Enumeration of Clar covers of parallelogram chains

The number of Clar covers, the number of Kekule structures, and the Clar covering polynomials (aka Zhang-Zhang or ZZ polynomials) of benzenoid parallelogram chains M-k (m, n) formed by merging k benzenoid parallelograms M (m, n) are characterized in terms of analogous quantities of the elementary building block, M (m, n). The appropriate formulas are compactly expressed as determinants of highly structured, tridiagonal, Toeplitz k x k matrices. All the 2(k) distinct parallelogram chains M-k (m, n) = M-1 M-2 ... M-k of constant length k, where M-i is an element of {R equivalent to M (m, n), L equivalent to M (n, m)}, share the same ZZ polynomial and consequently possess the same number of Clar covers and Kekule structures. The presented results constitute the first attempt to express the Clar theory of complex benzenoid moieties in terms of elementary benzenoids. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This study characterizes the number of Clar covers, Kekule structures, and Clar covering polynomials of benzenoid parallelogram chains formed by merging k benzenoid parallelograms. The results, expressed as determinants of structured matrices, provide insight into complex benzenoid moieties in terms of elementary benzenoids.