The Clar covering polynomial of hexagonal systems I

AbstractIn this paper the Clar covering polynomial of a hexagonal system is introduced. In fact it is a kind of F polynomial [4] of a graph, and can be calculated by recurrence relations. We show that the number of aromatic sextets (in a Clar formula), the number of Clar formulas, the number of Kekulé structures and the first Herndon number for any Kekuléan hexagonal system can be easily obtained by its Clar covering polynomial. In addition, we give some theorems to calculate the Clar covering polynomial of a hexagonal system. As examples we finally derive the explicit expressions of the Clar covering polynomials for some small hexagonal systems and several types of catacondensed hexagonal systems. A relation between the resonance energy and the Clar covering polynomial of a hexagonal system is considered in the next paper

ZZPolyCalc: An open-source code with fragment caching for determination of Zhang-Zhang polynomials of carbon nanostructures

Determination of topological invariants of graphene flakes, nanotubes, and fullerenes constitutes a challenging task due to its time-intensive nature and exponential scaling. The invariants can be organized in a form of a combinatorial polynomial commonly known as the Zhang-Zhang (ZZ) polynomial or the Clar covering polynomial. We report here a computer program, ZZPolyCalc, specifically designed to compute ZZ polynomials of large carbon nanostructures. The curse of exponential scaling is avoided for a broad class of nanostructures by employing a sophisticated bookkeeping algorithm, in which each fragment appearing in the recursive decomposition is stored in the cache repository of molecular fragments indexed by a hash of the corresponding adjacency matrix. Although exponential scaling persists for the remaining nanostructures, the computational time is reduced by a few orders of magnitude owing to efficient use of hash-based fragment bookkeeping. The provided benchmark timings show that ZZPolyCalc allows for treating much larger carbon nanostructures than previously envisioned.Comment: 8 pages, 7 figures; submitted to "Comput. Phys. Commu

Topological Properties of Benzenoid Systems. XXI. Theorems, Conjectures, Unsolved Problems

The main known mathematical results (in the form of 32 theorems and 5 conjectures) about benzenoid systems are collected. A few new results (seven theorems) are proved. Seven unsolved problems are also pointed out. The paper contains results on the basic properties of benzenoid graphs, on the number of Kekule structures and on Clar\u27s resonant sextet formulas

Topological Properties of Benzenoid Systems. XXI. Theorems, Conjectures, Unsolved Problems

The main known mathematical results (in the form of 32 theorems and 5 conjectures) about benzenoid systems are collected. A few new results (seven theorems) are proved. Seven unsolved problems are also pointed out. The paper contains results on the basic properties of benzenoid graphs, on the number of Kekule structures and on Clar\u27s resonant sextet formulas