"The" graph is the path graph on two vertices: .
An -graph for and is a generalization of a generalized Petersen graph and has vertex set
and edge set
where the subscripts are read modulo (Bouwer et al. 1988, itnik et al. ). Such graphs can be constructed by graph expansion on .
If the restriction is relaxed to allow and to equal , gives the ladder rung graph and gives the sunlet graph .
Two -graphs and are isomorphic iff there exists an integer relatively prime to such that either or (Boben et al. 2005, Horvat et al. 2012, itnik 2012).
The graph is connected iff . If , then the graph consists of copies of (itnik et al. 2012).
The -graph corresponds to copies of the graph
The following table summarizes special named -graphs and classes of named -graphs.
graph | |
cubical graph | |
Petersen graph | |
Dürer graph | |
Möbius-Kantor graph | |
dodecahedral graph | |
Desargues graph | |
Nauru graph | |
prism graph | |
generalized Petersen graph |
All -graphs with have a non-vertex degenerate unit-distance representation in the plane, and with the exception of the families and , the representations can be constructed with -fold rotational symmetry (itnik et al. 2012). While some of these may be vertex-edge degenerate (i.e., an edge passes over a vertex to which it is not incident), computer searching has found only four distinct such cases (, , , and ), and in each case, a different indexing of the I graph gives a unit-distance embedding that is not degenerate in this way (itnik et al. 2012).