A prism graph is a graph corresponding to the skeleton of an 
-prism. Prism
graphs are therefore both planar and polyhedral.
An 
-prism graph has 
nodes and 
edges. The 
-prism graph is isomorphic to the generalized
Petersen graph 
as well as to the graph Cartesian product 
, where 
is the path graph on two nodes and 
is the cycle graph on 
nodes.
Prism graphs are therefore a special case of KC graph.
There appears to be no standard notation for the prism graph. It is variously denoted 
(this work), 
(Gallian 1987), 
(Hladnik et al. 2002), or 
(Gross and Yellen 1999, p. 14, standing for "circular
ladder").
For odd 
,
the 
-prism is isomorphic to the circulant
graph 
,
as can be seen by rotating the inner graph cycle by
and increasing its radius
to equal that of the outer graph cycle in the top
drawings above. More generally, for odd 
, 
is isomorphic to the circulant graph 
whenever 
, where 
is the greatest common
divisor.
The prism graph 
is equivalent to the Cayley graph of the dihedral
group 
with respect to the generating set 
(Biggs 1993, p. 126).
The prism graph 
is the line graph of the complete
bipartite graph 
.
The prism graph 
is isomorphic to the cubical graph. The 
-prism graph is isomorphic to the Haar
graph 
.
As a result of being the graph Cartesian product of unit-distance graphs, a prism graph is itself a unit-distance graph (Horvat and Pisanski 2010).
Prism graphs are graceful (Gallian 1987, Frucht and Gallian 1988, Gallian 2025).
The numbers of directed Hamiltonian paths on the 
-prism graph for 
, 4, ... are 60, 144, 260, 456, 700, 1056, 1476, ... (OEIS
A124350), which has the beautiful closed form
 |
where 
is the floor function (M. Alekseyev, pers.
comm., Feb. 7, 2008).
The numbers of graph cycles on the 
-prism graph for 
, 4, ... are 14, 28, 52, 94, 170, ... (OEIS A077265),
illustrated above for 
.
The graph Cartesian product 
is isomorphic to the torus
grid graph 
.
The bipartite double graph of prism graph 
for 
odd is the prism graph 
.
Precomputed properties of prism graphs are available in the Wolfram Language as GraphData[
"Prism", n
].
The generalization of the prism graph to the graph Cartesian product 
may be known as a stacked prism graph.
