A polyhedral 6-vertex graph is a polyhedral graph on six vertices. There are seven distinct polyhedral 6-vertex graphs (illustrated above) which, through duality, correspond to seven convex hexahedra. Steiner (1828) enumerated the six-faced polyhedra, as documented by Duijvestijn and Federico (1981).
Three of the polyhedral 6-vertex graphs correspond to the skeletons of the pentagonal pyramid (i.e., the wheel
graph 
),
triangular prism, and octahedron
(square dipyramid, triangular antiprism).
The name "hexahedral graph" is also used for this class (Gayathri 2018, p. 50). This usage differs from the use of the same name for the eight-vertex cubical graph (The Sage Developers).
