Count the paths along the edges from a vertex of a polyhedron to one of its adjacent vertices. You may take a detour, but may not pass through the same vertex twice.
| Polyhedron | Charactaristic of the edge (a) (b) | Number of paths | |
|---|---|---|---|
| Tetrahedron | [3,3] | ⟨3,3⟩ | 5 |
| Truncated tetrahedron | [3,6] | ⟨3,3⟩ | 38 |
| Truncated tetrahedron | [6,6] | ⟨3,3⟩ | 49 |
| Cube | [4,4] | ⟨3,3⟩ | 15 |
| Octahedron | [3,3] | ⟨4,4⟩ | 26 |
| Cuboctahedron | [3,4] | ⟨4,4⟩ | 546 |
| Truncated cube | [3,8] | ⟨3,3⟩ | 1178 |
| Truncated cube | [8,8] | ⟨3,3⟩ | 1569 |
| Truncated octahedron | [4,6] | ⟨3,3⟩ | 1738 |
| Truncated octahedron | [6,6] | ⟨3,3⟩ | 1953 |
| Small rhombicuboctahedron | [3,4] | ⟨4,4⟩ | 315050 |
| Small rhombicuboctahedron | [4,4] | ⟨4,4⟩ | 345326 |
| Great rhombicuboctahedron | [4,6] | ⟨3,3⟩ | 2821146 |
| Great rhombicuboctahedron | [4,8] | ⟨3,3⟩ | 2951002 |
| Great rhombicuboctahedron | [6,8] | ⟨3,3⟩ | 3291385 |
| Snub cube | [3,3] † | ⟨5,5⟩ | 21480681 |
| Snub cube | [3,3] ‡ | ⟨5,5⟩ | 21485947 |
| Snub cube | [3,4] | ⟨5,5⟩ | 22129907 |
| Dodecahedron | [5,5] | ⟨3,3⟩ | 561 |
| Icosahedron | [3,3] | ⟨5,5⟩ | 4151 |
| Icosidodecahedron | [3,5] | ⟨4,4⟩ | 7545354 |
| Truncated dodecahedron | [3,10] | ⟨3,3⟩ | 47057714 |
| Truncated dodecahedron | [10,10] | ⟨3,3⟩ | 62743617 |
| Truncated icosahedron | [5,6] | ⟨3,3⟩ | 181045596 |
| Truncated icosahedron | [6,6] | ⟨3,3⟩ | 188580177 |
| Small rhombicosidodecahedron | [3,4] | ⟨4,4⟩ | 89202613227098 |
| Small rhombicosidodecahedron | [4,5] | ⟨4,4⟩ | 101218512140962 |
| Great rhombicosidodecahedron | [4,6] | ⟨3,3⟩ | 16648953820232074 |
| Great rhombicosidodecahedron | [4,10] | ⟨3,3⟩ | 17761151206070874 |
| Great rhombicosidodecahedron | [6,10] | ⟨3,3⟩ | 19735132861167641 |
| Snub dodecahedron | [3,3] † | ⟨5,5⟩ | 4314181324075421543 |
| Snub dodecahedron | [3,3] ‡ | ⟨5,5⟩ | 4333711459240318005 |
| Snub dodecahedron | [3,5] | ⟨5,5⟩ | 4522374719995599256 |