The Frucht graph is a graph on 12 vertices and 18 edges that is one of five smallest (and two smallest planar) cubic identity graphs (cf. Skiena 1990, p. 185). It is illustrated above in a number of embeddings.
The Frucht graph is implemented in the Wolfram Language as GraphData["FruchtGraph"].
The Frucht graph is Hamiltonian and unit-distance.
It has three inequivalent order-1 LCF notations: [
, 
, 
,
2, 5, 
,
2, 5, 
,
, 4, 2], [
, 
,
2, 3, 
,
4, 
, 5, 2, 
, 
,
2], and [
,
2, 
, 
, 2, 3, 
, 5, 
, 2, 4, 
].
The cubic symmetric graph 
, which was the first known cubic 1-arc-transitive
graph, is another graph associated with Frucht (Frucht 1952).
