Completely Regular Graph

A Polyhedral Graph is completely regular if the Dual Graph is also Regular. There are only five types. Let $\rho$ be the number of Edges at each node, $\rho^*$ the number of Edges at each node of the Dual Graph, $V$ the number of Vertices, $E$ the number of Edges, and $F$ the number of faces in the Platonic Solid corresponding to the given graph. The following table summarizes the completely regular graphs.

Type	$\rho$	$\rho^*$	$V$	$E$	$F$
Tetrahedral	3	3	4	6	4
Cubical	3	4	8	12	6
Dodecahedral	3	5	20	39	12
Octahedral	4	3	6	12	8
Icosahedral	5	3	12	30	20