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.
Canonical polyhedra corresponding to fullerenes
on 20 to 34 vertices are illustrated above.
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.
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. Algorithms23,
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."
Carbon30, 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. Acta78, 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."
,
graph on 26 vertices given by Gosil and Royle (2001, p. 208), truncated
icosahedral graph, and stable molecule 
(Babić et al. 2002), illustrated above.
vertices is 
-regular, and it has precisely 12 odd chordless cycles,
all of them of order 5.
,
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.
-Goldberg
polyhedra) and type II (isomorphic to the skeletons
of 
-Goldberg
polyhedra) are implemented in the Wolfram
Language as BuckyballGraph[n,
"I"] and BuckyballGraph[n,
"II"], respectively.
