A cubic polyhedral graph is a graph that is both cubic and polyhedral. The numbers of cubical polyhedral
graphs on 
,
4, ... nodes are 0, 1, 1, 2, 5, 14, 50, 233, 1249, ... (OEIS A000109).
The following table summarizes some named cubical polyhedral graphs and classes of graphs, some of which are illustrated above.
 |  |
 |  -prism graph |
 |  -generalized Petersen graph |
| 4 | tetrahedral graph |
| 8 | cubical graph |
| 12 | Frucht graph |
| 12 | truncated tetrahedral graph |
| 18 | truncated prism graph |
| 20 | dodecahedral graph |
| 24 | truncated cubical graph |
| 24 | truncated octahedral graph |
| 38 | Barnette-Bosák-Lederberg graph |
| 42 | Faulkner-Younger graph 42 |
| 42 | Grinberg graph 42 |
| 44 | Faulkner-Younger graph 44 |
| 44 | Grinberg graph 44 |
| 46 | Grinberg graph 46 |
| 46 | Tutte's graph |
| 48 | great rhombicuboctahedral graph |
| 60 | truncated dodecahedral graph |
| 60 | truncated icosahedral graph |
| 94 | Thomassen graph 94 |
| 120 | great rhombicosidodecahedral graph |
| 124 | Grünbaum graph 124 |
Tabulations of cubic polyhedral graphs are commonly limited to those that are triangle-free (e.g., Read and Wilson 1998). The numbers of 
-node triangle-free cubic polyhedral graphs on 
, 2, ... nodes are 0, 0, 0, 1, 1, 2, 5, 12, 34, (OEIS A000103).
The figure above illustrates triangle-free cubic polyhedral graphs up to 18 vertices, together with their notation in Read and Wilson (1998).
