The 13 Archimedean dual graphs are the skeletons of the Archimedean dual solids, illustrated above. Since they are polyhedral graphs, they are also planar. However, none of them are regular.
The following table summarizes properties of the Archimedean dual graphs.
graph | Hamiltonian | Eulerian | vertex-transitive | edge-transitive | |||
deltoidal hexecontahedral graph | 62 | 120 | 120 | no | no | no | no |
deltoidal icositetrahedral graph | 26 | 48 | 48 | no | no | no | no |
disdyakis dodecahedral graph | 26 | 72 | 48 | yes | yes | no | no |
disdyakis triacontahedral graph | 62 | 180 | 120 | yes | yes | no | no |
pentagonal hexecontahedral graph | 92 | 150 | 60 | yes | no | no | no |
pentagonal icositetrahedral graph | 38 | 60 | 24 | yes | no | no | no |
pentakis dodecahedral graph | 32 | 90 | 120 | yes | no | no | no |
rhombic dodecahedral graph | 14 | 24 | 48 | no | no | no | yes |
rhombic triacontahedral graph | 32 | 60 | 120 | no | no | no | yes |
small triakis octahedral graph | 14 | 36 | 48 | no | no | no | no |
tetrakis hexahedral graph | 14 | 36 | 48 | yes | yes | no | no |
triakis icosahedral graph | 32 | 90 | 120 | no | no | no | no |
triakis tetrahedral graph | 8 | 18 | 24 | yes | no | no | no |