TY - JOUR

T1 - Enumeration of Clar covers of parallelogram chains

AU - He, Bing Hau

AU - Witek, Henryk A.

N1 - Publisher Copyright:
© 2021 Elsevier B.V.

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

UR - http://www.scopus.com/inward/record.url?scp=85093658792&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2021.07.003

DO - 10.1016/j.dam.2021.07.003

M3 - Article

AN - SCOPUS:85093658792

SN - 0166-218X

VL - 302

SP - 221

EP - 233

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -