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Open AccessArticle

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

1
Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300092, Taiwan
2
Center for Emergent Functional Matter Science, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300092, Taiwan
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(10), 1599; https://doi.org/10.3390/sym12101599
Received: 11 September 2020 / Revised: 20 September 2020 / Accepted: 22 September 2020 / Published: 25 September 2020
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. View Full-Text
Keywords: enumeration of Clar covers; ZZ polynomials; Clar covering polynomials; parallelogram-shaped benzenoids enumeration of Clar covers; ZZ polynomials; Clar covering polynomials; parallelogram-shaped benzenoids
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MDPI and ACS Style

Witek, H.A.; Langner, J. Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms. Symmetry 2020, 12, 1599. https://doi.org/10.3390/sym12101599

AMA Style

Witek HA, Langner J. Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms. Symmetry. 2020; 12(10):1599. https://doi.org/10.3390/sym12101599

Chicago/Turabian Style

Witek, Henryk A.; Langner, Johanna. 2020. "Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms" Symmetry 12, no. 10: 1599. https://doi.org/10.3390/sym12101599

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