[2] The Frucht graph is a pancyclic, Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4.

The Frucht graph can be constructed from the LCF notation: [−5,−2,−4,2,5,−2,2,5,−2,−5,4,2].

The Frucht graph is one of the five smallest cubic graphs possessing only a single graph automorphism, the identity[3] (that is, every vertex can be distinguished topologically from every other vertex).

Frucht's theorem states that any group can be realized as the group of symmetries of a graph,[2] and a strengthening of this theorem also due to Frucht states that any group can be realized as the symmetries of a 3-regular graph;[4] the Frucht graph provides an example of this realization for the trivial group.

The characteristic polynomial of the Frucht graph is