The Cycle Indices of the Symmetry Groups of Fullerenes

Abstract

Various forms, including multi-dimensional forms of the cycle index of the symmetry group of the fullerene C₆₀ acting on its sets of vertices, faces, edges etc. are computed and applied for the enumeration of essentially different colourings of the truncated icosahedron. Furthermore we determine the 3-dimensional cycle index of a polyhedron under the leapfrog operation introduced by P.W. Fowler.