A polyhedral graph is completely regular if the dual graph is also regular.
There are only five types. Let 
be the number of graph edges
at each node, 
the number of graph edges at each node of the dual
graph, 
the number of graph vertices, 
the number of graph edges, and
the number of faces in the Platonic solid corresponding
to the given graph. The following table summarizes the completely regular graphs,
which are simply equivalent to the Platonic graphs.
| type |  |  |  |  |  |
| tetrahedral graph | 3 | 3 | 4 | 6 | 4 |
| cubical graph | 3 | 4 | 8 | 12 | 6 |
| dodecahedral graph | 3 | 5 | 20 | 39 | 12 |
| octahedral graph | 4 | 3 | 6 | 12 | 8 |
| icosahedral graph | 5 | 3 | 12 | 30 | 20 |
