Geodesic
The Steiner Wiener Index of A Graph
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 2, pp. 455-465

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The Wiener index W(G) of a connected graph G, introduced by Wiener in 1947, is defined as W(G) = Σ_ u,v ∈ V(G) d(u, v) where d_G(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 SW_k(G) of G is defined by Σ_ S ⊆ V(G) |S| = k d(S). Expressions for SW_k for some special graphs are obtained. We also give sharp upper and lower bounds of SW_k 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.
Keywords: distance, Steiner distance, Wiener index, Steiner Wiener k- index 
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     title = {The {Steiner} {Wiener} {Index} of {A} {Graph}},
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Li, Xueliang; Mao, Yaping; Gutman, Ivan. The Steiner Wiener Index of A Graph. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 2, pp. 455-465. http://geodesic.mathdoc.fr/item/DMGT_2016_36_2_a15/