Skip to main content
SpringerLink
Log in

Enumeration of Möbius type cyclic polyazulenoids

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

A Möbius type cyclic polyazulenoid is a kind of nonalternant conjugated hydrocarbon, consisting of a cyclic series of alternatingly fused five membered rings and seven membered rings, which forms a “Möbius belt of rings” or say, “Möbius belt of azulenes”, more precisely. As a sequel of the recent paper (Deng and Zhang in J Math Chem 54(2):416–427, 2016), which counts “belt type” cyclic polyazulenoid, this article studies the enumeration problem of Möbius type cyclic polyazulenoid. We obtain the exact counting formula for the number of Möbius type cyclic polyazulenoids with n azulene units, by a method based on coding technique and a generalization of ‘Burnside’ lemma. And we give the numerical results for n not larger than 20, which may contribute to the fully synthesis of Möbius type cyclic polyazulenoids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. M.J.S. Dewar, The Molecular Orbital Theory of Organic Chemistry (McGraw-Hill, New York, 1959)

    Google Scholar 

  2. B.A. Hess Jr., L.J. Schaad, Hückel molecular orbital \(\pi \) resonance energies. The nonalternant hydrocarbons. J. Org. Chem. 36(22), 3418–3423 (1971)

    Article  CAS  Google Scholar 

  3. J. Aihara, General graph theory of superaromaticity. Bull. Chem. Soc. Jpn. 66, 57–60 (1993)

    Article  CAS  Google Scholar 

  4. J. Aihara, Non-superaromatic reference defined by graph theory for a super-ring molecule. J. Chem. Soc. Faraday Trans. 91(2), 237–239 (1995)

    Article  CAS  Google Scholar 

  5. M. Randić, A graph theoretical approach to conjugation and resonance energies of hydrocarbons. Tetrahedron 33, 1905–1920 (1977)

    Article  Google Scholar 

  6. M. Randić, Aromaticity of polycyclic conjugated hydrocarbons. Chem. Rev. 103, 3449–3605 (2003)

    Article  Google Scholar 

  7. J. Aihara, A simple method for estimating the superaromatic stabilization energy of a super-ring molecule. J. Phys. Chem. A 112, 4382–4385 (2008)

    Article  CAS  Google Scholar 

  8. A. Ciesielski, M.K. Cyrański, T.M. Krygowski, P.W. Fowler, M. Lillington, Super-delocalized valence isomer of coronene. J. Org. Chem. 71, 6840–6845 (2006)

    Article  CAS  Google Scholar 

  9. D. Orlikowski, M.B. Nardelli, J. Bernholc, C. Roland, Theoretical STM signatures and transport properties of native defects in carbon nanotubes. Phys. Rev. B. 61, 14194–14203 (2000)

    Article  CAS  Google Scholar 

  10. D.P. Craig, A novel type of aromaticity. Nature 181, 1052–1053 (1958)

    Article  CAS  Google Scholar 

  11. D.P. Craig, Delocalization in \(p\pi \)-\(d\pi \) bonds. J. Chem. Soc. (Resumed) 202, 997–1001 (1959)

    Article  Google Scholar 

  12. E. Heilbronner, Hückel molecular orbitals of Möbius-type conformations of annulenes. Tetrahedron Lett. 5(29), 1923–1928 (1964)

    Article  Google Scholar 

  13. H.E. Zimmerman, Moebius-Hueckel concept in organic chemistry. Application of organic molecules and reactions. Acc. Chem. Res. 4, 272–280 (1971)

    Article  CAS  Google Scholar 

  14. D.M. Walba, M. Richards, R.C. Haltiwanger, Total synthesis of the first molecular Möbius strip. J. Am. Chem. Soc. 104, 3219–3221 (1982)

    Article  CAS  Google Scholar 

  15. D.M. Walba, T.C. Homan, R.M. Richards, R.C. Haltiwanger, Topological stereochemistry. 9. Synthesis and cutting “in half” of a molecular Möblus strip. New J. Chem. 17, 661–681 (1993)

    CAS  Google Scholar 

  16. D. Ajami, O. Oeckler, A. Simon, R. Herges, Synthesis of a Möbius aromatic hydrocarbon. Nature 426, 819–821 (2003)

    Article  CAS  Google Scholar 

  17. D. Ajami, K. Hess, F. Köhler, C. Näther, O. Oeckler, A. Simon, C. Yamamoto, Y. Okamoto, R. Herges, Synthesis and properties of the first Möbius annulenes. Chem.-Eur. J. 12, 5434–5445 (2006)

    Article  CAS  Google Scholar 

  18. K.C. Deng, X.L. Zhang, Enumeration of cyclic polyazulenoids. J. Math. Chem. 54(2), 416–427 (2016)

    Article  CAS  Google Scholar 

  19. D. Walba, R. Richards, R.C. Haltiwanger, Total synthesis of the first molecular Möbius strip. J. Am. Chem. Soc. 104(11), 3219–3221 (1982)

    Article  CAS  Google Scholar 

  20. J. Simon, Topological chirality of cerntain molecules. Topology 25(2), 229–235 (1986)

    Article  Google Scholar 

  21. E.F. Beckenbach, Applied Combinatorial Mathematics (Wiley, New York, 1964)

    Google Scholar 

  22. D.J. Klein, A. Misra, Topological isomer generation and enumeration: application for polyphenacenes. MATCH-Commun. Math. Comp. Chem. 46, 45–69 (2002)

    CAS  Google Scholar 

  23. A. Misra, D.J. Klein, Characterization of cyclo-polyphenacenes. J. Chem. Inf. Comp. Sci. 42, 1171–1175 (2002)

    Article  CAS  Google Scholar 

  24. J.G. Qian, F.J. Zhang, Counting the cyclocized polyphenacenes. J. Comput. Chem. 31(14), 2577–2584 (2010)

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Sciences, Quanzhou Normal University, Quanzhou, 362000, Fujian, People’s Republic of China

    Xiaoling Zhang

  2. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, Fujian, People’s Republic of China

    Kecai Deng

Authors
  1. Xiaoling Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kecai Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kecai Deng.

Additional information

Research supported by Science Foundation of Fujian Province, China (No. 2015J05013).

Research supported by Science Foundation of Fujian Province, China (No. 2016J05009).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, X., Deng, K. Enumeration of Möbius type cyclic polyazulenoids. J Math Chem 55, 132–141 (2017). https://doi.org/10.1007/s10910-016-0675-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-016-0675-y

Keywords

Mathematics Subject Classification

Search

Navigation