Frucht graph
{{Short description|Cubic graph with 12 vertices and 18 edges}}
{{infobox graph
| name = Frucht graph
| image = 200px
| image_caption = The Frucht graph
| namesake = Robert Frucht
| vertices = 12
| edges = 18
| automorphisms = 1 ({id})
| girth = 3
| radius = 3
| diameter = 4
| chromatic_number = 3
| chromatic_index = 3
| properties = Cubic
Halin
Pancyclic
}}
In the mathematical field of graph theory, the Frucht graph is a cubic graph with 12 vertices, 18 edges, and no nontrivial symmetries.{{MathWorld|urlname=FruchtGraph|title=Frucht Graph|mode=cs2}} It was first described by Robert Frucht in 1949.
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) and Hamiltonian, with girth 3. Its independence number is 5.
The Frucht graph can be constructed from the LCF notation: {{math|[−5,−2,−4,2,5,−2,2,5,−2,−5,4,2]}}.
Algebraic properties
The Frucht graph is one of the five smallest cubic graphs possessing only a single graph automorphism, the identity{{citation|last1=Bussemaker|first1=F. C.|last2=Cobeljic|first2=S.|last3=Cvetkovic|first3=D. M.|last4=Seidel|first4=J. J.|publisher=Dept. of Mathematics and Computing Science, Eindhoven University of Technology|series=EUT report|title=Computer investigation of cubic graphs|url=https://research.tue.nl/en/publications/computer-investigation-of-cubic-graphs|volume=76-WSK-01|year=1976}} (that is, every vertex can be distinguished topologically from every other vertex). 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,{{Citation | last1=Frucht | first1=R. | title=Herstellung von Graphen mit vorgegebener abstrakter Gruppe. | url=http://www.numdam.org/item?id=CM_1939__6__239_0 | language=German | zbl=0020.07804 | year=1939 | journal=Compositio Mathematica | issn=0010-437X | volume=6 | pages=239–250}}. 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.{{Citation | last1=Frucht | first1=R. | title=Graphs of degree three with a given abstract group | doi = 10.4153/CJM-1949-033-6 | mr=0032987 | year=1949 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=1 | issue=4 | pages=365–378| s2cid=124723321 | doi-access=free }}. The Frucht graph provides an example of this strengthened realization for the trivial group.
The characteristic polynomial of the Frucht graph is .
Gallery
File:Frucht_graph_3COL.svg|The chromatic number of the Frucht graph is 3.
File:Frucht Lombardi.svg| The Frucht graph is Hamiltonian.
References
{{commonscat|Frucht graph}}
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