# Inverse Problem on the Steiner Wiener Index

• Volume: 38, Issue: 1, page 83-95
• ISSN: 2083-5892

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## Abstract

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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) $S{W}_{k}\left(G\right)={\sum }_{\underset{|S|=k}{S\subseteq V\left(G\right)}}d\left(S\right)$ . 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.

## How to cite

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Xueliang Li, Yaping Mao, and Ivan Gutman. "Inverse Problem on the Steiner Wiener Index." Discussiones Mathematicae Graph Theory 38.1 (2018): 83-95. <http://eudml.org/doc/288411>.

@article{XueliangLi2018,
abstract = {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) $SW_k (G) = \sum \nolimits _\{\mathop \{S \subseteq V(G)\}\limits _\{|S| = k\} \} \{d(S)\}$ . 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 SWk(G) = w (or SWk(T) = w)? In this paper, we give some solutions to this problem.},
author = {Xueliang Li, Yaping Mao, Ivan Gutman},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {distance; Steiner distance; Wiener index; Steiner Wiener index},
language = {eng},
number = {1},
pages = {83-95},
title = {Inverse Problem on the Steiner Wiener Index},
url = {http://eudml.org/doc/288411},
volume = {38},
year = {2018},
}

TY - JOUR
AU - Xueliang Li
AU - Yaping Mao
AU - Ivan Gutman
TI - Inverse Problem on the Steiner Wiener Index
JO - Discussiones Mathematicae Graph Theory
PY - 2018
VL - 38
IS - 1
SP - 83
EP - 95
AB - 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) $SW_k (G) = \sum \nolimits _{\mathop {S \subseteq V(G)}\limits _{|S| = k} } {d(S)}$ . 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 SWk(G) = w (or SWk(T) = w)? In this paper, we give some solutions to this problem.
LA - eng
KW - distance; Steiner distance; Wiener index; Steiner Wiener index
UR - http://eudml.org/doc/288411
ER -

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