Colourings of the C60H60-molecule

## Colourings of the C60H60-molecule

The cycle indices for the action on the set of vertices can be applied to investigate the C60 H60-molecule. Consider a colouring of the 60 vertices C with 2 colours H or Cl. In other words a colouring is a function f from the set of vertices of the truncated icosahedron into the set {H,Cl} . Then Pólyas theorem says that the number of C60HkCl60-k molecules is given as the coefficient of xk in the expansion of the cycle index of the symmetry group R or S acting on the set of vertices when all the indeterminates xi are replaced by 1+xi. (In mathematical terms we are speaking of weighted enumeration when we define the weight of a colouring as a product weight. Let w(H)=x and w(Cl)=1, where x is an indeterminate over the set of rationals. Then the product weight of the colouring f is defined to be
w(f):=Õvf(v),
where the product runs over all vertices v.) In table you can find the numbers of different molecules C60HkCl60-k both for the symmetry groups R and S.

 k S R 0, 60 1 1 1, 59 1 1 2, 58 23 37 3, 57 303 577 4, 56 4190 8236 5, 55 45718 91030 6, 54 418470 835476 7, 53 3.220218 6.436782 8, 52 21.330558 42.650532 9, 51 123.204921 246.386091 10, 50 628.330629 1256.602779

Number of C60HkCl60-k molecules

The substitution xi -> 1+xi into the cycle index is implemented in SYMMETRICA as well. It is called `polya_sub(a,b,c)`, where `a` is the cycle index, `b` is the number of indeterminates xi that should be replaced by 1+xi and `c` is the result of the expansion of the cycle index.

More or less in the same way the numbers of hetero fullerenes, these are fullerenes where some of the carbon atoms are replaced by other atoms, can be computed from the cycle index of the symmetry group acting on the set of vertices. For instance replacing some C atoms by B atoms the numbers in table can be interpreted as the numbers of different C60-kBk molecules.

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

 Colourings of the C60H60-molecule