Symmetric graphs constructed as the state spaces of rolling dice of different shapes: https://math.stackexchange.com/questions/2972454/rolling-icosahedron-hamiltonian-path

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?