Here is an alternative method for generating an idealized model of the newly discovered
B
80
buckyball. First, generate the vertex coordinates and face indices for the truncated icosahedron, using PolyhedronData:
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verts=PolyhedronData["TruncatedIcosahedron","VertexCoordinates"];
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fac=PolyhedronData["TruncatedIcosahedron","FaceIndices"];
Get the circumradius of the truncated icosahedron:
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ρ=PolyhedronData["TruncatedIcosahedron","Circumradius"]
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1
4
58+18
5
Pick the hexagonal faces:
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hexid=Select[fac,Length[#]==6&];
To find the 20 new vertices, take the centroids of the hexagonal faces, and project them to the circumsphere of the truncated icosahedron:
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np=ToRadicals[RootReduce[ρNormalize[Mean[verts[[#]]]]]&/@hexid];
We can now build the required Polyhedron through a bit of reorganization:
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nvi=Length[verts]+Range[Length[np]];
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tfac=Flatten[MapThread[Map[Append[#2],Partition[#1,2,1,1]]&,{hexid,nvi}],1];
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b80=Polyhedron[Join[verts,np],Join[Select[fac,Length[#]==5&],tfac]]
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Polyhedron
Number of points: 80
Number of faces: 132

Show the polyhedron:
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Graphics3D[b80,BoxedFalse]
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Use MeshConnectivityGraph to generate the skeleton graph corresponding to the polyhedron:
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b80g=MeshConnectivityGraph[b80,0]
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We can now proceed to do graph operations on the skeletal graph, like visualizing the adjacency matrix:
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MatrixPlot[AdjacencyMatrix[b80g]]
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