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 [Austin, S.J, and J. Baker, P. W. Fowler, D. E. Manolopoulos, Bond-stretch Isomerism and the Fullerenes, J. Chem. Soc. Perkin Trans. 2 (1994), 2319–2323] and [Rogers, K.M., and P. W. Fowler, Leapfrog fullerenes, Huckel bond order and Kekulé structures, J. Chem. Soc. Perkin Trans. 2 (2001), 18–22]. All fullerenes with at least ten symmetries were studied in [Graver, J.E. The Structure of Fullerene Signature, DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64, AMS (2005), 137–166.] and a complete catalog was given in [Graver, J. E. Catalog of All Fullerene with Ten or More Symmetries DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64 AMS (2005), 167–188]. Starting from this catalog in [Bogaerts, M., and G. Mazzuoccolo, G.Rinaldi, Totally symmetric Kekulé structures in fullerene graphs with ten or more symmetries, MATCH Communications in Mathematical and in Computer Chemistry 69 (2013), 677–705] we established exactly which of them have at least one totally symmetric Kekulé structure.