A fullerene graph is a three-regular and three-connected plane graph exactly 12 faces of which are pentagons and the remaining faces are hexagons. Let F n be a fullerene graph with n vertices. The Clar number c(F n ) of F n is the maximum size of sextet patterns, the sets of disjoint hexagons which are all M-alternating for a perfect matching (or Kekulé structure) M of F n . A sharp upper bound of Clar number for any fullerene graphs is obtained in this article: \(c(F_n)\leqslant \lfloor \frac {n-12} 6\rfloor\). Two famous members of fullerenes C60 (Buckministerfullerene) and C70 achieve this upper bound. There exist infinitely many fullerene graphs achieving this upper bound among zigzag and armchair carbon nanotubes.
Similar content being viewed by others
References
Kroto H.W., Heath J.R., Obrien S.C., Curl R.F., Smalley R.E. (1985) C60: Buckminsterfullerene. Nature 318: 162–163
Krätschmer W., Lamb L.D., Fostiropoulos K., Huffman D.R. (1990) Solid C60: a new form of carbon. Nature 347: 354
R. Taylor, J.P. Hare, A.K. Abdul-Sada and H.W. Kroto, Isolation, seperation and characterisation of the fullerenes C60 and C70: the third form of carbon, J. Chem. Soc. Chem. Commum. (1990) 1423.
Fowler P.W., Manolopoulos D.E. (1995). An Atlas of Fullerenes. Oxford Univ. Press, Oxford
Brinkmann G., Dress A. (1997) A constructive enumeration of fullerenes. J. Algorithms 23: 345–358
Clar E. (1972). The Aromatic Sextet. Wiley, New York
Hansen P., Zheng M. (1992) Upper bounds for the Clar number of benzenoid hydrocarbons. J. Chem. Soc. Faraday Trans. 88: 1621–1625
Hansen P., Zheng M. (1994) The Clar number of a benzenoid hydrocarbon and linear programming. J. Math. Chem. 15: 93–107
H. Abeledo and G. Atkinson, The Clar and Fries problems for benzenoid hydrocarbons are linear programs, in: Discrete Mathematical Chemistry, DIMACS Series, Vol. 51, eds. P. Hansen, P. Fowler and M. Zheng (American Mathematical Society, Providence, RI, 2000), pp. 1–8.
Klavžar S., Žigert P., Gutman I. (2002). Clar number of catacondensed benzenoid hydrocarbons. J. Mol. Struct. (Theochem) 586: 235–240
Salem K., Gutman I. (2004) Clar number of hexagonal chains. Chem. Phys. Lett. 394: 283–286
Zhang F., Li X. (1989) The Clar formulas of a class of hexagonal systems. Match 24: 333–347
Zhang H. (1993) The Clar formula of a type of benzenoid systems. J. Xinjiang Univ. (Natural Science, In Chinese) 10: 1–7
Zhang H. (1995) The Clar formula of hexagonal polyhexes. J. Xinjiang Univ. (Natural Science) 12: 1–9
Zhang H. (1995) The Clar formula of regular t-tier strip benzenoid systems. Sys. Sci. Math. Sci. 8(4): 327–337
Zhang F., Wang L. (2004) k-resonance of open-end carbon nanotubes. J. Math. Chem. 35(2): 87–103
El-Basil S. (2000) Clar sextet theory of buckminsterfullere (C60). J. Mol. Struct. (Theochem) 531: 9–21
Shiu W.C., Lam P.C.B., Zhang H. (2003) Clar and sextet polynomials of buckminsterfullerene. J. Mol. Struct. (Theochem) 662: 239–248
Zhang H., He J. (2005) A comparison between 1-factor count and resonant pattern count in plane non-bipartite graphs. J. Math. Chem. 38(3): 315–324
Gutman I., Cyvin S.J. (1989). Introduction to the Theory of Benzenoid Hydrocarbons. Springer-Verlag, Berlin
Došlić T. (2003) Cyclical edge-connectivity of fullerene graphs and (k,6)-cages. J. Math. Chem. 33: 103–112
Saito R., Dresselhaus M.S., Dresselhaus G. (1998). Physical Properties of Carbon Nanotubes. Imperial College Press, London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, H., Ye, D. An Upper Bound for the Clar Number of Fullerene Graphs. J Math Chem 41, 123–133 (2007). https://doi.org/10.1007/s10910-006-9061-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-006-9061-5