Fullerene patches II

Graver, Jack E.; Graves, Christina; Graves, Stephen J.

doi:10.26493/1855-3974.391.a0d

Cited by 8 publications

(13 citation statements)

References 10 publications

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“…The stand of the mathematical understanding of graphene is comparably less developed. All available results are extremely recent and concern the modeling of transport properties of electrons in graphene sheets [3,6,13,16,25,34,35], homogenization [8,33], atomistic-to-continuum passage for nanotubes [14], geometry of monolayers under Gaussian perturbations [11], external charges [27] or magnetic fields [10], combinatorial description of graphene patches [22], and numerical simulation of dynamics via nonlocal elasticity theory [44]. Remarkably, the determination of the equilibrium shapes and the Wulff shapes of graphene samples and graphene nanostructures is still a challenging problem [1,5,19].…”

Section: Introductionmentioning

confidence: 99%

Wulff shape emergence in graphene

Davoli

Piovano

Stefanelli

2016

Math. Models Methods Appl. Sci.

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Graphene samples are identified as minimizers of configurational energies featuring both two- and three-body atomic-interaction terms. This variational viewpoint allows for a detailed description of ground-state geometries as connected subsets of a regular hexagonal lattice. We investigate here how these geometries evolve as the number [Formula: see text] of carbon atoms in the graphene sample increases. By means of an equivalent characterization of minimality via a discrete isoperimetric inequality, we prove that ground states converge to the ideal hexagonal Wulff shape as [Formula: see text]. Precisely, ground states deviate from such hexagonal Wulff shape by at most [Formula: see text] atoms, where both the constant [Formula: see text] and the rate [Formula: see text] are sharp.

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Section: Introductionmentioning

confidence: 99%

Wulff shape emergence in graphene

Davoli

Piovano

Stefanelli

2016

Math. Models Methods Appl. Sci.

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“…We will present a mathematical formalisation which is based on the treatment of coronoids and benzenoids in the previous section of this paper. As we will see later, our definition of a fullerene patch is compatible with Graver's definition [12,13]. There is also a notion of a (m, k)-patch which received a lot of attention in the past years [2,3,11,18].…”

Section: Figure 18: a Patchmentioning

confidence: 58%

“…Let G be the subdivided cube in Figure 23. Let C 1 = (1, 2, 3, 4, 7, 6, 5), 1,5,9,10,14,8) and C 3 = (10, 11,12,13,16,15,14). Then (G; C 1 , C 2 , C 3 ) is an admissible structure.…”

Section: Altans Generalised Altans and Iterated Altansmentioning

confidence: 99%

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Coronoids, patches and generalised altans

2016

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In this paper we revisit coronoids, in particular multiple coronoids. We consider a mathematical formalisation of the theory of coronoid hydrocarbons that is solely based on incidence between hexagons of the infinite hexagonal grid in the plane. In parallel, we consider perforated patches, which generalise coronoids: in addition to hexagons, other polygons may also be present. Just as coronoids may be considered as benzenoids with holes, perforated patches are patches with holes. Both cases, coronoids and perforated patches, admit a generalisation of the altan operation that can be performed at several holes simultaneously. A formula for the number of Kekulé structures of a generalised altan can be derived easily if the number of Kekulé structures is known for the original graph. Pauling Bond Orders for generalised altans are also easy to derive from those of the original graph.

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“…In the final section we give closed expressions for the modified Wiener index of two infinite families of nanocones. These chemical graphs belong to the family of the so-called fullerene patches [4,5].…”

Section: Introductionmentioning

confidence: 99%

Modified Wiener index via canonical metric representation, and some fullerene patches

Ghorbani

Klavžar

2015

Ars Math. Contemp.

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A variation of the classical Wiener index, the modified Wiener index, that was introduced in 1991 by Graovac and Pisanski, takes into account the symmetries of a given graph. In this paper it is proved that the computation of the modified Wiener index of a graph G can be reduced to the computation of the Wiener indices of the appropriately weighted quotient graphs of the canonical metric representation of G. The computation simplifies in the case when G is a partial cube. The method developed is applied to two infinite families of fullerene patches.

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Fullerene patches II

Abstract: In this paper, we show that fullerene patches with nice boundaries containing between 1 and 5 pentagons fall into several equivalence classes; furthermore, any two fullerene patches in the same class can be transformed into the same minimal configuration using combinatorial alterations.

Cited by 8 publications

References 10 publications

Wulff shape emergence in graphene

Wulff shape emergence in graphene

Coronoids, patches and generalised altans

Modified Wiener index via canonical metric representation, and some fullerene patches

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