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Fullerene

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Fullerenes

A fullerene is a cubic polyhedral graph having all faces bounded by graph cycles of length 5 or 6. Examples include the 20-vertex dodecahedral graph, 24-vertex generalized Petersen graph GP(12,2), graph on 26 vertices given by Godsil and Royle (2001, p. 208), truncated icosahedral graph, and stable molecule C_(70) (Babić et al. 2002), illustrated above.

Every fullerene has exactly twelve graph cycles of length 5. The graph complement of a fullerene on n vertices is a regular graph of vertex degree n-4, and it has precisely 12 odd chordless cycles, all of them of length 5.

The numbers of fullerenes on n=20, 22, 24, ... vertices (counting enantiomers as equivalent) are given by 1, 0, 1, 1, 2, 3, 6, 6, 15, 17, 40, 45, 89, ... (OEIS A007894). Fullerenes can be enumerated and generated using the programs plantri and fullgen.

FullereneCanonicalPolyhedra

Canonical polyhedra corresponding to fullerenes on 20 to 34 vertices are illustrated above.

Fullerenes of type I (isomorphic to the skeletons of (n+1,0)-Goldberg polyhedra) and type II (isomorphic to the skeletons of (n,n)-Goldberg polyhedra) are implemented in the Wolfram Language as BuckyballGraph[n, "I"] and BuckyballGraph[n, "II"], respectively.

The following table lists some small fullerenes whose fractional chromatic number exceeds 5/2 (indexed according to Brinkmann and McKay).

fullerenechi^*
(24, 1)8/3
(28, 1)68/27
(28, 2)28/11
(30, 2)28/11

See also

Benzenoid, Fusene, Truncated Icosahedral Graph

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References

Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance-Distance Matrix: A Computational Algorithm and Its Application." Int. J. Quant. Chem. 90, 166-176, 2002. https://doi.org/10.1002/qua.10057.Balaban, A. T.; Liu, X.; Klein, D. J.; Babić, D.; Schmalz, T. G.; Seitz, W. A.; and Randić, M. "Graph Invariants for Fullerenes." J. Chem. Inf. Comput. Sci. 35, 396-404, 1995.Brinkmann, G. and Dress, A. W. M. "A Constructive Enumeration of Fullerenes." J. Algorithms 23, 345-358, 1997. https://doi.org/10.1006/jagm.1996.0806.Brinkmann, G. and McKay, B. "plantri and fullgen." https://users.cecs.anu.edu.au/~bdm/plantri/.Faulon, J. L.; Visco, D.; and Roe, D. "Enumerating Molecules." In Reviews in Computational Chemistry, Vol. 21 (Ed. K. Lipkowitz.) New York: Wiley-VCH, 2005.Fowler, P. W. and Manolopoulos, D. E. An Atlas of Fullerenes. New York: Dover, 2007.Fowler, P. W.; Manolopoulos, D. E.; and Ryan, R. P. "Isomerization of Fullerenes." Carbon 30, 1235, 1992.Godsil, C. and Royle, G. "Fullerenes." §9.8 in Algebraic Graph Theory. New York: Springer-Verlag, pp. 208-210, 2001.House of Graphs. "Fullerenes." https://houseofgraphs.org/meta-directory/fullerenes.Livshits, A. M. and Lozovik, Yu. E. "Cut-And-Unfold Approach to Fullerene Enumeration." J. Chem. Inf. Comput. Sci. 44, 1517-1520, 2004.Milicevic, A. and Trinajstic, N. "Combinatorial Enumeration in Chemistry." Chem. Modell. 4, 405-469, 2006.Petkovšek, M. and Pisanski, T. "Counting Disconnected Structures: Chemical Trees, Fullerenes, I-Graphs and Others." Croatica Chem. Acta 78, 563-567, 2005.Update a linkRoyle, G. "Fullerenes." http://school.maths.uwa.edu.au/~gordon/remote/fullerenes/Sloane, N. J. A. Sequence A007894 in "The On-Line Encyclopedia of Integer Sequences."

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Fullerene

Cite this as:

Weisstein, Eric W. "Fullerene." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Fullerene.html

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