Invariant Kekulé structures in fullerene graphs

Fullerene graphs are trivalent plane graphs with only hexagonal and pentagonal faces. They are often used to model large carbon molecules. A totally symmetric Kekule structure in a fullerene graph is a set of independent edges which is fixed by each automorphism of the fullerene. Starting from the complete catalog of all fullerenes with at least ten symmetries, we establish exactly which of them have at least one totally symmetric Kekule structure.