The Steiner Wiener Index of A Graph
Abstract The Wiener index W(G) of a connected graph G, introduced by Wiener in 1947, is defined as W(G) u,vV(G) d(u, v) where dG(u, v) is the distance between vertices u and v of G. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least 2 and S V (G) the Steiner distance d(S) of the vertices of S is the minimum size of a connected subgraph whose vertex set is S. We now introduce the concept of the Steiner Wiener index of a graph. The Steiner k Wiener index SWk(G) of G is defined by Expressions for SWk for some special graphs are obtained. We also give sharp upper and lower bounds of SWk of a connected graph, and establish some of its properties in the case of trees. An application in chemistry of the Steiner Wiener index is reported in our another paper.
Discuss. Math. Graph Theory