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Clar structures vs Fries structures in hexagonal systems

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  • Zhai, Shaohui
  • Alrowaili, Dalal
  • Ye, Dong

Abstract

A hexagonal system H is a 2-connected bipartite plane graph such that all inner faces are hexagons, which is often used to model the structure of a benzenoid hydrocarbon or graphen. A perfect matching of H is a set of disjoint edges which covers all vertices of H. A resonant set S of H is a set of hexagons in which every hexagon is M-alternating for some perfect matching M. The Fries number of H is the size of a maximum resonant set and the Clar number of H is the size of a maximum independent resonant set (i.e. all hexagons are disjoint). A pair of hexagonal systems with the same number of vertices is called a contra-pair if one has a larger Clar number but the other has a larger Fries number. In this paper, we investigates the Fries number and Clar number for hexagonal systems, and show that a catacondensed hexagonal system has a maximum resonant set containing a maximum independent resonant set, which is conjectured for all hexagonal systems. Further, our computation results demonstrate that there exist many contra-pairs.

Suggested Citation

  • Zhai, Shaohui & Alrowaili, Dalal & Ye, Dong, 2018. "Clar structures vs Fries structures in hexagonal systems," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 384-394.
    • Handle: RePEc:eee:apmaco:v:329:y:2018:i:c:p:384-394
    DOI: 10.1016/j.amc.2018.02.014
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    References listed on IDEAS

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    1. Zhao, Shuang & Zhang, Heping, 2016. "Forcing polynomials of benzenoid parallelogram and its related benzenoids," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 209-218.
    2. Li, Wei & Qin, Zhongmei & Zhang, Heping, 2016. "Extremal hexagonal chains with respect to the coefficients sum of the permanental polynomial," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 30-38.
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    Cited by:

    1. Shi, Lingjuan & Zhang, Heping, 2019. "Counting Clar structures of (4, 6)-fullerenes," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 559-574.

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