A prism graph, denoted , (Gallian 1987), or (Hladnik et al. 2002), and sometimes also called a circular ladder graph and denoted (Gross and Yellen 1999, p. 14), 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, and is equivalent to the generalized Petersen graph . For odd , the -prism is isomorphic to the circulant graph , as can be seen by rotating the inner cycle by and increasing its radius to equal that of the outer cycle in the top embeddings above. In addition, for odd , is isomorphic to , , ..., .
is isomorphic to the graph Cartesian product , where is the path graph on two nodes and is the cycle graph on nodes. As a result, it is a unit-distance graph (Horvat and Pisanski 2010).
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 with the cubical graph. The -prism graph is isomorphic to the Haar graph .
Prism graphs are graceful (Gallian 1987, Frucht and Gallian 1988, Gallian 2018).
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 ismorphic 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].