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{{infobox graph
| name = Frucht graph
| image = [[File:Frucht planar Lombardi.svg|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 graph|Cubic]]<br>[[Planar graph|Planar]]<br>[[Hamiltonian graph|Hamiltonian]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Frucht graph''' is a 3-[[regular graph]] with 12 vertices, 18 edges, and no nontrivial [[graph automorphism|symmetries]].<ref>{{MathWorld|urlname=FruchtGraph|title=Frucht Graph}}</ref> It was first described by [[Robert Frucht]] in 1939.<ref name="f38"/>


The Frucht graph is a [[Halin graph]] with [[chromatic number]] 3, [[chromatic index]] 3, radius 3, diameter 4 and [[Girth (graph theory)|girth]] 3. As with every Halin graph, the Frucht graph is [[planar graph|planar]], 3-[[k-vertex-connected graph|vertex-connected]], and [[polyhedral graph|polyhedral]]. It is also a 3-[[k-edge-connected graph|edge-connected graph]].  Its [[Glossary_of_graph_theory#Independence|independence number]] is 5.


The Frucht graph is [[Hamiltonian graph|Hamiltonian]] and can be constructed from the [[LCF notation]]: [−5,−2,−4,2,5,−2,2,5,−2,−5,4,2].
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==Algebraic properties==
The Frucht graph is one of the two smallest [[cubic graph]]s possessing only a single [[graph automorphism]], the identity<ref>Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990</ref> (that is, every vertex can be distinguished topologically from every other vertex). Such graphs are called [[asymmetric graph|asymmetric]] (or identity) graphs. [[Frucht's theorem]] states that any [[group (mathematics)|group]] can be realized as the group of symmetries of a graph,<ref name="f38">{{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}}.</ref> 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;<ref>{{Citation | last1=Frucht | first1=R. | title=Graphs of degree three with a given abstract group | url=http://cms.math.ca/cjm/v1/p365 | doi = 10.4153/CJM-1949-033-6 | mr=0032987 | year=1949 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=1 | pages=365–378}}.</ref> the Frucht graph provides an example of this realization for the [[trivial group]].
 
The [[characteristic polynomial of a graph|characteristic polynomial]] of the Frucht graph is <math>(x-3) (x-2) x (x+1) (x+2) (x^3+x^2-2 x-1) (x^4+x^3-6 x^2-5 x+4)</math>.
 
==Gallery==
<gallery>
File:Frucht_graph_3COL.svg|The [[chromatic number]] of the Frucht graph is&nbsp;3.
File:Frucht Lombardi.svg| The Frucht graph is [[Hamiltonian graph|Hamiltonian]].
</gallery>
 
== See also ==
{{commonscat|Frucht graph}}
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]
[[Category:Planar graphs]]

Revision as of 02:51, 21 February 2014


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