### Abstract

A Kekulé structure for a benzenoid or a fullerene Γ is a set of edges K such that each vertex of Γ is incident with exactly one edge in K, i.e. a perfect matching. All fullerenes admit a Kekulé structure; however, this is not true for benzenoids. In this paper, we develop methods for deciding whether or not a given benzenoid admits a Kekulé structure by constructing Kekulé structures that have a high density of benzene rings. The benzene rings of the Kekulé structure K are the faces in Γ that have exactly three edges in K. The Fries number of Γ is the maximum number of benzene rings over all possible Kekulé structures for Γ and the set of benzene rings giving the Fries number is called a Fries set. The Clar number is the maximum number of independent benzene rings over all possible Kekulé structures for Γ and the set of benzene rings giving the Clar number is called a Clar set. Our method of constructing Kekulé structures for benzenoids generally gives good estimates for the Clar and Fries numbers, often the exact values.

Original language | English (US) |
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Pages (from-to) | 977-989 |

Number of pages | 13 |

Journal | Journal of Mathematical Chemistry |

Volume | 52 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2014 |

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### Keywords

- Benzene faces
- Benzene rings
- Benzenoids
- Clar number
- Conjugated 6-circuits
- Fries number
- Fullerenes
- Graphene patches
- Kekulé structure

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Chemistry*,

*52*(3), 977-989. https://doi.org/10.1007/s10910-013-0304-y