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 connectediff . 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.

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).

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