The Wiener index W(G) of a connected graph G, introduced by Wiener in 1947, is defined as W(G) =∑u,v∈V (G) dG(u, v), where dG(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 SWk(G) of G is defined as [...] SWk(G)=∑S⊆V(G)|S|=kd(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 SWk(G) = w (or SWk(T) = w)? In this paper, we give some solutions to this problem.
@article{bwmeta1.element.doi-10_7151_dmgt_2000,
author = {Xueliang Li and Yaping Mao and Ivan Gutman},
title = {Inverse Problem on the Steiner Wiener Index},
journal = {Discussiones Mathematicae Graph Theory},
volume = {38},
year = {2018},
pages = {83-95},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_2000}
}
Xueliang Li; Yaping Mao; Ivan Gutman. Inverse Problem on the Steiner Wiener Index. Discussiones Mathematicae Graph Theory, Tome 38 (2018) pp. 83-95. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_2000/