Prism Graph

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].

REFERENCES

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993.

Gallian, J. "Labeling Prisms and Prism Related Graphs." Congr. Numer. 59, 89-100, 1987.

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018.

 https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Gross, J. T. and Yellen, J. Graph Theory and Its Applications. Boca Raton, FL: CRC Press, p. 14, 1999.

Frucht R. and Gallian, J. A. "Labeling Prisms." Ars Combin. 26, 69-82, 1988.

Hladnik, M.; Marušič, D.; and Pisanski, T. "Cyclic Haar Graphs." Disc. Math. 244, 137-153, 2002.

Horvat, B. and Pisanski, T. "Products of Unit Distance Graphs." Disc. Math. 310, 1783-1792, 2010.

Hosoya, H. and Harary, F. "On the Matching Properties of Three Fence Graphs." J. Math. Chem. 12, 211-218, 1993

Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, p. 263 and 270, 1998.

Sloane, N. J. A. Sequences A077265 and A124350 in "The On-Line Encyclopedia of Integer Sequences."