Nordhaus–Gaddum-type results for the Steiner Wiener index of graphs

Mao, Yaping; Wang, Zhao; Gutman, Iván; Li, He

doi:10.1016/j.dam.2016.11.014

Cited by 24 publications

(12 citation statements)

References 14 publications

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“…It is usual to consider SW k for 2 ≤ k ≤ n − 1, but the above definition implies SW 1 (G) = 0 and SW n (G) = n − 1. We refer to [18,19,[21][22][23][24] for more details on Steiner Wiener index.…”

Section: Y Maomentioning

confidence: 99%

Steiner Harary index

Mao¹

2018

Kragujevac J Math

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The Harary index H(G) of a connected graphs G is defined as H(G) = u,v∈V (G) 1 d G (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 G (S) of the vertices of S is the minimum size of all connected subgraphs whose vertex set contain S. Recently, Furtula, Gutman, and Katanić introduced the concept of Steiner Harary index and give its chemical applications. The k-center Steiner Harary index SH k (G) of G is defined by SH k (G) = S⊆V (G), |S|=k 1 d G (S). Expressions for SH k for some special graphs are obtained. We also give sharp upper and lower bounds of SH k of a connected graph, and establish some of its properties in the case of trees.

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Section: Y Maomentioning

confidence: 99%

Steiner Harary index

Mao¹

2018

Kragujevac J Math

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“…The weighting of multiplication of degree or expected degree has also been extensively explored in, for example, the field of random graphs [19,20] and proves to be very prolific. For more results on Steiner Wiener index, Steiner degree distance, and Steiner Gutman index, we refer to the reader to [13,16,18,[21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

et al. 2020

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Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk(G) of G is defined by SGutk(G)=∑S⊆V(G),|S|=k∏v∈SdegG(v)dG(S), in which dG(S) is the Steiner distance of S and degG(v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk. We obtain sharp upper and lower bounds of SGutk(G)+SGutk(G¯) and SGutk(G)·SGutk(G¯) for a connected graph G of order n, m edges, maximum degree Δ and minimum degree δ.

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“…in trees. As a generalization of the Wiener index the Steiner Wiener index has received much attention in the past few years [6,7,9,15,17,18,19,23]. We will consider the extremal problems with respect to the Steiner Wiener index in trees with a given segment sequence and show that the starlike tree and the quasi-caterpillar are still extremal.…”

Section: Introductionmentioning

confidence: 99%

The Steiner Wiener index of trees with a given segment sequence

Zhang

Wang

Zhang

2019

Applied Mathematics and Computation

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The Steiner distance of vertices in a set S is the minimum size of a connected subgraph that contain these vertices. The sum of the Steiner distances over all sets S of cardinality k is called the Steiner k-Wiener index and studied as the natural generalization of the famous Wiener index in chemical graph theory. In this paper we study the extremal structures, among trees with a given segment sequence, that maximize or minimize the Steiner k-Wiener index. The same extremal problems are also considered for trees with a given number of segments.

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Nordhaus–Gaddum-type results for the Steiner Wiener index of graphs

Cited by 24 publications

References 14 publications

Steiner Harary index

Steiner Harary index

Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

The Steiner Wiener index of trees with a given segment sequence

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