The skeleton of a trapezohedron may be termed a trapezohedral graph. 
-trapezohedral graphs are illustrated above for 
to 10 in circular drawings with the two interior points
corresponding to the trapezohedron apices. The case 
corresponds to the cubical
graph.
The trapezohedral graphs are bipartite, Hamiltonian, perfect, planar, polyhedral, traceable, triangle-free, and uniquely colorable.
The 
-trapezohedral
graph has the following property counts.
| property | count |
| graph cycle count |  |
| edge count |  |
| graph face count |  |
| Hamiltonian cycle count |  |
| Hamiltonian path count |  |
| graph path count |  |
| vertex count |  |
