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

Henryk Arnold Witek; Johanna Langner

doi:10.3390/sym12101599

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

Henryk Arnold Witek^*, Johanna Langner

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekulean 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 2x2 determinant bearing close structural resemblance to analogous formulas for the number of Kekule structures derived from the John-Sachs theory of Kekule structures.

Original language	English
Article number	1599
Number of pages	20
Journal	Symmetry-Basel
Volume	12
Issue number	10
DOIs	https://doi.org/10.3390/sym12101599
State	Published - Oct 2020

Keywords

enumeration of Clar covers
ZZ polynomials
Clar covering polynomials
parallelogram-shaped benzenoids
ZHANG-ZHANG POLYNOMIALS
POLYCYCLIC AROMATIC-HYDROCARBONS
SINGLE ZIGZAG CHAINS
CLOSED-FORM FORMULAS
HEXAGONAL SYSTEMS
INTERFACE THEORY
ALGORITHM
KEKULE
ZZDECOMPOSER
EQUIVALENCE

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@article{34f31404ee26460090f4ce8dcd730658,

title = "Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms",

abstract = "We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekulean 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 2x2 determinant bearing close structural resemblance to analogous formulas for the number of Kekule structures derived from the John-Sachs theory of Kekule structures.",

keywords = "enumeration of Clar covers, ZZ polynomials, Clar covering polynomials, parallelogram-shaped benzenoids, ZHANG-ZHANG POLYNOMIALS, POLYCYCLIC AROMATIC-HYDROCARBONS, SINGLE ZIGZAG CHAINS, CLOSED-FORM FORMULAS, HEXAGONAL SYSTEMS, INTERFACE THEORY, ALGORITHM, KEKULE, ZZDECOMPOSER, EQUIVALENCE",

author = "Witek, {Henryk Arnold} and Johanna Langner",

year = "2020",

month = oct,

doi = "10.3390/sym12101599",

language = "English",

volume = "12",

journal = "Symmetry-Basel",

publisher = "MDPI AG",

number = "10",

}

TY - JOUR

T1 - Clar Covers of Overlapping Benzenoids

T2 - Case of Two Identically-Oriented Parallelograms

AU - Witek, Henryk Arnold

AU - Langner, Johanna

PY - 2020/10

Y1 - 2020/10

N2 - We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekulean 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 2x2 determinant bearing close structural resemblance to analogous formulas for the number of Kekule structures derived from the John-Sachs theory of Kekule structures.

AB - We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekulean 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 2x2 determinant bearing close structural resemblance to analogous formulas for the number of Kekule structures derived from the John-Sachs theory of Kekule structures.

KW - enumeration of Clar covers

KW - ZZ polynomials

KW - Clar covering polynomials

KW - parallelogram-shaped benzenoids

KW - ZHANG-ZHANG POLYNOMIALS

KW - POLYCYCLIC AROMATIC-HYDROCARBONS

KW - SINGLE ZIGZAG CHAINS

KW - CLOSED-FORM FORMULAS

KW - HEXAGONAL SYSTEMS

KW - INTERFACE THEORY

KW - ALGORITHM

KW - KEKULE

KW - ZZDECOMPOSER

KW - EQUIVALENCE

U2 - 10.3390/sym12101599

DO - 10.3390/sym12101599

M3 - Article

VL - 12

JO - Symmetry-Basel

JF - Symmetry-Basel

IS - 10

M1 - 1599

ER -

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

Abstract

Keywords

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