The Cycle Index of the Symmetry Group of the Fullerene C60

Harald Fripertinger 1

February 19, 1996

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. The implementation of these cycle index methods in the computer algebra system SYMMETRICA is discussed, and finally cycle indices of the symmetry groups of some further fullerenes are derived.
  • Introduction
  • Some cycle indices
  • The symmetry group of the fullerene C60
  • The action on the set of vertices
  • The action on the set of edges
  • The action on the set of faces
  • Some counting problems
  • Colourings of the fullerene C60
  • Colourings of the C60H60-molecule
  • The resonance structure of the fullerene C60
  • A multi-dimensional cycle index
  • Implementation in SYMMETRICA
  • Some further fullerenes
  • The fullerene C20
  • The Leapfrog principle
  • The fullerene C70
  • Further fullerenes
  • Some numerical examples using SYMMETRICA
  • Generators of the symmetry groups of fullerenes
  • Cycle indices of the symmetry groups of fullerenes
  • Number of different colourings of fullerenes
  • All different colourings of fullerenes with k colours
  • Some multi-dimensional cycle indices
  • The resonance structure of C60
  • References
  • Address of the author
  • Footnotes

  • harald.fripertinger@kfunigraz.ac.at,
    last changed: January 23, 2001