Zhang-Zhang Polynomials of Ribbons

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 Kekulé 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 $Or\left(m,n\right)$.

16 pages, 12 figures