

‎Distance (in graph)‎ , ‎Steiner distance (in graph)‎ , ‎Steiner Wiener index‎ , ‎product (of graphs)