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arXiv · 2010.03895

Zhang-Zhang Polynomials of Ribbons

Bing-Hau He·Chien-Pin Chou·Johanna Langner·Henryk A. Witek
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Abstract

We report a closed-form formula for the Zhang-Zhang polynomial (aka ZZ polynomial or Clar covering polynomial) of an important class of elementary pericondensed benzenoids $Rb\left(n_{1},n_{2},m_{1},m_{2}\right)$ usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem. 84, 143--176 (2020)]. The discovered formula provides compact expressions for various topological invariants of $Rb\left(n_{1},n_{2},m_{1},m_{2}\right)$: the number of Kekul\'e structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes $O\left(k,m,n\right)$ and oblate rectangles $Or\left(m,n\right)$.

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Bing-Hau He, Chien-Pin Chou, Johanna Langner, Henryk A. Witek. 2020-10-08. Zhang-Zhang Polynomials of Ribbons. https://arxiv.org/abs/2010.03895

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