Steiner Wiener index of graph products

Abstract

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 u and v 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.

Keywords

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