The skeleton of a trapezohedron may be termed a trapezohedral graph. 
-trapezohedral graphs are illustrated above for 
to 10 in circular embeddings 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 |
| cycle count |  |
| edge count |  |
| face count |  |
| Hamiltonian cycle count |  |
| Hamiltonian path count |  |
| path count |  |
| vertex count |  |
