Advertisement

Journal of Mathematical Chemistry

, Volume 54, Issue 2, pp 416–427 | Cite as

Enumeration of cyclic polyazulenoids

  • Kecai DengEmail author
  • Xiaoling Zhang
Original Paper

Abstract

A cyclic polyazulenoid structure 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 “belt of rings” or say, “belt of azulenes”, more precisely. We give the exact counting formula for the number of cyclic polyazulenoid isomers according to the number of azulene units, by a method based on a generalization of Pólya’s Theorem.

Keywords

Cyclic polyazulenoids Enumeration Pólya’s Theorem 

Mathematics Subject Classification

92E10 05C30 05C90 

References

  1. 1.
    A. Cayley, On the theory of the analytical forms called trees. Philos. Mag. 13, 172–176 (1857)Google Scholar
  2. 2.
    G. Pálya, R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (Springer, Berlin, 1987)CrossRefGoogle Scholar
  3. 3.
    T. Imada, S. Ota, H. Nagamochi, T. Akutsu, Efficient enumeration of stereoisomers of outerplanar chemical graphs using dynamic programming. J. Chem. Inf. Model. 51, 2788–2807 (2011)CrossRefGoogle Scholar
  4. 4.
    J.L. Faulon, D. Visco, D. Roe, Enumerating molecules, in Reviews in Computational Chemistry, ed. by K.B. Lipkowitz, R. Larter, T.R. cundari (Wiley-VCH, Hoboken, 2005), pp. 209–257CrossRefGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    B.A. Hess Jr, L.J. Schaad, Hückel molecular orbital \(\pi \) resonance energies. The nonalternant hydrocarbons. J. Org. Chem. 36(22), 3418–3423 (1971)CrossRefGoogle Scholar
  7. 7.
    J. Aihara, General graph theory of superaromaticity. Bull. Chem. Soc. Jpn. 66, 57–60 (1993)CrossRefGoogle Scholar
  8. 8.
    J. Aihara, Non-superaromatic reference defined by graph theory for a super-ring molecule. J. Chem. Soc. Faraday Trans. 91(2), 237–239 (1995)CrossRefGoogle Scholar
  9. 9.
    M. Randić, A graph theoretical approach to conjugation and resonance energies of hydrocarbons. Tetrahedron 33, 1905–1920 (1977)CrossRefGoogle Scholar
  10. 10.
    M. Randić, Aromaticity of polycyclic conjugated hydrocarbons. Chem. Rev. 103, 3449–3605 (2003)CrossRefGoogle Scholar
  11. 11.
    J. Aihara, A simple method for estimating the superaromatic stabilization energy of a super-ring molecule. J. Phys. Chem. A 112, 4382–4385 (2008)CrossRefGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    N.G. de Bruijn, Pólya’s theorem of counting, in Applied Combinatorial Mathematics, ed. by E.F. Beckenbach (John Wiley & Sons, New York, 1964), pp. 144–172Google Scholar
  14. 14.
    J.G. Qian, F.J. Zhang, Counting the cyclocized polyphenacenes. J. Comput. Chem. 31(14), 2577–2584 (2010)CrossRefGoogle Scholar
  15. 15.
    K.C. Deng, J.G. Qian, F.J. Zhang, Enumerating the total colorings of a polyhedron and application to polyhedral links. J. Math. Chem. 50(6), 1693–1705 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaqiao UniversityQuanzhouPeople’s Republic of China
  2. 2.School of Mathematical and Computer SciencesQuanzhou Normal UniversityQuanzhouPeople’s Republic of China

Personalised recommendations