Fullerene
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 
, graph on 26 vertices given by
Gosil and Royle (2001, p. 208), truncated
icosahedral graph, and stable molecule 
(Babić
et al. 2002), illustrated above.
Every fullerene has exactly twelve 5-cycles. The complement of a fullerene on 
vertices is 
-regular, and
it has precisely 12 odd chordless cycles, all of them of order 5.
The numbers of fullerenes on 
, 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.
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.
| fullerene |  |
| (24, 1) | 8/3 |
| (28, 1) | 68/27 |
| (28,
2) | 28/11 |
| (30, 2) | 28/11 |
SEE ALSO: Benzenoid,
Fusene,
Truncated Icosahedral Graph
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." https://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." https://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
Contact the MathWorld Team
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