The resonance
structure of the fullerene C_{60} |

U | |U| | |Ũ | | # | U | |U| | |Ũ | | # |

C_{1} |
1 | 1 | 70 | C_{3v} |
6 | 10 | 3 |

C_{i} |
2 | 1 | 0 | D_{2h} |
8 | 5 | 0 |

C_{2} |
2 | 15 | 19 | C_{5v} |
10 | 6 | 1 |

C_{s} |
2 | 15 | 36 | D_{5} |
10 | 6 | 0 |

C_{3} |
3 | 10 | 7 | C_{5i} |
10 | 6 | 0 |

D_{2} |
4 | 5 | 3 | T | 12 | 5 | 1 |

C_{2v} |
4 | 15 | 5 | D_{3d} |
12 | 10 | 3 |

C_{2h} |
4 | 15 | 3 | D_{5d} |
20 | 6 | 2 |

C_{5} |
5 | 6 | 0 | T_{h} |
24 | 5 | 1 |

D_{3} |
6 | 10 | 2 | I | 60 | 1 | 0 |

C_{3i} |
6 | 10 | 1 | I_{h} |
120 | 1 | 1 |

Resonance structures of the C_{60}
fullerene

The SYMMETRICA routine
`all_orbits_right_from_vector(a,b,c)`

computes a
complete list `c`

of these representatives. The
permutation group acting on the domain of the functions is given by
`a`

, which is a `VECTOR`

of generators, which
must be `PERMUTATION`

-objects. The `VECTOR`

`b`

is a list of all the functions which must be tested
to be a canonical representative or not. The list of all canonical
representatives will be computed as the `VECTOR`

`c`

. For our problem of determining all the
*different* resonance structures we have to take for
`a`

a `VECTOR`

of the generators of the
symmetry group acting on the set of edges of the truncated
icosahedron. And for `b`

we have to take the
`VECTOR`

of all the 12500 resonance structures generated
by the backtrack algorithm described above.

Balasubramanian extensively applied Pólya theory for the
enumeration of isomers. He published a review on chemical and
spectroscopic applications of this theory in [1]. In [2][3][7] he computes the cycle
indices of the symmetry group of C_{60} acting on its sets
of vertices, edges or faces, and he demonstrates how to enumerate
isomers of the form C_{60}H_{n} and
C_{60}H_{n}D_{m}. Furthermore he computes
the numbers of face and edge colourings of C_{60} and
determines the nuclear-spin statistics for C_{60} and
C_{60}H_{60}. When actually computing the numbers
of isomers given in his papers he reports that he had to face
complexity problems and arithmetic overflows occurred. So he had to
implement a double precision arithmetic into his algorithm. When
using SYMMETRICA all these problems do not occur, since SYMMETRICA
is working with integers of arbitrary length and with rational
numbers stored as fractions.

Fujita [15] computes the
numbers of colourings of the truncated icosahedron by
*stabilizer type*. He derives the cycle index of S acting on
the set of vertices of C_{60} by summing over so called
*partial cycle indices* for certain subgroups of S. (For
more details on partial cycle indices see [16].)

harald.fripertinger "at" uni-graz.at, May 16, 2011

The resonance
structure of the fullerene C_{60} |