Frucht graph

Frucht graph
The Frucht graph
Named after	Robert Frucht
Vertices	12
Edges	18
Radius	3
Diameter	4
Girth	3
Automorphisms	1 ({id})
Chromatic number	3
Chromatic index	3
Properties	Cubic Halin Pancyclic
Table of graphs and parameters

In the mathematical field of graph theory, the Frucht graph is a cubic graph with 12 vertices, 18 edges, and no nontrivial symmetries.^[1] It was first described by Robert Frucht in 1949.^[2]

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: every vertex can be distinguished topologically from every other vertex.^[3] Such graphs are called asymmetric (or identity) graphs. Frucht's theorem states that any finite group can be realized as the group of symmetries of a graph,^[4] and a strengthening of this theorem, also due to Frucht, states that any finite group can be realized as the symmetries of a 3-regular graph.^[2] The Frucht graph provides an example of this strengthened realization for the trivial group.

The characteristic polynomial of the Frucht graph is ${\displaystyle (x-3)(x-2)x(x+1)(x+2)(x^{3}+x^{2}-2x-1)(x^{4}+x^{3}-6x^{2}-5x+4)}$.

The Frucht graph is a pancyclic, Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4. Like every Halin graph, the Frucht graph is polyhedral (planar and 3-vertex-connected)^[5] and Hamiltonian, with girth 3. Its independence number is 5.

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  The chromatic number of the Frucht graph is 3.
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  The Frucht graph is Hamiltonian.

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