Number of ways to choose 30 nonadjacent edges from Truncated Icosahedron

Question

While working with a molecular design of Buckminsterfullerene C60 which I can relate to Truncated Icosahedron, I found an interesting fact. If you see the picture below it has 30 yellow edges that are representing double bonds and 60 red edges that are representing single bonds. The rule here is every 60 vertex should connect to 1 yellow edge (double bond) and 2 red edges(single bonds). Until now I thought the model below was the only solution that satisfies the rule.

However I recently found another model that is linked to Wikipedia page as an interactive 3D model that has a different arrangement. You can see some yellow edges(double bonds) on a pentagon side whereas in the first model all edges making up pentagons were red(single bond).

Although edges(bonds) can be arranged in many different ways, eventually they satisfy these rules:

1. Total number of yellow edges(double bonds) are exactly 30.
2. None of the 30 yellow edges(double bonds) shares the same vertex with one another. In other words, none of the yellow edges(double bonds) are adjacent to each other.

But for the sake of this question, distinguishing between single bonds and double bonds or coloring them seems irrelevant.

So my question is:
How can I get the total number of ways to choose 30 edges from Truncated Icosahedron so that all chosen edges are nonadjacent? While counting all the arrangements that can be derived by rotating another arrangement as 1.

Answer 1

This is a partial answer. According to Hosoya's paper Matching and symmetry of graphs, the number of perfect matchings on the truncated icosahedral graph is equal to $$2^2\cdot 5^5=12500.$$ How many of them are distinct up to rotations?

The rotation group $G$ of the truncated icosahedron has $60$ elements:

• identity
• 12 rotations by 72°
• 12 rotations by 144°
• 20 rotations by 120°
• 15 rotations by 180°

Hence, by Burnside's Lemma, the number of distinct matchings is $$\frac{12500+\sum_{g\in G, g\not=\text{id}}|\text{Fix}(g)|}{60}.$$

Computing $|\text{Fix}(g)|$, i.e. the cardinality of the set of matchings which are invariant by $g$ for each $g\in G$ when $g$ is not the identity might be a bit annoying.

P.S. The matching given in your first picture is invariant for all $g\in G$. On the other hand, the matching given in your second picture is not invariant for the 12 rotations by 72° and for the 12 rotations by 144°.