C60 double bonding networks

C60 (or buckminsterfullerene) has no hydrogens, so it must have quite a few double bonds. I am beginning to understand that bond orders are in some sense a simplification, and I suppose that the bonding is in some way delocalised across the sphere. However, here is a picture of two different bonding patterns:

(click for bigger, as usual). The 'ChEBI' bonding pattern on the left is from the molfile in a ChEBI entry while the 'radial' bonding one on the right is bonded according to schemes from this site which has an interesting graph-theory perspective on fullerenes.

The radial bonding version has a simpler, layered structure like this:

Ok, so that's a slightly comical picture of the slices. What is also nice is the quotient graph for the ChEBI structure:

I didn't color this, but what is great is that it looks like a subgraph of a fullerene! Apart from the loops along the top and bottom, of course.