A survey of the automorphism groups of some fulleroids

Khaksari, A.; Hakimi-Nezhaad, Mardjan; Ori, O.; Ghorbani, Modjtaba

doi:10.1080/1536383x.2017.1402007

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…In [4], it was proven that it is possible to consider at intermediate steps only simple polytopes with 5-, 6-and one 7-gonal face adjacent to a 5-gon. It seems that the appearance of a 7-gon was not extensively studied by mathematicians (although it was considered in [30], and other mathematical generalizations of fullerenes were introduced; see [32,[77][78][79]), but there are many works in physics and chemistry of fullerenes, which state that this is natural in the formation of fullerenes (see, for example, [80][81][82]).…”

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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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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“…For more details about the GP number of nanostructures, linear polymers, some classes of fullerenes and fullerene-like molecules, see [8,[10][11][12][13][14]19,20,[22][23][24][25][26][27][28][29][30][31]. However, the following result is crucial in the whole of this paper.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

et al. 2020

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The Graovac–Pisanski (GP) index of a graph is a modified version of the Wiener index based on the distance between each vertex x and its image α(x), where α is an automorphism of graph. The aim of this paper is to compute the automorphism group of some classes of cubic polyhedral graphs and then we determine their Wiener index. In addition, we investigate the GP-index of these classes of graphs.

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A survey of the automorphism groups of some fulleroids

Cited by 2 publications

References 18 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

Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

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