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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




© 1996-9 Eric W. Weisstein
1999-05-26