Geodesic
Inverse Problem on the Steiner Wiener Index
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 83-95

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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_G(u, v), where d_G(u, v) is the distance (the length a shortest path) between the vertices u and v in G. For S ⊆ V (G), the Steiner distance d(S) of the vertices of S, introduced by Chartrand et al. in 1989, is the minimum size of a connected subgraph of G whose vertex set contains S. The k-th Steiner Wiener index SW_k(G) of G is defined as SW_k(G)= Σ_ S ⊆ V(G) |S|=k d(S). We investigate the following problem: Fixed a positive integer k, for what kind of positive integer w does there exist a connected graph G (or a tree T) of order n ≥ k such that SW_k(G) = w (or SW_k(T) = w)? In this paper, we give some solutions to this problem.
Keywords: distance, Steiner distance, Wiener index, Steiner Wiener index 
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Li, Xueliang; Mao, Yaping; Gutman, Ivan. Inverse Problem on the Steiner Wiener Index. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 83-95. http://geodesic.mathdoc.fr/item/DMGT_2018_38_1_a6/