Growing fullerenes from seed: Growth transformations of fullerene polyhedra

Brinkmann, Gunnar; Franceus, Dieter; Fowler, Patrick W.; Graver, Jack E.

doi:10.1016/j.cplett.2006.07.040

Cited by 9 publications

(13 citation statements)

References 16 publications

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“…The incomplete lists of fullerenes were also used in another article of Brinkmann et al [4]. All reducibility results given there remain true, except for for 186 and 190 vertices and 2 too small for 194 vertices.…”

Section: Testing and Resultsmentioning

confidence: 99%

“…For energetical reasons, patch replacement as a chemical mechanism to grow fullerenes would need very small patches. Brinkmann et al [4] investigated replacements of small patches and introduced two infinite families of operations. These operations can generate all fullerenes up to at least 200 vertices, but -as already shown in their paper -fail in general.…”

Section: Introductionmentioning

confidence: 99%

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The Generation of Fullerenes

Brinkmann

Goedgebeur

McKay

2012

J. Chem. Inf. Model.

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We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes - fullgen - and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are Hamiltonian and verify that the smallest counterexample to the spiral conjecture has 380 vertices.

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Section: Testing and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Generation of Fullerenes

Brinkmann

Goedgebeur

McKay

2012

J. Chem. Inf. Model.

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“…Later, a new method based on growing fullerenes from seed (see [68]) appeared. The idea was to build any fullerene from a finite set of seeds by growth operations substituting a new patch for a patch on a fullerene with a lesser number of faces and the same boundary.…”

Section: Discussionmentioning

confidence: 99%

“…Our methods and results developed jointly with V.M. Buchstaber [4,13,23,26,48] combine both methods and results on the construction of fullerenes by growth operations in [49,68,69] and on the construction of c5-connected graphs in [4,8,9,16]. Namely, (5, 0)-nanotubes are connected sums of copies of the 5-barrel.…”

Section: Discussionmentioning

confidence: 99%

Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face

Erokhovets

2018

Symmetry

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A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3-and 4-gons and may have any prescribed numbers of k-gons, k ≥ 7. Any simple polytope with only 5-, 6-and at most one 7-gon is Pogorelov. For any other prescribed numbers of k-gons, k ≥ 7, we give an explicit construction of a Pogorelov and a non-Pogorelov polytope. Any Pogorelov polytope different from k-barrels (also known as Löbel polytopes, whose graphs are biladders on 2k vertices) can be constructed from the 5-or the 6-barrel by cutting off pairs of adjacent edges and connected sums with the 5-barrel along a 5-gon with the intermediate polytopes being Pogorelov. For fullerenes, there is a stronger result. Any fullerene different from the 5-barrel and the (5, 0)-nanotubes can be constructed by only cutting off adjacent edges from the 6-barrel with all the intermediate polytopes having 5-, 6-and at most one additional 7-gon adjacent to a 5-gon. This result cannot be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the 5-barrel and 3 new operations, which are compositions of cutting off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons.

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“…Thus, the important problem is to define and study different structures and operations on the set of all combinatorial types of fullerenes. The well-known problem [9,11,12,13,14,15,16] is to find a simple set of operations sufficient to construct arbitrary fullerene from the dodecahedron C 20 .…”

Section: Introductionmentioning

confidence: 99%

Finite sets of operations sufficient to construct any fullerene from C 20

Buchstaber

Erokhovets

2016

Struct Chem

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We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene C 20 ). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fullerenes is needed to construct any fullerene from the dodecahedron. We prove that if we allow a polytope to contain one exceptional facet, which is a quadrangle or a heptagon, then a finite set of growth operation is sufficient. We analyze pairs of objects: a finite set of operations, and a family of acceptable polytopes containing fullerenes such that any polytope of the family can be obtained from the dodecahedron by a sequence of operations from the corresponding set. We describe explicitly three such pairs. First two pairs contain seven operations, and the last -eleven operations. Each of these operations corresponds to a finite set of growth operations and is a composition of edge-and two edges-truncations.

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Growing fullerenes from seed: Growth transformations of fullerene polyhedra

Cited by 9 publications

References 16 publications

The Generation of Fullerenes

The Generation of Fullerenes

Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face

Finite sets of operations sufficient to construct any fullerene from C 20

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