The average Steiner distance of a graph

Dankelmann, Peter; Oellermann, Ortrud R.; Swart, Henda C.

doi:10.1002/(sici)1097-0118(199605)22:1<15::aid-jgt3>3.0.co;2-o

Cited by 52 publications

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“…It was observed by Plesník [29] that among all trees of given order the star minmimises the Wiener index. It was shown by Dankelmann, Oellermann and Swart [6], and by Li, Mao and Gutman [21] that the same statement holds also for the Steiner k-Wiener index. Plesník [29] also showed that among all trees of order n and diameter d, the Wiener index is minimised by the tree obtained from a path on d + 1 vertices by attaching n − d − 1 vertices to a centre vertex of the path.…”

Section: Introductionmentioning

confidence: 61%

“…with equality if and only if G is a path. Dankelmann, Oellermann and Swart [6] and Li, Mao, Gutman [21] showed that this result extends to the Steiner k-Wiener index.…”

Section: Introductionmentioning

confidence: 93%

“…The Steiner k-Wiener index was introduced by Li, Mao and Gutman [21]. A closely related graph parameter, the average Steiner k-distance of G, denoted by µ k (G) and defined as the average of the Steiner distances of all k-element subsets of V (G), was introduced in [6] and further investigated in [8]. In the same way in which the Steiner k-Wiener index generalises the Wiender index, the average k-distance generalises the average distance since µ 2 (G) = µ(G).…”

Section: Introductionmentioning

confidence: 99%

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The Steiner k-Wiener index of graphs with given minimum degree

Dankelmann

2019

Discrete Applied Mathematics

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Section: Introductionmentioning

confidence: 61%

“…with equality if and only if G is a path. Dankelmann, Oellermann and Swart [6] and Li, Mao, Gutman [21] showed that this result extends to the Steiner k-Wiener index.…”

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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The Steiner k-Wiener index of graphs with given minimum degree

Dankelmann

2019

Discrete Applied Mathematics

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“…The average Steiner distance µ k (G) of a graph G, introduced by Dankelmann, Oellermann and Swart in [12], is defined as the average of the Steiner distances of all k-subsets of V (G), i.e. For more details on average Steiner distance, we refer to [12,13].…”

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A note on the Steiner -diameter of a graph

Mao

Melekian

Cheng

2018

International Journal of Computer Mathematics: Computer Systems

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The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou 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) among the vertices of S is the minimum size among all connected subgraphs whose vertex sets contain S. Let n and k be two integers with 2 ≤ k ≤ n. Then the Steiner k-eccentricityIn 2011, Chartrand, Okamoto, Zhang showed that k − 1 ≤ sdiam k (G) ≤ n − 1. In this paper, graphs with sdiam k (G) = ℓ for k = n, n − 1, n − 2, n − 3 and k − 1 ≤ ℓ ≤ n − 1 are characterized, respectively.

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“…In fact, no such algorithm is even known for other classes of graphs such as the partial 2-trees, Halin graphs, interval graphs, or distance hereditary graphs for which the Steiner problem has an efficient solution. The ideas presented in [1] initiated the study of the average Steiner n-distance of a graph (see [5]).…”

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On Steiner centers and Steiner medians of graphs

Oellermann

1999

Networks

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Let G be connected graph and S a set of vertices of G. Then a Steiner tree for S is a connected subgraph of G of smallest size (number of edges) that contains S. The size of such a subgraph is called the Steiner distance for S and is denoted by d(S). For a vertex v of G, and integer n, 2 ≤n ≤ |V(G)|, the Steiner n‐eccentricity en(v) of v is defined as en(v) = max{d(S)|S ⊆V(G), |S| = n, and v ∈ S}. The Steiner n‐radius radnG and Steiner n‐diameter diamnG are defined as the minimum and maximum n‐eccentricity respectively, taken over all vertices of G. Relationships between radnG and diamnG are given if G is a tree, and a conjecture (with some supporting results) is made that relates these parameters for general graphs. The subgraph induced by these vertices with n‐eccentricity radnG is called the Steiner n‐center of G and is denoted by Cn(G). It is shown that every graph is the Steiner n‐center of some graph. The Steiner n‐distance of a vertex v, denoted by dn(v), is defined by dn(v) = ∑{d(S)|v ∈S, |S| =n}. The Steiner n‐median Mn(G) of G is the subgraph induced by those vertices with minimum Steiner n‐distance. Algorithms for finding Cn(T) and Mn(T) for a tree T are described. It is shown that the distance between the Cn(T) and Mn(T) for a tree T can be arbitrarily large. Eccentricity measures are defined that extend those of the Steiner n‐eccentricity and Steiner n‐distance of a vertex. Then it is shown that every vertex on a shortest path between the Steiner n‐center and Steiner n‐median of a tree belongs to a “center” relative to one of these eccentricity measures. © 1999 John Wiley & Sons, Inc. Networks 34: 258–263, 1999

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The average Steiner distance of a graph

Cited by 52 publications

References 0 publications

The Steiner k-Wiener index of graphs with given minimum degree

The Steiner k-Wiener index of graphs with given minimum degree

A note on the Steiner -diameter of a graph

On Steiner centers and Steiner medians of graphs

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