Zhang–Zhang Polynomials of Ribbons

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids Rbn1,n2,m1,m2, 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.2020, 84, 143–176]. The discovered formula provides compact expressions for various topological invariants of Rbn1,n2,m1,m2: 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 Ok,m,n and oblate rectangles Orm,n.

On the Quasi-Ordering of Catacondensed Hexagonal Systems with Respective to their Clar Covering Polynomials

In this paper, we discuss the quasi-ordering of hexagonal systems with respective to the coefficients of their Clar covering polynomials (also known as Zhang-Zhang polynomials). The last six minimal catacondensed hexagonal systems and the hexagonal chains with the maximum Clar covering polynomial are determined. Furthermore, the smallest pair of incomparable catacondensed hexagonal systems is given.