Mathematical aspects of fullerenes

We consider extremal values of sum-Balaban index among graphs on n vertices. We determine that the upper bound for the minimum value of the sum-Balaban index is at most 4.47934 when n goes to infinity. For small values of n we determine the extremal graphs and we observe that they are similar to dumbbell graphs, in most cases having one extra edge added to the corresponding extreme for the usual Balaban index. We show that in the class of balanced dumbbell graphs, those with clique sizes 4 √ 2 log 1 + √ 2 √ n + o( √ n) have asymptotically the smallest value of sum-Balaban index. We pose several conjectures and problems regarding this topic.

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“…where c b and c a are some values such that c b ∈ [ 1 2 , 1] and c a ∈ [2,4]. Estimating m/(m − n + 2) analogously as in (5) we get…”

Section: Claim 1 It Holds a ∈ O(n)mentioning

confidence: 99%

On the minimum value of sum-Balaban index

Knor

Kranjc²,

krekovski

et al. 2017

Applied Mathematics and Computation

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“…Therefore, it would be interesting to know, which fullerenes are ndistance-balanced at least for some values of n (for example, for n ∈ {1, 2, D}, where D is the diameter of a fullerene in question). For more on fullerenes see [2,5,18].…”

Section: -Distance-balanced Lexicographic Productmentioning

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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“…They are considered as thermodynamic stable fullerene compounds. Description of mathematical properties of fullerene graphs can be found in [4,7,11,19,20,36].…”

Section: Introductionmentioning

confidence: 99%

On the Wiener Complexity and the Wiener Index of Fullerene Graphs

Dobrynin

Vesnin

2019

Mathematics

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Fullerenes are molecules in the form of cage-like polyhedra, consisting solely of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex v of a graph is the sum of distances from v to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order n ≤ 216 are presented. Structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed and formulas for the Wiener index of several families of graphs are obtained.

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Mathematical aspects of fullerenes

Cited by 47 publications

References 75 publications

On the minimum value of sum-Balaban index

On the minimum value of sum-Balaban index

On 2-distance-balanced graphs

On the Wiener Complexity and the Wiener Index of Fullerene Graphs

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