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 |