Boundary uniqueness of fusenes

Guo, Xiaofeng; Hansen, Pierre; Zheng, Maolin

doi:10.1016/s0166-218x(01)00180-9

Cited by 25 publications

(32 citation statements)

References 11 publications

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“…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]. By our definition, faces may have a range of different degrees, but m = 3.…”

Section: Figure 18: a Patchmentioning

confidence: 56%

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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Section: Figure 18: a Patchmentioning

confidence: 56%

Coronoids, patches and generalised altans

2016

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“…A result by Guo, Hansen and Zheng [21] shows that even this restricted problem is not as easy as was first expected: in Fig. 1 their example is shown which shows that also in this case, different patches may exist with the same boundary code.…”

mentioning

confidence: 78%

“…It is a hexagonal patch if all inner faces have length 6. Hexagonal patches are also known as fusenes [21], hexagonal systems [12], polyhexes [20] and (6, 3)-polycycles [14] in the literature. A sequence x 0 , .…”

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confidence: 99%

“…Our drawing of this graph is taken from [10]. (In [21] it is also shown that although multiple solutions [21] and Graver [20] give conditions for when solutions are unique, if they exist. Deza et al [12] give an algorithm for deciding whether at least one solution exists.…”

mentioning

confidence: 99%

“…The problem HEXAGONAL PATCH is also interesting by itself, since hexagonal patches model benzenoid hydrocarbons and graphite fragments (see e.g. [21] and the references therein). In addition, our results also have implications outside of computational chemistry, since our algorithm is based on the following idea: with a few intermediate steps, we transform the problem HEXAGONAL PATCH to the problem of counting maximum independent sets in circle graphs.…”

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confidence: 99%

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Counting Hexagonal Patches and Independent Sets in Circle Graphs

Bonsma

Breuer

2011

Algorithmica

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A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the enumeration of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm for this problem. It is also shown how to subsequently generate hexagonal patches.

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The topology of fullerenes

Schwerdtfeger

Wirz

Avery

2014

WIREs Comput Mol Sci

184

168

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Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207Conflict of interest: The authors have declared no conflicts of interest for this article.For further resources related to this article, please visit the WIREs website.

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Boundary uniqueness of fusenes

Cited by 25 publications

References 11 publications

Coronoids, patches and generalised altans

Coronoids, patches and generalised altans

Counting Hexagonal Patches and Independent Sets in Circle Graphs

The topology of fullerenes

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