I Graph

"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, Žitnik 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, Žitnik 2012).

The graph is connected iff . If , then the graph consists of copies of (Žitnik 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 (Žitnik 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 (Žitnik et al. 2012).