Enumeration of Clar covers of parallelogram chains

Bing Hau He, Henryk A. Witek*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish
Pages (from-to)221-233
Number of pages13
JournalDiscrete Applied Mathematics
StatePublished - 30 Oct 2021


  • Benzenoid
  • Clar cover
  • Kekulé structure
  • Zhang–Zhang polynomial


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