The Frucht graph is smallest cubic identity graph (Skiena 1990, p. 185). It is implemented in the Wolfram Language as GraphData["FruchtGraph"]. It has 12 vertices and 18 edges. It is also both planar and Hamiltonian.
It has inequivalent 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).
