Frucht graph
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In 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: Template:Math. This describes it as a cubic graph in which two of the three adjacencies of each vertex form part of a Hamiltonian cycle and the numbers specify how far along the cycle to find the third neighbor of each vertex.[3]
Properties
The Frucht graph is a cubic graph, because three vertices are incident to every vertex, thereby the degree of every vertex is 3. It 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.[4] 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,[5] 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 Frucht graph is a Halin graph, a type of planar graph formed from a tree with no degree-two vertices by adding a cycle connecting its leaves.[1] Every Halin graph is 3-vertex-connected: deleting two of its vertices cannot disconnect it. By Steinitz's theorem, the Frucht graph is hence polyhedral, meaning its 12 vertices and 18 edges form the skeleton of a convex polyhedron.[6] It is also Hamiltonian.
It is pancyclic,[7] with chromatic number 3, chromatic index 3, radius 3, and diameter 4. Its girth 3. Its independence number is 5.
The characteristic polynomial of the Frucht graph is .
Gallery
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The chromatic number of the Frucht graph is 3. -
The Frucht graph is Hamiltonian.
