This entry is the first in a series of two posts on applications of topological graph theory to chemistry.
What is a Fullerene?
The geometry of a (pre-2006) football is familiar to most: it is a truncated icosahedron with 12 pentagonal and 20 hexagonal faces. What is maybe less familiar is that the truncated icosahedron appears naturally as a carbon molecule, the C60 Buckminster-fullerene.
Figure 1. A C60 fullerene (the Buckminster-fullerene). Author: Benjah-bmm27.
A fullerene is a 2-dimensional lattice of carbon atoms folded into a 3-dimensional shape like that of a ball or a cylinder. The cylindrical fullerenes are better known as carbon nanotubes.
We have additional restrictions on the lattice: every atom of carbon has to bond with exactly 3 other atoms, and the rings formed by the carbon atoms can only be pentagonal or hexagonal.
Figure 2. A lattice for a C26 fullerene. This lattice is folded over a sphere in such a way that the “outer” face forms a pentagon.
The C60 Buckminster-fullerene was the first fullerene to be discovered; its discovery is due to Kroto, Heath, O’Brien, Curl and Smalley, in 1985. In 1996, Kroto, Curl and Smalley were awarded the Nobel Prize in chemistry for their study of fullerenes.
Combinatorial Properties of Fullerenes
As mathematicians we probably cannot give much insight into the chemistry of fullerenes, however we can say a lot about their combinatorial properties.
The spherical fullerenes with V atoms, E bonds and F=P+H pentagonal and hexagonal faces satisfy the relation
known as the Euler’s polyhedron formula. This formula already tells us a lot about fullerenes: suppose that a spherical fullerene has P>0 pentagonal and H>0 hexagonal faces. Then the number of atoms must be V=(5P+6H)/3 since we’re counting every atom three times; the number of bonds must be E=(5P+6H)/2 since every bond is counted twice, and the number of faces is clearly F=P+H. Furthermore, the Euler’s polyhedron formula tells us that
Hence spherical fullerenes must have exactly 12 pentagonal faces!
This result greatly simplifies the enumeration of all spherical fullerenes. Using additional concepts from topological graph theory, Brinkmann and Dress devised a sufficiently fast algorithm for enumerating spherical fullerenes; their results are presented in Table 1.
Note that the C60 fullerene has 1812 chemical isomeres; the one discovered by Kroto et al. is special among those: it is the only isomer in which the pentagonal faces are not adjacent.
Such fullerenes are called IPR-fullerenes, where IPR stands for Isolated Pentagon Rule, and are chemically more stable. The algorithm of Brinkmann and Dress enumerates IPR-fullerenes as well; their results are presented in Table 2.
 G. Brinkmann, A. Dress. A Constructive Enumeration of Fullerenes. Journal of Algorithms 23, 345-358 (1997).
 E.A. Lord, A.L. Mackay, S. Ranganathan. New Geometries for New Materials. Cambridge University Press, Cambridge, 2006.
Author: Goran Malic, email@example.com