Skip to main content
SpringerLink
Log in

k-resonance in Toroidal Polyhexes*

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

Abstract

This paper considers the k -resonance of a toroidal polyhex (or toroidal graphitoid) with a string (p, q, t) of three integers (p ≥ 2, q ≥ 2, 0 ≤ t ≤ p − 1). A toroidal polyhex G is said to be k-resonant if, for 1≤ i ≤ k, any i disjoint hexagons are mutually resonant, that is, G has a Kekulé structure (perfect matching) M such that these hexagons are M-alternating (in and off M). Characterizations for 1, 2 and 3-resonant toroidal polyhexes are given respectively in this paper.

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

Similar content being viewed by others

References

  1. G.G. Cash (1998) ArticleTitleSimple means of computing the Kekulé structure count for toroidal polyhex fullerenes J. Chem. Inf. Comput Sci. 38 58 Occurrence Handle10.1021/ci970057d

    Article  Google Scholar 

  2. R. Chen X. Guo (1993) ArticleTitlek-coverable coronoid systems J. Math. Chem 12 147 Occurrence Handle10.1007/BF01164632

    Article  Google Scholar 

  3. E. Clar (1972) The Aromatic Sextet Wiley London

    Google Scholar 

  4. M. Deza P.W Fowler A. Rassat K.M. Rogers (2000) ArticleTitleFullerenes as filings of surfaces J. Chem. Inf. Comput. Sci. 40 550 Occurrence Handle10.1021/ci990066h

    Article  Google Scholar 

  5. X. Guo F. Zhang (2001) ArticleTitlek-resonant benzenoid systems and k-cycle resonant graphs J. Chem. Inf. Comput. Sci. 41 IssueID3 480 Occurrence Handle10.1021/ci000080q

    Article  Google Scholar 

  6. Gutman I., Covering hexagonal systems with hexagons, Proceeding of 4th Yugoslav Seminar on Graph Theory, Novi Sad, 1983, 151.

  7. H. Hosoya T. Yamaguchi (1975) ArticleTitleSextet polynomials. A new enumeratin and proof technique for the resonance theory applied to the aromatic hydrocarbons Tetrahedron Lett. 52 4659 Occurrence Handle10.1016/S0040-4039(00)91045-1

    Article  Google Scholar 

  8. P.E. John (1998) ArticleTitleKekulé count in toroidal hexagonal carbon cages Croat. Chem. Acta 71 IssueID3 435

    Google Scholar 

  9. E.C. Kirby R.B. Mallion P. Pollak (1993) ArticleTitleToridal polyhexes J Chem. Soc. Faraday Trans. 89 IssueID12 1945 Occurrence Handle10.1039/ft9938901945

    Article  Google Scholar 

  10. E.C. Kirby P. Pollak (1998) ArticleTitleHow to enumerate the connectional isomers of a toridal polyhex fullerene J. Chem. Inf. Comput Sci. 38 66 Occurrence Handle10.1021/ci970072i

    Article  Google Scholar 

  11. D.J. Klein (1994) ArticleTitleElemental benzenoids J. Chem. Inf. Comput. Sci 34 453 Occurrence Handle10.1021/ci00018a037

    Article  Google Scholar 

  12. D.J. Klein H. Zhu (1996) ArticleTitleResonance in elemental benzenoids Discrete Appl. Math. 67 157 Occurrence Handle10.1016/0166-218X(95)00017-L

    Article  Google Scholar 

  13. K. Lin R. Chen (1996) ArticleTitlek-coverable polyhex graphs Ars Combin 43 33

    Google Scholar 

  14. J. Liu H.J. Dai J.H. Hafner D.T. Colbert D.T.R.E. Smalley S.J. Tans C. Dekker (1997) ArticleTitleFullerene ‘crop circles’ Nature 385 780 Occurrence Handle10.1038/385780b0 Occurrence Handle9039908

    Article  PubMed  Google Scholar 

  15. Lovász L., Plummer M.D. (1986). Matching Theory, Annals of Discrete Math . 29. North-Hollard, Amsterdam.

  16. D. Marušič T. Pisanski (2000) ArticleTitleSymmetries of hexagonal molecular graphs on the torus Croat. Chem. Acta 73 IssueID4 969

    Google Scholar 

  17. S. Negami (1983) ArticleTitleUniqueness and faithfulness of embedding of toroidal graphs Discrete Math. 44 161 Occurrence Handle10.1016/0012-365X(83)90057-2

    Article  Google Scholar 

  18. M. Randić (2003) ArticleTitleAroznaticity of polycyclic conjugated hydrocarbons Chem. Reviews 103 IssueID9 3449 Occurrence Handle10.1021/cr9903656

    Article  Google Scholar 

  19. M. Randić (1976) ArticleTitleConjugated circuits and resonance energies of benzenoid hydrocarbons Chem. Phys. Lett. 38 68 Occurrence Handle10.1016/0009-2614(76)80257-6

    Article  Google Scholar 

  20. M. Randić (1977) ArticleTitleAromaticity and conjugation J. Am. Chem. Soc 99 444 Occurrence Handle10.1021/ja00444a022

    Article  Google Scholar 

  21. T.G. Schmalz W.A. Seitz D.J. Klein G.E. Hite (1988) ArticleTitleElemental carbon cages J. Am. Chem. Soc. 110 1113 Occurrence Handle10.1021/ja00212a020

    Article  Google Scholar 

  22. C. Thomassen (1991) ArticleTitleTilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface Trans. Amer. Math Soc. 323 605

    Google Scholar 

  23. F. Zhang L. Wang (2004) ArticleTitlek-resonance of open-ended carbon nanotubes J. Math. Chem. 35 IssueID2 87 Occurrence Handle10.1023/B:JOMC.0000014306.86197.22

    Article  Google Scholar 

  24. F. Zhang R. Chen (1991) ArticleTitleWhen each hexagon of a hexagonal system covers it Discrete Appl. Math. 30 63 Occurrence Handle10.1016/0166-218X(91)90014-N

    Article  Google Scholar 

  25. H. Zhang F. Zhang (1996) ArticleTitleThe Clar covering polynomial of hexagonal systems I Discrete Appl. Math. 69 147 Occurrence Handle10.1016/0166-218X(95)00081-2

    Article  Google Scholar 

  26. H. Zhang F. Zhang (2000) ArticleTitlePlane elementary bipartite graphs Discrete Appl. Math. 105 291 Occurrence Handle10.1016/S0166-218X(00)00204-3

    Article  Google Scholar 

  27. M. Zheng (1991) ArticleTitlek-resonant benzenoid systems J. Mol. Struct (Theochem) 231 321 Occurrence Handle10.1016/0166-1280(91)85230-5

    Article  Google Scholar 

  28. M. Zheng (1992) ArticleTitleConstruction of 3-resonant benzenoid systems J. Mol Struct. (Theochem) 277 1 Occurrence Handle10.1016/0166-1280(92)87125-J

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, P. R. China

    Wai Chee Shiu & Peter Che Bor Lam

  2. School of Mathematics and Statistics, Lanzhou University, Gansu, Lanzhou, 730000, P. R. China

    Heping Zhang

Authors
  1. Wai Chee Shiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Che Bor Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Heping Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wai Chee Shiu.

Additional information

*This work is supported by FRG, Hong Kong Baptist University; NSFC and TRAPOYT.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shiu, W.C., Lam, P.C.B. & Zhang, H. k-resonance in Toroidal Polyhexes*. J Math Chem 38, 451–466 (2005). https://doi.org/10.1007/s10910-004-6897-4

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-004-6897-4

Keywords

AMS 2000 MSC

Search

Navigation