×

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.

MSC:

05C90 Applications of graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.