A hexahedral graph is a polyhedral graph on six vertices. There are seven distinct hexahedral graphs (illustrated above) which, through
duality, correspond to seven convex hexahedra. The
hexahedral graphs were first enumerated by Steiner (1828; Duijvestijn and Federico
1981).

Three of the hexahedral graphs correspond to the skeletons of the pentagonal pyramid (i.e., the wheel
graph ),
triangular prism, and octahedron
(square dipyramid, triangular antiprism).
An additional hexahedron can be obtained by truncation of two of the four apexes
of the tetrahedron, producing a solid composed of
2 triangles, 2 quadrilaterals, and two pentagons which, like the cube, has 6 faces,
8 vertices, 12 edges and cubic connectivity.

Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (3-Connected Planar) Graphs." Math. Comput.37,
523-532, 1981.Guy, R. K. Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.Steiner,
J. "Problème de situation." Ann. de Math19, 36, 1828.
Reprinted in Jacob
Steiner's gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 227, 1971.