Zhang-Zhang Polynomials of Ribbons

TL;DR

This paper derives a closed-form formula for the Zhang-Zhang polynomial of ribbons, enabling efficient computation of various topological invariants of these benzenoid structures.

Contribution

It provides the first explicit formula for the ZZ polynomial of ribbons, advancing the understanding of their topological properties.

Findings

Closed-form formula for ZZ polynomial of ribbons

Efficient calculation of Kekulé structures and Clar covers

Determination of Clar number and Clar structures

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 $Or\left(m,n\right)$.