Totally Symmetric Kekule structures in fullerene graphs with ten or more symmetries

Graph Theoretic fullerenes are designed to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric kekule structure in a fullerene graph is a set of independent edges which is fixed by all symmetries of the fullerene. It was recently suggested that molecules with totally symmetric kekule structures could have physical and chemical properties. Starting from a catalog given by J.Graver, we study all graph theoretic fullerenes with at least ten symmetries and we establish exaclty which of them have a totally symmetric kekule structure.