They may also be constructed as the Cartesian product of a cycle graph with a single edge.
It can be generated by two elements, a rotation by an angle of 2π/n and a single reflection, and its Cayley graph with this generating set is the prism graph.
(where r is a rotation and f is a reflection or flip) and the Cayley graph has r and f (or r, r−1, and f) as its generators.
However, this construction does not work for even values of n.[1] The graph of an n-gonal prism has 2n vertices and 3n edges.
[3] The number of spanning trees of an n-gonal prism graph is given by the formula[4] For n = 3, 4, 5, ... these numbers are The n-gonal prism graphs for even values of n are partial cubes.
They form one of the few known infinite families of cubic partial cubes, and (except for four sporadic examples) the only vertex-transitive cubic partial cubes.
[5] The pentagonal prism is one of the forbidden minors for the graphs of treewidth three.
If the two cycles of a prism graph are broken by the removal of a single edge in the same position in both cycles, the result is a ladder graph.