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

Abstract

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.

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

“…Application of ZZDecomposer to M(10, 10) yields the desired result within a few seconds. (For details, see Figure 2) Another exemplary application of ZZDecomposer is the otherwise very tedious determination of ZZ polynomials of all the isomers of carbon fullerenes also presented in the recent volume of "Symmetry" [28]. More importantly, ZZDecomposer can be used for discovering closed-form formulas for ZZ polynomials of whole families of isostructural benzenoids.…”

Hexagonal flakes as fused parallelograms: A determinantal formula for Zhang‐Zhang polynomials of the O(2, m, n) benzenoids

“…For hexagons, the closed-form ZZ polynomial formulas are readily available when one of the indices is equal to 1, because in this case the hexagonal flake O (1, m, n) reduces to a parallelogram M(m, n). 16,27,40 Closed-form formulas are also available for a few selected fixed values of k and m and arbitrary values of n. (It should be remembered that owing to the high symmetry of O(k, m, n), the discussion below applies to any permutation of the indices k, m, and n.) For (k = 2, m = 2) and (k = 3, m = 3), the closed-form ZZ polynomial formulas were given originally by eqs 35 13) for (k = 3, m = 3)). All the known formulas can be summarized concisely as follows:…”

“…No computer program is available to date based on these concepts, but we hope that such a code—enabling one to surpass the limit of 500 carbon atoms in ZZ polynomial calculations for pericondensed benzenoids—will be made available soon. In addition to the obvious brute‐force calculations, ZZDecomposer has been used in numerous applications 15–17,26–37 to find closed‐form formulas of ZZ polynomials for various families of basic catacondensed and pericondensed benzenoids, substantially extending the total body of previously available results 4,9–13,38–43 . The rapid development of the Clar theory stimulated by these discoveries in recent years leads to many new interesting applications and connections to other branches of chemistry, graph theory, and combinatorics 7,8,18,31,41–63 …”

Zhang–Zhang Polynomials of Ribbons

“…(See, for example, Properties 1-7 in [3].) Consequently, the ZZ polynomial of an arbitrary benzenoid B or fullerene [36] can be efficiently computed using recursive decomposition algorithms [3,14,37] or determined using interface theory of benzenoids [38][39][40][41]. A useful practical tool for determination of ZZ polynomials is ZZDecomposer [10,37].…”