TOPICS
Search

Fullerene


Fullerenes

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

Every fullerene has exactly twelve 5-cycles. The complement of a fullerene on n vertices is (n-4)-regular, and it has precisely 12 odd chordless cycles, all of them of order 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). Brinkmann and McKay have written programs for the enumeration and generation of fullerenes.

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.

While almost all small fullerenes have fractional chromatic number 5/2, those listed in the following table (indexed according to Brinkmann and McKay) do not.

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

See also

Benzenoid, Fusene, Truncated Icosahedral Graph

Explore with Wolfram|Alpha

References

Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem. 60, 161-176, 2002.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, W. M. "A Constructive Enumeration of Fullerenes." J. Algorithms 23, 245-358, 1997.Brinkmann, G. and McKay, B. "plantri and fullgen." http://cs.anu.edu.au/~bdm/plantri.Faulon, J. L.; Visco, D.; and Roe, D. "Enumerating Molecule." 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.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.Royle, 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."

Cite this as:

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

Subject classifications