A -crossed prism graph for positive even (a term introduced here for the first time), is a graph obtained by taking two disjoint cycle graphs and adding edges and for , 3, ..., .
The crossed prism graphs are cubic vertex-transitive (and hence appear in Read and Wilson 1998, though without any designation indicating membership in a special graph family), weakly regular, Hamiltonian, and Hamilton-laceable. The -crossed prism graphs are toroidal for (E. Weisstein, May 9, 2023).
Simmons (2014) used the term "polygonal bigraph on vertices" for graphs isomorphic to the -crossed prism graph and investigated the Hamilton-laceability and structure of Hamiltonian paths in these graphs.
The first few crossed prism graphs and some of their properties are implemented in the Wolfram Language as GraphData["CrossedPrism", n].
The -crossed prism graph has independence polynomial
which has recurrence equation
The -crossed prism graph is isomorphic to the Haar graph and to the -honeycomb toroidal graph. Other special cases are summarized in the following table.
-crossed prism graph | |
4 | cubical graph |
6 | Franklin graph |
8 | cubic vertex-transitive graph Ct19 |
10 | cubic vertex-transitive graph Ct29 |
12 | cubic vertex-transitive graph Ct42 |
14 | cubic vertex-transitive graph Ct54 |
16 | cubic vertex-transitive graph Ct74 |