there are four archimedean solids with 60 vertices: the truncated icosahedron, the rhombicosidodecahedron, the snub dodecahedron, and the truncated dodecahedron. the skeletons of these polyhedra form 60-node vertex-transitive graphs. there are also two archimedean solids with 60 edges: the snub cube and the icosidodecahedron. the skeleton of the icosidodecahedron forms a 60-edge symmetric graph.