Journal of Mathematical Chemistry

, Volume 53, Issue 8, pp 1702–1724 | Cite as

Recursive generation of IPR fullerenes

  • Jan GoedgebeurEmail author
  • Brendan D. McKay
Original Paper


We describe a new construction algorithm for the recursive generation of all non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm stays entirely within the class of IPR fullerenes, that is: every IPR fullerene is constructed by expanding a smaller IPR fullerene unless it belongs to a limited class of irreducible IPR fullerenes that can easily be made separately. The class of irreducible IPR fullerenes consists of 36 fullerenes with up to 112 vertices and 4 infinite families of nanotube fullerenes. Our implementation of this algorithm is faster than other generators for IPR fullerenes and we used it to compute all IPR fullerenes up to 400 vertices.


IPR fullerene Nanotube cap Fullerene patch Recursive construction Computation 



Most computations for this work were carried out using the Stevin Supercomputer Infrastructure at Ghent University. We would like to thank Gunnar Brinkmann and Jack Graver for useful suggestions.


  1. 1.
    E. Albertazzi, C. Domene, P.W. Fowler, T. Heine, G. Seifert, C. Van Alsenoy, F. Zerbetto, Pentagon adjacency as a determinant of fullerene stability. Phys. Chem. Chem. Phys. 1(12), 2913–2918 (1999)CrossRefGoogle Scholar
  2. 2.
    J. Bornhöft, G. Brinkmann, J. Greinus, Pentagon-hexagon-patches with short boundaries. Eur. J. Comb. 24(5), 517–529 (2003)CrossRefGoogle Scholar
  3. 3.
    G. Brinkmann, Zur mathematischen Behandlung gestörter periodischer Pflasterungen. PhD thesis, Universität Bielefeld (1990)Google Scholar
  4. 4.
    G. Brinkmann, K. Coolsaet, J. Goedgebeur, H. Mélot, House of graphs: a database of interesting graphs. Discret. Appl. Math. 161(1–2),311–314 (2013).
  5. 5.
    G. Brinkmann, O.D. Friedrichs, S. Lisken, A. Peeters, N. Van Cleemput, CaGe—a virtual environment for studying some special classes of plane graphs—an update. MATCH Commun. Math. Comput. Chem. 63(3), 533–552 (2010).
  6. 6.
    G. Brinkmann, O.D. Friedrichs, U. von Nathusius, Numbers of faces and boundary encodings of patches, in Graphs and Discovery, volume 69 of DIMACS Series in Discrete Mathematics and Theoretical Computer Sciences, pp. 27–38 (2005)Google Scholar
  7. 7.
    G. Brinkmann, A.W.M. Dress, A constructive enumeration of fullerenes. J. Algorithms 23, 345–358 (1997)CrossRefGoogle Scholar
  8. 8.
    G. Brinkmann, J. Goedgebeur, B.D. McKay, Homepage of buckygen.
  9. 9.
    G. Brinkmann, J. Goedgebeur, B.D. McKay, The generation of fullerenes. J. Chem. Inf. Model. 52(11), 2910–2918 (2012)CrossRefGoogle Scholar
  10. 10.
    G. Brinkmann, J. Goedgebeur, B.D. McKay, The smallest fullerene without a spiral. Chem. Phys. Lett. 522(2), 54–55 (2012)CrossRefGoogle Scholar
  11. 11.
    G. Brinkmann, J.E. Graver, C. Justus, Numbers of faces in disordered patches. J. Math. Chem. 45(2), 263–278 (2009)CrossRefGoogle Scholar
  12. 12.
    G. Brinkmann, B.D. McKay, Fast generation of planar graphs. MATCH Commun. Math. Comput. Chem. 58(2), 323–357 (2007)Google Scholar
  13. 13.
    G. Brinkmann, U. von Nathusius, A.H.R. Palser, A constructive enumeration of nanotube caps. Discret. Appl. Math. 116(1–2), 55–71 (2002)CrossRefGoogle Scholar
  14. 14.
    J. Goedgebeur, Generation Algorithms for Mathematical and Chemical Problems. PhD thesis, Ghent University, Belgium, May (2013)Google Scholar
  15. 15.
    J. Goedgebeur, B.D. McKay, Fullerenes with distant pentagons, in Preparation Google Scholar
  16. 16.
    X. Guo, P. Hansen, M. Zheng, Boundary uniqueness of fusenes. Discret. Appl. Math. 118(3), 209–222 (2002)CrossRefGoogle Scholar
  17. 17.
    M. Hasheminezhad, H. Fleischner, B.D. McKay, A universal set of growth operations for fullerenes. Chem. Phys. Lett. 464, 118–121 (2008)CrossRefGoogle Scholar
  18. 18.
    C. Justus, Transformationen zwischen Fullerenen und die Flächenzahl von Patches mit gleichem Rand. Master’s thesis, Universität Bielefeld (2003). (Advisor: G. Brinkmann)Google Scholar
  19. 19.
    H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Smalley, \(C_{60}\): buckminsterfullerene. Nature 318(6042), 162–163 (1985)CrossRefGoogle Scholar
  20. 20.
    X. Liu, D.J. Klein, T.G. Schmalz, W.A. Seitz, Generation of carbon cage polyhedra. J. Comput. Chem. 12(10), 1252–1259 (1991)CrossRefGoogle Scholar
  21. 21.
    D.E. Manolopoulos, P.W. Fowler, Molecular graphs, point groups, and fullerenes. J. Chem. Phys. 96(10), 7603–7614 (1992)CrossRefGoogle Scholar
  22. 22.
    D.E. Manolopoulos, J.C. May, Theoretical studies of the fullerenes: \({C_{34}}\) to \({C_{70}}\). Chem. Phys. Lett. 181, 105–111 (1991)CrossRefGoogle Scholar
  23. 23.
    B.D. McKay, Isomorph-free exhaustive generation. J. Algorithms 26(2), 306–324 (1998)CrossRefGoogle Scholar
  24. 24.
    C.H. Sah, Combinatorial construction of fullerene structures. Croat. Chem. Acta 66, 1–12 (1993)Google Scholar
  25. 25.
    R. Saito, G. Dresselhaus, M.S. Dresselhaus (eds.), Physical Properties of Carbon Nanotubes, vol 4 (Imperial College Press, London, 1998)Google Scholar
  26. 26.
    T.G. Schmalz, W.A. Seitz, D.J. Klein, G.E. Hite, Elemental carbon cages. J. Am. Chem. Soc. 110(4), 1113–1127 (1988)CrossRefGoogle Scholar
  27. 27.
    M. Yoshida, E. Osawa, Formalized drawing of fullerene nets. 1. Algorithm and exhaustive generation of isomeric structures. Bull. Chem. Soc. Jpn. 68, 2073–2081 (1995)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Computer Science and StatisticsGhent UniversityGhentBelgium
  2. 2.Research School of Computer ScienceAustralian National UniversityCanberraAustralia

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