An upper bound for the Clar number of fullerene graphs. (English) Zbl 1213.05259

Summary: 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: \(c(F_n)\leq\lfloor\frac{n-12}{6}\rfloor\). Two famous members of fullerenes \(C_{60}\) (Buckministerfullerene) and \(C_{70}\) achieve this upper bound. There exist infinitely many fullerene graphs achieving this upper bound among zigzag and armchair carbon nanotubes.


05C90 Applications of graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
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[1] Kroto H.W., Heath J.R., Obrien S.C., Curl R.F., Smalley R.E. (1985) C60: Buckminsterfullerene. Nature 318: 162–163
[2] Krätschmer W., Lamb L.D., Fostiropoulos K., Huffman D.R. (1990) Solid C60: a new form of carbon. Nature 347: 354
[3] 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.
[4] Fowler P.W., Manolopoulos D.E. (1995). An Atlas of Fullerenes. Oxford Univ. Press, Oxford
[5] Brinkmann G., Dress A. (1997) A constructive enumeration of fullerenes. J. Algorithms 23: 345–358
[6] Clar E. (1972). The Aromatic Sextet. Wiley, New York
[7] Hansen P., Zheng M. (1992) Upper bounds for the Clar number of benzenoid hydrocarbons. J. Chem. Soc. Faraday Trans. 88: 1621–1625
[8] Hansen P., Zheng M. (1994) The Clar number of a benzenoid hydrocarbon and linear programming. J. Math. Chem. 15: 93–107
[9] 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. · Zbl 0973.90048
[10] Klavžar S., Žigert P., Gutman I. (2002). Clar number of catacondensed benzenoid hydrocarbons. J. Mol. Struct. (Theochem) 586: 235–240
[11] Salem K., Gutman I. (2004) Clar number of hexagonal chains. Chem. Phys. Lett. 394: 283–286
[12] Zhang F., Li X. (1989) The Clar formulas of a class of hexagonal systems. Match 24: 333–347
[13] Zhang H. (1993) The Clar formula of a type of benzenoid systems. J. Xinjiang Univ. (Natural Science, In Chinese) 10: 1–7 · Zbl 0964.92519
[14] Zhang H. (1995) The Clar formula of hexagonal polyhexes. J. Xinjiang Univ. (Natural Science) 12: 1–9 · Zbl 0964.92521
[15] Zhang H. (1995) The Clar formula of regular t-tier strip benzenoid systems. Sys. Sci. Math. Sci. 8(4): 327–337 · Zbl 0851.05090
[16] Zhang F., Wang L. (2004) k-resonance of open-end carbon nanotubes. J. Math. Chem. 35(2): 87–103 · Zbl 1045.92052
[17] El-Basil S. (2000) Clar sextet theory of buckminsterfullere (C60). J. Mol. Struct. (Theochem) 531: 9–21
[18] Shiu W.C., Lam P.C.B., Zhang H. (2003) Clar and sextet polynomials of buckminsterfullerene. J. Mol. Struct. (Theochem) 662: 239–248
[19] 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 · Zbl 1217.05192
[20] Gutman I., Cyvin S.J. (1989). Introduction to the Theory of Benzenoid Hydrocarbons. Springer-Verlag, Berlin · Zbl 0722.05056
[21] Došlić T. (2003) Cyclical edge-connectivity of fullerene graphs and (k,6)-cages. J. Math. Chem. 33: 103–112 · Zbl 1018.92036
[22] Saito R., Dresselhaus M.S., Dresselhaus G. (1998). Physical Properties of Carbon Nanotubes. Imperial College Press, London · Zbl 0752.68069
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