Frucht graph
In the mathematical field of graph theory, the Frucht graph is a 3-regular graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1939.
The Frucht graph is a pancyclic Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4. As with every Halin graph, the Frucht graph is polyhedral and Hamiltonian, with girth 3. Its independence number is 5.
The Frucht graph can be constructed from the LCF notation: .The Frucht graph is one of the five smallest cubic graphs possessing only a single graph automorphism, the identity. Such graphs are called asymmetric graphs. Frucht's theorem states that any group can be realized as the group of symmetries of a graph, 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; the Frucht graph provides an example of this realization for the trivial group.
The characteristic polynomial of the Frucht graph is.Gallery
