Enumeration of Clar covers of parallelogram chains

Bing Hau He; Henryk A. Witek

doi:10.1016/j.dam.2021.07.003

Enumeration of Clar covers of parallelogram chains

Bing Hau He, Henryk A. Witek^*

^*Corresponding author for this work

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

6 Scopus citations

Abstract

The number of Clar covers, the number of Kekulé structures, and the Clar covering polynomials (aka Zhang–Zhang or ZZ polynomials) of benzenoid parallelogram chains M_km,n formed by merging k benzenoid parallelograms Mm,n are characterized in terms of analogous quantities of the elementary building block, Mm,n. The appropriate formulas are compactly expressed as determinants of highly structured, tridiagonal, Toeplitz k×k matrices. All the 2^k distinct parallelogram chains M_km,n≡M₁M₂…M_k of constant length k, where M_i∈R≡Mm,n,L≡Mn,m, share the same ZZ polynomial and consequently possess the same number of Clar covers and Kekulé structures. The presented results constitute the first attempt to express the Clar theory of complex benzenoid moieties in terms of elementary benzenoids.

Original language	English
Pages (from-to)	221-233
Number of pages	13
Journal	Discrete Applied Mathematics
Volume	302
DOIs	https://doi.org/10.1016/j.dam.2021.07.003
State	Published - 30 Oct 2021

Keywords

Benzenoid
Clar cover
Kekulé structure
Zhang–Zhang polynomial

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10.1016/j.dam.2021.07.003

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title = "Enumeration of Clar covers of parallelogram chains",

abstract = "The number of Clar covers, the number of Kekul{\'e} structures, and the Clar covering polynomials (aka Zhang–Zhang or ZZ polynomials) of benzenoid parallelogram chains Mkm,n formed by merging k benzenoid parallelograms Mm,n are characterized in terms of analogous quantities of the elementary building block, Mm,n. The appropriate formulas are compactly expressed as determinants of highly structured, tridiagonal, Toeplitz k×k matrices. All the 2k distinct parallelogram chains Mkm,n≡M1M2…Mk of constant length k, where Mi∈R≡Mm,n,L≡Mn,m, share the same ZZ polynomial and consequently possess the same number of Clar covers and Kekul{\'e} structures. The presented results constitute the first attempt to express the Clar theory of complex benzenoid moieties in terms of elementary benzenoids.",

keywords = "Benzenoid, Clar cover, Kekul{\'e} structure, Zhang–Zhang polynomial",

author = "He, {Bing Hau} and Witek, {Henryk A.}",

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year = "2021",

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doi = "10.1016/j.dam.2021.07.003",

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journal = "Discrete Applied Mathematics",

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TY - JOUR

T1 - Enumeration of Clar covers of parallelogram chains

AU - He, Bing Hau

AU - Witek, Henryk A.

PY - 2021/10/30

Y1 - 2021/10/30

N2 - The number of Clar covers, the number of Kekulé structures, and the Clar covering polynomials (aka Zhang–Zhang or ZZ polynomials) of benzenoid parallelogram chains Mkm,n formed by merging k benzenoid parallelograms Mm,n are characterized in terms of analogous quantities of the elementary building block, Mm,n. The appropriate formulas are compactly expressed as determinants of highly structured, tridiagonal, Toeplitz k×k matrices. All the 2k distinct parallelogram chains Mkm,n≡M1M2…Mk of constant length k, where Mi∈R≡Mm,n,L≡Mn,m, share the same ZZ polynomial and consequently possess the same number of Clar covers and Kekulé structures. The presented results constitute the first attempt to express the Clar theory of complex benzenoid moieties in terms of elementary benzenoids.

AB - The number of Clar covers, the number of Kekulé structures, and the Clar covering polynomials (aka Zhang–Zhang or ZZ polynomials) of benzenoid parallelogram chains Mkm,n formed by merging k benzenoid parallelograms Mm,n are characterized in terms of analogous quantities of the elementary building block, Mm,n. The appropriate formulas are compactly expressed as determinants of highly structured, tridiagonal, Toeplitz k×k matrices. All the 2k distinct parallelogram chains Mkm,n≡M1M2…Mk of constant length k, where Mi∈R≡Mm,n,L≡Mn,m, share the same ZZ polynomial and consequently possess the same number of Clar covers and Kekulé structures. The presented results constitute the first attempt to express the Clar theory of complex benzenoid moieties in terms of elementary benzenoids.

KW - Benzenoid

KW - Clar cover

KW - Kekulé structure

KW - Zhang–Zhang polynomial

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U2 - 10.1016/j.dam.2021.07.003

DO - 10.1016/j.dam.2021.07.003

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SN - 0166-218X

VL - 302

SP - 221

EP - 233

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -

Enumeration of Clar covers of parallelogram chains

Abstract

Keywords

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