Symmetric graphs constructed as the state spaces of rolling dice of different shapes:
It doesn't say so in the post, but Ed Pegg pointed out separately to me that if you do this with a regular octahedron (d8) you get the Nauru graph. A dodecahedron (d12) should get you a nice 5-regular 120-vertex graph (because each face has 10 orientations) – anyone have any idea what's known about this graph?

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