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: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric Kekulé structure in a fullerene graph is a set of independent edges which is fixed by all symmetries of the fullerene and molecules with totally symmetric Kekulé structures could have special physical and chemical properties, as suggested in [1] and [8]. All fullerenes with at least ten symmetries were studied in [4] and a complete catalog was given in [5]. Starting from this catalog in [2] we established exactly which of them have at least one totally symmetric Kekulé structure.