  After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry,
fullerenes have been subject of much research. One part of that research is the
prediction of a fullerene's stability using topological descriptors. It was
mainly done by considering the distribution of the twelve pentagonal facets on
its surface, calculations mostly were performed on all isomers of $C_{40},
C_{60}$ and $C_{80}$. This paper suggests a novel method for the classification
of combinatorial fullerene isomers using spectral graph theory. The
classification presupposes an invariant scheme for the facets based on the
Schlegel diagram. The main idea is to find clusters of isomers by analyzing
their graph structure of hexagonal facets only. We also show that our
classification scheme can serve as a formal stability criterion, which became
evident from a comparison of our results with recent quantum chemical
calculations. We apply our method to classify all isomers of $C_{60}$ and give
an example of two different cospectral isomers of $C_{44}$. Calculations are
done with MATLAB. The only input for our algorithm is the vector of positions
of pentagons in the facet spiral. These vectors and Schlegel diagrams are
generated with the software package Fullerene.
