Equal opportunity networks, distance-balanced graphs, and Wiener game

Balakrishnan, Kalpana; Brešar, Boštjan; Changat, Manoj; Klavžar, Sandi; Vesel, Aleksander; Pleteršek, Petra Žigert

doi:10.1016/j.disopt.2014.01.002

Cited by 37 publications

(62 citation statements)

References 16 publications

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“…A concept of Wiener-type indices of graphs was recently introduced in [95] as follows. Let G be a (connected) graph and let X ⊆ V (G).…”

Section: Colored Wiener Index In Graphsmentioning

confidence: 99%

Cut Method: Update on Recent Developments and Equivalence of Independent Approaches

Nadjafi-Arani

2015

COC

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The cut method is a powerful tool for the investigation of distance-based (and some other) molecular structure-descriptors. In this paper a survey on the recent developments of the method is given. The instances of the standard cut method for the Wiener index, the Szeged index, the PI index, the generalized terminal Wiener index, the Gutman index, the edge-Wiener index, and the edge-Szeged index are described, where a standard cut method is a method that applies to partial cubes. It is pointed out that the standard cut method was recently independently discovered a couple of times. Numerous proper extensions of the standard cut method are presented. The method extends to 1-graphs, graphs with a non-trivial canonical metric representation, graphs with transitive relation Θ, and partial Hamming graphs. The instances of these extended cut methods include the Wiener index, the degree distance, distance moments, and the colored Wiener index.

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“…A concept of Wiener-type indices of graphs was recently introduced in [95] as follows. Let G be a (connected) graph and let X ⊆ V (G).…”

Section: Colored Wiener Index In Graphsmentioning

confidence: 99%

Cut Method: Update on Recent Developments and Equivalence of Independent Approaches

Nadjafi-Arani

2015

COC

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“…Equal opportunity graphs are precisely those graphs having the opportunity index equal to zero. Furthermore, in [3] the authors show that equal opportunity graphs are precisely distance-balanced graphs (of even order) that correspond to a class of graphs first studied in [16] in the case of partial cubes. Other interesting applications of distance-balanced graphs can be found in [19].…”

Section: Introductionmentioning

confidence: 99%

“…Actually, in [10] the author considers a situation when there are two firms competing in a common market with an objective to minimize the cost of transport between all the vertices (clients) of the network. It is desirable to design a network in such a way that its vertex set can be divided into two subsets so that neither of the two involved firms has an advantage to the other likewise reducing the cost of distribution and transportation of goods [3]. Another interesting application is in Social Network Analysis, where one wants to identify (connected) clusters of members with homogenous features.…”

Section: Introductionmentioning

confidence: 99%

Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

Cavaleri

Donno

Scozzari

2019

Electron. J. Combin.

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In the last decades much attention has turned towards centrality measures on graphs. The Wiener index and the total distance are key tools to investigate the median vertices, the distance-balanced property and the opportunity index of a graph. This interest has recently been addressed to graphs obtained via classical graph products like the Cartesian, the direct, the strong and the lexicographic product. We extend this study to a relatively new graph product, that is, the wreath product. In this paper, we compute the total distance for the vertices of an arbitrary wreath product graph $G\wr H$ in terms of the total distances in $H$ and of some distance-based indices of $G$. We explicitly compute these indices for the star graph $S_n$, providing a closed formula for the total distances in $S_n\wr H$ when $H$ is complete or a star. As a consequence, we obtain the Wiener index of these graphs, we characterize the median and the central vertices, and finally we give an upper and a lower bound for the opportunity index of $S_n\wr S_m$ in terms of tail conditional expectations of an associated binomial distribution.

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“…In the literature these graphs were studied from various purely graph-theoretic aspects such as symmetry [13], connectivity [9,16] or complexity aspects of algorithms related to such graphs [6], to name just a few. However, it turns out that these graphs have applications in other areas, such as mathematical chemistry (see for instance [3,11,12]) and communication networks (see for instance [3]). …”

mentioning

confidence: 99%

“…In [1], distance-balanced graphs are called transmission regular. Finally, even order distance-balanced graphs possess yet another nice property, making them what are called equal opportunity graphs (see [3] for the definition).…”

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confidence: 99%

On 2-distance-balanced graphs

Frelih

Miklavič

2018

Ars Math. Contemp.

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Let n denote a positive integer. A graph Γ of diameter at least n is said to be n-distancebalanced whenever for any pair of vertices u, v of Γ at distance n, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. In this article we consider n = 2 (e.g. we consider 2-distance-balanced graphs). We show that there exist 2-distance-balanced graphs that are not 1-distance-balanced (e.g. distance-balanced). We characterize all connected 2-distance-balanced graphs that are not 2-connected. We also characterize 2-distance-balanced graphs that can be obtained as cartesian product or lexicographic product of two graphs.Keywords: n-distance-balanced graph, cartesian product, lexicographic product.Math. Subj. Class.: 05C12, 05C76 Introductory remarksA graph Γ is distance-balanced if for each pair u, v of adjacent vertices of Γ the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Although these graphs are interesting from the purely graph-theoretical point of view, they also have applications in other areas of research, such as mathematical chemistry and communication networks. It is for that reason that they have been studied from various different points of view in the literature.Distance-balanced graphs were first studied by Handa [9] in 1999. The name distancebalanced, however, was introduced nine years later by Jerebic, Klavžar and Rall [12]. The * We thank the two anonymous referees for many useful comments and suggestions that have greatly improved our initial manuscript, especially for pointing out the mistake in Theorem 5.4.

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Equal opportunity networks, distance-balanced graphs, and Wiener game

Cited by 37 publications

References 16 publications

Cut Method: Update on Recent Developments and Equivalence of Independent Approaches

Cut Method: Update on Recent Developments and Equivalence of Independent Approaches

Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

On 2-distance-balanced graphs

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