## Extremal fullerene graphs with the maximum Clar number (2009)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Ye09extremalfullerene,

author = {Dong Ye and Heping Zhang},

title = {Extremal fullerene graphs with the maximum Clar number},

year = {2009}

}

### OpenURL

### Abstract

A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar number is no more than ⌊n−12 6 ⌋. Many fullerenes with experimental evidence attain the upper bound, for instance, C60 and C70. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal n−12 6. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including C60, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.

### Citations

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(Show Context)
Citation Context ...all vertices and edges incident with hexagons in H. A maximum sextet pattern is also called a Clar formula. The cardinality of a Clar formula is the Clar number of F, denoted by c(F). In Clar’s model =-=[4]-=-, a Clar formula is designated by depicting circles within their hexagons (see Figure 1). Figure 1: A Clar formula of C60. The Clar number is originally defined for benzenoid systems based on the Clar... |

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Citation Context ...agonal faces. Fullerene graphs correspond to the fullerene molecule frames in chemistry. Let Fn be a fullerene graph with n vertices. It is well known that Fn exists for any even n ≥ 20 except n = 22 =-=[2, 6]-=-. For small n, a constructive enumeration of fullerene isomers with n vertices was given [2]. For example, there are 1812 distinct fullerene graphs with 60 vertices including the famous C60 synthesize... |

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Citation Context ...exagons (see Figure 1). Figure 1: A Clar formula of C60. The Clar number is originally defined for benzenoid systems based on the Clar sextet theory [4] and related to Randić conjugated circuit model =-=[17]-=-. It is effective to measure the molecule stability of benzenoid hydrocarbons. For two isomeric benzenoid hydrocarbons, the one with larger Clar number is more stable. Clar numbers of benzenoid hydroc... |

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Citation Context ...ponents such that each of them contains at least one cycle. The cyclic edge-connectivity of graph G, denoted by cλ(G), is the maximum integer k such that G is cyclically k-edge-connected. Lemma 2.1. (=-=[5, 16]-=-) Let F be a fullerene graph. Then cλ(F) = 5. From now on, let F be a fullerene graph. Let C be a cycle of F. Lemma 2.1 implies that the size of C is larger than 4. The subgraph consisting of C togeth... |

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Citation Context ...ph with n vertices is no more than ⌊n−12 ⌋, for which equality holds 6 for infinitely many fullerene graphs, including C60 and C70. We would like to mention here that a recent paper of Kardoˇs et al. =-=[12]-=- obtained a exponentially bound of perfect matching 2numbers of fullerene graphs. In fact, they applied Four-Color Theorem to show that a fullerene graph with n ≥ 380 vertices has a sextet pattern wi... |

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Citation Context ...tigated and computed in many papers [10, 11, 13, 19, 20, 21, 22]. Hansen and Zheng [11] introduced an integer linear program to compute the Clar number of benzenoid hydrocarbons. Abeledo and Atkinson =-=[1]-=- showed that relaxing the integer-restrictions in such a program always yields an integral solution. Up to now there has been no an effective method to compute Clar numbers of fullerene graphs. The Cl... |

4 |
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Citation Context ...f disjoint faces including all vertices, an extension of a fully-Clar structure. Some relationships among the Clar number, the maximum face independent number and Fries number are presented by Graver =-=[9]-=-. A lower bound for the Clar numbers of leapfrog fullerenes with icosahedral symmetry was also given in [9]. The same authors of this paper [23] showed that the Clar number of a fullerene graph with n... |

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Citation Context ...zenoid hydrocarbons. For two isomeric benzenoid hydrocarbons, the one with larger Clar number is more stable. Clar numbers of benzenoid hydrocarbons have been investigated and computed in many papers =-=[10, 11, 13, 19, 20, 21, 22]-=-. Hansen and Zheng [11] introduced an integer linear program to compute the Clar number of benzenoid hydrocarbons. Abeledo and Atkinson [1] showed that relaxing the integer-restrictions in such a prog... |

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Citation Context ...zenoid hydrocarbons. For two isomeric benzenoid hydrocarbons, the one with larger Clar number is more stable. Clar numbers of benzenoid hydrocarbons have been investigated and computed in many papers =-=[10, 11, 13, 19, 20, 21, 22]-=-. Hansen and Zheng [11] introduced an integer linear program to compute the Clar number of benzenoid hydrocarbons. Abeledo and Atkinson [1] showed that relaxing the integer-restrictions in such a prog... |

3 |
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Citation Context ...agonal faces. Fullerene graphs correspond to the fullerene molecule frames in chemistry. Let Fn be a fullerene graph with n vertices. It is well known that Fn exists for any even n ≥ 20 except n = 22 =-=[2, 6]-=-. For small n, a constructive enumeration of fullerene isomers with n vertices was given [2]. For example, there are 1812 distinct fullerene graphs with 60 vertices including the famous C60 synthesize... |

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Citation Context ...voids double bonds in pentagons and has the possibly maximal number of conjugated hexagons (n/3). Fullerene graphs with a Fries structure are equivalent to leapfrog fullerenes or Clar type fullerenes =-=[7, 15]-=-. The latter means that they have a set of disjoint faces including all vertices, an extension of a fully-Clar structure. Some relationships among the Clar number, the maximum face independent number ... |

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Citation Context ...een no an effective method to compute Clar numbers of fullerene graphs. The Clar polynomial and sextet polynomial of C60 for counting Clar structures and sextet patterns respectively were computed in =-=[18]-=-. This implies that C60 has 5 Clar formulas and Clar number 8 [3]. In addition, C60 has a Fries structure [8], i.e. a Kekulé structure of C60 which avoids double bonds in pentagons and has the possibl... |

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Citation Context ...ependent number and Fries number are presented by Graver [9]. A lower bound for the Clar numbers of leapfrog fullerenes with icosahedral symmetry was also given in [9]. The same authors of this paper =-=[23]-=- showed that the Clar number of a fullerene graph with n vertices is no more than ⌊n−12 ⌋, for which equality holds 6 for infinitely many fullerene graphs, including C60 and C70. We would like to ment... |

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Citation Context ...aphs. The Clar polynomial and sextet polynomial of C60 for counting Clar structures and sextet patterns respectively were computed in [18]. This implies that C60 has 5 Clar formulas and Clar number 8 =-=[3]-=-. In addition, C60 has a Fries structure [8], i.e. a Kekulé structure of C60 which avoids double bonds in pentagons and has the possibly maximal number of conjugated hexagons (n/3). Fullerene graphs w... |

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Citation Context ...voids double bonds in pentagons and has the possibly maximal number of conjugated hexagons (n/3). Fullerene graphs with a Fries structure are equivalent to leapfrog fullerenes or Clar type fullerenes =-=[7, 15]-=-. The latter means that they have a set of disjoint faces including all vertices, an extension of a fully-Clar structure. Some relationships among the Clar number, the maximum face independent number ... |

2 |
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Citation Context ...ial of C60 for counting Clar structures and sextet patterns respectively were computed in [18]. This implies that C60 has 5 Clar formulas and Clar number 8 [3]. In addition, C60 has a Fries structure =-=[8]-=-, i.e. a Kekulé structure of C60 which avoids double bonds in pentagons and has the possibly maximal number of conjugated hexagons (n/3). Fullerene graphs with a Fries structure are equivalent to leap... |

2 |
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Citation Context ...zenoid hydrocarbons. For two isomeric benzenoid hydrocarbons, the one with larger Clar number is more stable. Clar numbers of benzenoid hydrocarbons have been investigated and computed in many papers =-=[10, 11, 13, 19, 20, 21, 22]-=-. Hansen and Zheng [11] introduced an integer linear program to compute the Clar number of benzenoid hydrocarbons. Abeledo and Atkinson [1] showed that relaxing the integer-restrictions in such a prog... |

1 |
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1 |
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1 |
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