I am trying to do some Monte Carlo modeling of the properties of particles based on molecules on their surface. There may be from 1,000 to 100,000 surface molecules for a given particle. I need to be able to determine some molecular properties based on adjacency, and then total those properties over all of these surface molecules. It strikes me that to model this, I need a graph that corresponds to the adjacencies of all of the molecules. I am, unfortunately, new to graph theory.
So it seems to me that I need a planar connected graph of size n (where n ranges from 1,000 to 100,000). I think it should be a sextic graph (put a coin on a table and surround it with six coins of the same size to get a close-packed arrangement) to represent the six nearest-neighbors. Although to curve into a sphere we need 12 pentagons, so I again suppose that 12 of the vertices should be pentic. And each face should be of size 3 to model this close packing. Naturally, there will be many such graphs, so to maximize how close the particle is to spherical, I would want to maximize the distance apart the 12 pentagons are.
It strikes me as well from the reading I have done that this graph I am looking for should be the dual of the graph of a fullerene (a triptic planar graph with twelve pentagonal faces and the rest of the faces hexagonal, where the desired n would be the number of faces rather than the number of vertices).
But... I don't know how to generate either graph, or how to find the dual of the fullerene graph if that's the easier route to take. Any suggestions?