Connectivity Graphs for Single Zigzag Chains and their Application for Computing ZZ Polynomials

Zhang-Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John-Sachs Theorem

Multiple zigzag chains Zm,n of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently encoded as the coefficients of a combinatorial polynomial, usually referred to as the ZZ polynomial of multiple zigzag chains Zm,n. The current study reports a novel method for determination of these ZZ polynomials based on a hypothesized extension to John–Sachs theorem, used previously to enumerate Kekulé structures of various benzenoid hydrocarbons. We show that the ZZ polynomial of the Zm,n multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size m2×m2 consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains Zkm,n, i.e., derivatives of Zm,n with a single attached polyacene chain of length k. All presented formulas are accompanied by formal proofs. The developed theoretical machinery is applied for predicting aromaticity distribution patterns in large and infinite multiple zigzag chains Zm,n and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains Zm,n.

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.

Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms

We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekuléan benzenoids that can be obtained by overlapping two parallelograms: generalized ribbons Rb, parallelograms M, vertically overlapping parallelograms MvM, horizontally overlapping parallelograms MhM, and intersecting parallelograms MxM. All formulas have the form of multiple sums over binomial coefficients. Three of the formulas are given with a proof based on the interface theory of benzenoids, while the remaining two formulas are presented as conjectures verified via extensive numerical tests. Both of the conjectured formulas have the form of a 2×2 determinant bearing close structural resemblance to analogous formulas for the number of Kekulé structures derived from the John-Sachs theory of Kekulé structures.

ZZ Polynomials for Isomers of (5,6)-Fullerenes Cn with n = 20–50

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes Cn with n = 20–50 is presented. The ZZ polynomials concisely summarize the most important topological invariants of the fullerene isomers: the number of Kekulé structures K, the Clar number Cl, the first Herndon number h1, the total number of Clar covers C, and the number of Clar structures. The presented results should be useful as benchmark data for designing algorithms and computer programs aiming at topological analysis of fullerenes and at generation of resonance structures for valence-bond quantum-chemical calculations.