Theory of polypentagons

Cyvin, S. J.; Cyvin, Bjørg N.; Brünvoll, J.; Brendsdal, E.; Zhang, Fuji; Xiaofeng, Guo; Tošić, R.

doi:10.1021/ci00013a027

Cited by 8 publications

(16 citation statements)

References 6 publications

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“…r-gons can be added around the perimeter, such that they will form a simple circuit. (They were found in [8]; such (6, 3)-polycycles are useful in Organic Chemistry.) All circumscribed (3, 5)-polycycles are the kernels of polycycles a i , 1 ≤ i ≤ 5, and c i , 1 ≤ i ≤ 4 of Figure 6.…”

Section: Lemmamentioning

confidence: 96%

“…Polyhexes are used widely (see, for example, [16], [3]) in Organic Chemistry: they represent completely condensed PAH (polycyclic aromatic hydrocarbons) C n H m with n vertices (atoms of the carbon C), including m vertices of degree 2, where atoms of the hydrogen H are adjoined. All 39 proper (5, 3)-polycycles were found in [8] in chemical context, but already in [18] were given all 3, 6, 9, 39, 263 proper spheric (r, q)-polycycles for (r, q) = (3, 3), (4, 3), (3,4), (5,3), (3,5), respectively.…”

Section: Definition and Examplesmentioning

confidence: 99%

“…Moreover, the (5, 3)-polycycles, which are reciprocal to any such extremal one, turn out to be also extremal. [8] asked about n(x) for x ≥ 12; this Section answer this question for any x and for all spheric (p, q). First, consider three trivial cases of (r, q).…”

Section: Maximal Number Of Interior Pointsmentioning

confidence: 99%

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Clusters of cycles

Deza¹,

Штогрин²

2002

Journal of Geometry and Physics

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A cluster of cycles (or (r, q)-polycycle) is a simple planar 2-connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r, q)-polycyclic realization P (G) on the plane. An (r, q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by p r ) are combinatorial r-gons, (iii) all vertices, edges and interior faces form a cell-complex.An example of (r, q)-polycycle is the skeleton of (r q ), i.e. of the q-valent partition of the sphere, Euclidean plane or hyperbolic plane by regular r-gons. Call spheric pairs (r, q) = (3, 3), (4, 3), (3, 4), (5, 3), (3, 5). Only for those five pairs, P ((r q )) is (r q ) without exterior face; otherwise, P ((r q )) = (r q ).Here we give a compact survey of results on (r, q)-polycycles. We start with the following general results for any (r, q)-polycycle G: (i) P (G) is unique, except of (easy) case when G is the skeleton of one of 5 Platonic polyhedra; (ii) P (G) admits a cell-homomorphism f into (r q ); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented.Call a polycycle proper if it is a partial subgraph of (r q ) and a helicene, otherwise. In [18] all proper spheric polycycles are given. An (r, q)-helicene exists if and only if p r > (q − 2)(r − 1) and (r, q) = (3, 3). We list the (4, 3)-, (3, 4)-helicenes and the number of (5, 3)-, (3, 5)-helicenes for first interesting p r . Any outerplanar (r, q)-polycycle G is a proper (r, 2q − 2)-polycycle and its projection f (P (G)) into (r 2q−2 ) is convex. Any outerplanar (3, q)-polycycle G is a proper (3, q + 2)-polycycle.

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Section: Lemmamentioning

confidence: 96%

Section: Definition and Examplesmentioning

confidence: 99%

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Clusters of cycles

Deza¹,

Штогрин²

2002

Journal of Geometry and Physics

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“…It is easy to check that Suppose that a is irreducible. Then for sequences g ij ( (a) to satisfy (2), it is necessary that For the sequences g ij ( (a) to be p-gonal, their leftness must satisfy (4). Clearly, if both sequences g ij ( (a) are p-gonal, then the sequence a is also p-gonal.…”

Section: Lemma 7 If a P-gonal Sequence Has No Waist Then The Operatmentioning

confidence: 99%

Allowed Boundary Sequences for Fused Polycyclic Patches and Related Algorithmic Problems

Deza

Fowler

Grishukhin

2001

J. Chem. Inf. Comput. Sci.

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We consider sequences that encode boundary circuits of fused polycycles made up of polygonal faces with p sides, p < or = 6. We give a constructive algorithm for recognizing such sequences when p = 5 or 6. A simpler algorithm is given for planar hexagonal sequences. Hexagonal and pentagonal sequences of length at most 8 are tabulated, the former corresponding to planar benzenoid hydrocarbons CxHy with y up to 14.

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“…A catapoly- q -gon is a catacondensed polygonal system consisting of exclusively q -gons. The algebraic solution for the numbers of nonisomorphic unbranched catapolypentagons ( q = 5) has been given , in the same style as the classical solution for unbranched catafusenes ( q = 6)…”

Section: Introductionmentioning

confidence: 99%

Enumeration of Unbranched Catacondensed Polygonal Systems: General Solution for Two Kinds of Polygons

Cyvín

Brunvoll

Cyvin

1997

J. Chem. Inf. Comput. Sci.

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An α-p-catapoly-q-gon is a catacondensed polygonal system with α p-gons and r − α q-gons, which represents a polycyclic conjugated hydrocarbon. Here r is the total number of polygons or rings. A complete mathematical solution for the numbers of nonisomorphic unbranched α-p-catapoly-q-gons is reported. It is expressed by complicated explicit formulas in r, α, p, and q, and it represents a generalization of several special cases studied previously. The solution was achieved by means of certain generalized triangular matrices with interesting mathematical properties.

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Theory of polypentagons

Cited by 8 publications

References 6 publications

Clusters of cycles

Clusters of cycles

Allowed Boundary Sequences for Fused Polycyclic Patches and Related Algorithmic Problems

Enumeration of Unbranched Catacondensed Polygonal Systems: General Solution for Two Kinds of Polygons

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