A trapezohedral graph is the skeleton of a trapezohedron. 
-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 |  |
In matching theory, the 
-trapezohedral graph, for 
, is also called a biwheel of order 
(Feng and Zhou 2026). It consists of a cycle
graph of length 
and two hub vertices, each
joined to one of the alternating color classes of the rim. It is a simple brace
graph with exactly 
perfect matchings. Adding
an edge between the hubs gives an extended biwheel
with 
perfect matchings (Feng and Zhou 2026). These graphs
attain the upper bound in the biwheel conjecture.
