The Clar and Fries structures of a fullerene I

Graver, Jack E.; Hartung, Elizabeth J.

doi:10.1016/j.dam.2016.07.016

Cited by 8 publications

(8 citation statements)

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“…In a perfect matching (Kekulé structure) of G, a benzenoid hexagon is a hexagon that contains three double bonds (matched edges). The Fries number of a molecular graph, F (G), is the maximum number of benzenoid hexagons, taken over all perfect matchings of G. The Clar number, C(G), is the maximum number of independent benzenoid hexagons, taken over all perfect matchings of G. Although Fries [30] and Clar [14] concepts originated as purely qualitative indications of thermodynamic and kinetic stability and reactivity of benzenoid hydrocarbons, and are still mainly applied to these molecules [11,12,26,34,36,63], evaluation of Fries and Clar numbers has also been applied to carbon nanostructures of many other types [44], and in particular to the fullerenes [3,31,35,40,41,57,62,64,65].…”

Section: Introductionmentioning

confidence: 99%

Clar and Fries structures for fullerenes

Fowler

Myrvold

Vandenberg

et al. 2022

Art Discrete Appl. Math.

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Fries and Clar numbers are qualitative indicators of stability in conjugated π systems. For a given Kekulé structure, call any hexagon that contains three double bonds benzenoid. The Fries number is the maximum number of benzenoid hexagons, whereas the Clar number is the maximum number of independent benzenoid hexagons, in each case taken over all Kekulé structures. A Kekulé structure that realises the Fries (Clar) number is a Fries (Clar) structure. For benzenoids, it is not known whether every Fries structure is also a Clar structure. For fullerenes C n , it is known that some Clar structures in large examples correspond to no Fries structure. We show that Fries structures that are not Clar occur early: examples where some Fries structure is not Clar start at C 34 , and examples where no Fries structure is Clar start at C 48 . Hence, it is unsafe to use fullerene Fries structures as routes to Clar number. However, Fries structures often describe the neutral fullerene better than a Clar structure, e.g. in rationalising bond lengths in the experimental isomer of C 60 . Conversely, an extension of Clar sextet theory suggests the notion of anionic Clar number for fullerene anions, where both pentagons and hexagons may support sextets.

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Section: Introductionmentioning

confidence: 99%

Clar and Fries structures for fullerenes

Fowler

Myrvold

Vandenberg

et al. 2022

Art Discrete Appl. Math.

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“…Let C be a maximal independent set of benzene rings in a Kekulé structure K, and let A be the set of edges of K that do not lie on any benzene ring. Then each vertex of G is incident with exactly one element from C ∪ A, so (C, A) forms a vertex covering of G. Such a face-edge vertex covering of a fullerene is called a Clar structure, and was introduced in [10] and further described in [9], [12]. The Clar number of a fullerene on v vertices is given by |C| = v 6 − |A| 3 , and therefore, finding a Kekulé structure that minimizes the number of edges in A is equivalent to finding the Clar number of a fullerene.…”

Section: Introductionmentioning

confidence: 99%

“…Properties of Clar chains and details of when two pentagons can be paired by a Clar chain are given in [9]. The number of edges of A in a Clar chain between two pentagons is linearly related to the distance between those pentagons.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, in a Clar structure that gives the Clar number, the pentagons paired by Clar chains are typically "close" to one another. Given a Clar structure (C, A) for a fullerene, the fullerene admits a partial face 3coloring in which the improperly colored faces are exactly those that have a bounding edge in A [9]. We call the faces with bounding edges in A the augmenting faces of the Clar chains.…”

Section: Introductionmentioning

confidence: 99%

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The Clar numbers of capped nanotubes

Graver¹,

Hartung²

2021

match

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A nanotube is a closed carbon molecule in the shape of a capped cylinder. The Clar number of a carbon molecule is the maximum number of independent benzene rings over all possible Kekulé structures. We prove that at most two Clar chains are required on nanotube cylinders, giving lower bounds on the Clar number of nanotubes. In other words, a fully conjugated π-system running along the nanotube's cylinder will be broken by at most two fracture lines. In [8], this double bond structure of capped nanotubes was described, but without a detailed mathematical proof that at most two Clar chains are required across a nanotube cylinder. We use this result to settle a conjecture in the case of long nanotubes. Carr, Wang and Ye proved that the Clar number for fullerenes on v vertices is bounded below by v−380 61 and further conjectured that the sharp lower bound is v−20 10 [4]. We prove that this sharp lower bound holds for nanotubes of sufficient length. We also give a formula for the maximum number of vertices in the cap of a nanotube with chiral indices (n, m), where the caps are defined as in [3].

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“…The basic properties of chains have been worked out in [2] and we summarize these basic properties here. Assume that two neighboring pentagons can be joined by a chain.…”

Section: Chains and The Clar And Fries Numbersmentioning

confidence: 99%

Similarity and a Duality for Fullerenes

Edmond

Graver

2015

Symmetry

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Fullerenes are molecules of carbon that are modeled by trivalent plane graphs with only pentagonal and hexagonal faces. Scaling up a fullerene gives a notion of similarity, and fullerenes are partitioned into similarity classes. In this expository article, we illustrate how the values of two important fullerene parameters can be deduced for all fullerenes in a similarity class by computing the values of these parameters for just the three smallest representatives of that class. In addition, it turns out that there is a natural duality theory for similarity classes of fullerenes based on one of the most important fullerene construction techniques: leapfrog construction. The literature on fullerenes is very extensive, and since this is a general interest journal, we will summarize and illustrate the fundamental results that we will need to develop similarity and this duality.

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The Clar and Fries structures of a fullerene I

Cited by 8 publications

References 3 publications

Clar and Fries structures for fullerenes

Clar and Fries structures for fullerenes

The Clar numbers of capped nanotubes

Similarity and a Duality for Fullerenes

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