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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Wiener index of the tensor product of a path and a cycle

100%
Pattabiraman K., Paulraja P.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 4
737-751
EN
The Wiener index, denoted by W(G), of a connected graph G is the sum of all pairwise distances of vertices of the graph, that is, $W(G) = ½Σ_{u,v ∈ V(G)} d(u,v)$. In this paper, we obtain the Wiener index of the tensor product of a path and a cycle.
2
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The Steiner Wiener Index of A Graph

80%
Li X., Mao Y., Gutman I.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 2
455-465
EN
The Wiener index W(G) of a connected graph G, introduced by Wiener in 1947, is defined as W(G) = ∑u,v∈V(G) d(u, v) where dG(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(S) of the vertices of S is the minimum size of a connected subgraph whose vertex set is S. We now introduce the concept of the Steiner Wiener index of a graph. The Steiner k-Wiener index SWk(G) of G is defined by [...] . Expressions for SWk for some special graphs are obtained. We also give sharp upper and lower bounds of SWk of a connected graph, and establish some of its properties in the case of trees. An application in chemistry of the Steiner Wiener index is reported in our another paper.
3
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Inverse Problem on the Steiner Wiener Index

80%
Li X., Mao Y., Gutman I.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
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nr 1
83-95
EN
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.
4
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Wiener index of generalized stars and their quadratic line graphs

80%
Dobrynin A., Mel'nikov L.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 1
161-175
EN
The Wiener index, W, is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of Δ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W(S) = W(L(L(S)) exist only for 4 ≤ Δ ≤ 6. Infinite families of generalized stars with this property are constructed.
5
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Wiener and vertex PI indices of the strong product of graphs

80%
Pattabiraman K., Paulraja P.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 4
749-769
EN
The Wiener index of a connected graph G, denoted by W(G), is defined as $½ ∑_{u,v ∈ V(G)}d_G(u,v)$. Similarly, the hyper-Wiener index of a connected graph G, denoted by WW(G), is defined as $½W(G) + ¼ ∑_{u,v ∈ V(G)} d²_G(u,v)$. The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, the exact formulae for Wiener, hyper-Wiener and vertex PI indices of the strong product $G ⊠ K_{m₀,m₁,...,m_{r -1}}$, where $K_{m₀,m₁,...,m_{r -1}}$ is the complete multipartite graph with partite sets of sizes $m₀,m₁, ...,m_{r -1}$, are obtained. Also lower bounds for Wiener and hyper-Wiener indices of strong product of graphs are established.
6
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Eccentric distance sum index for some classes of connected graphs

61%
Bielak H., Broniszewska K.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2017
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tom 71
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nr 2
EN
In this paper we show some properties of the eccentric distance sum index which is defined as follows \(\xi^{d}(G)=\sum_{v \in V(G)}D(v) \varepsilon(v)\). This index is widely used by chemists and biologists in their researches. We present a lower bound of this index for a new class of graphs.
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