Cyclic 7-edge-cuts in fullerene graphs

Kardoš, František; Krnc, Matjaž; Lužar, Borut; Škrekovski, Riste

doi:10.1007/s10910-009-9599-0

Cited by 16 publications

(10 citation statements)

References 9 publications

(9 reference statements)

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“…It is based on the recursive generation of simple partitions of a disk by three types of operations and a gluing of these disks into surfaces of polytopes with the possible addition of belts of 6-gons between them. This method is close to the method of F. Kardoš, M. Krnc, B. Lužar and R. Škrekovski [52,73] for the generation of cyclic k-edge cuts in fullerene graphs.…”

Section: Discussionmentioning

confidence: 73%

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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Section: Discussionmentioning

confidence: 73%

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

Erokhovets

2018

Symmetry

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“…Kardoš andŠkrekovski [7] characterised the cyclic 5-and 6-edge-cuts in fullerene graphs; the cyclic 5-edge-cuts were characterised independently by Kutnar and Marušič [9]. Furthermore, Kardoš et al [6] characterised the cyclic 7-edge-cuts in fullerene graphs. The characterisations together imply the following three lemmas (see Figures 1-3 for illustrations).…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

On the 2-Resonance of Fullerenes

Kaiser¹,

Stehlı́k²,

Škrekovski³

2011

SIAM J. Discrete Math.

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“…Cube (8,12,6) Dodecahedron (20,30,12) Tetrahedron (4,6,4) Octahedron (6,12,8) Icosahedron (12,30,20) Fig. 3.…”

Section: Platonic Solidsmentioning

confidence: 99%

“…In [29] cyclic 6-edge cuts were classified. In [30] degenerated cyclic 7-edge cuts and fullerenes with nondegenerated cyclic 7-edge cuts were classified, where a cyclic k-edge cut is called degenerated, if one of the connected components has less than 6 pentagonal facets, otherwise it is called non-degenerated.…”

Section: Cyclic K-edge Cutsmentioning

confidence: 99%

Fullerenes, Polytopes and Toric Topology

Buchstaber¹,

Erokhovets²

2017

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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The lectures are devoted to a remarkable class of 3-dimensional polytopes, which are mathematical models of the important object of quantum physics, quantum chemistry and nanotechnology -fullerenes. The main goal is to show how results of toric topology help to build combinatorial invariants of fullerenes. Main notions are introduced during the lectures. The lecture notes are addressed to a wide audience.

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Cyclic 7-edge-cuts in fullerene graphs

Cited by 16 publications

References 9 publications

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

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

On the 2-Resonance of Fullerenes

Fullerenes, Polytopes and Toric Topology

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