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{{infobox graph
| name = McGee Graph
| image = [[Image: McGee graph hamiltonian.svg|220px]]
| image_caption = The McGee Graph
| namesake = W. F. McGee
| vertices = 24
| edges = 36
| automorphisms = 32<ref name="Mathworld" />
| girth = 7<ref name="Mathworld" />
| diameter = 4<ref name="Mathworld" />
| radius = 4
| chromatic_number = 3<ref name="Mathworld" />
| chromatic_index = 3<ref name="Mathworld" />
| properties = [[Cubic graph|Cubic]]<br>[[Cage (graph theory)|Cage]]<br>[[Hamiltonian graph|Hamiltonian]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''McGee Graph''' or the '''(3-7)-cage''' is a 3-[[regular graph]] with 24 vertices and 36 edges.<ref name="Mathworld">{{MathWorld|urlname=McGeeGraph|title=McGee Graph}}</ref>


The McGeeGraph is the unique (3,7)-[[cage graph|cage]] (the smallest [[cubic graph]] of girth 7). It is also the smallest cubic cage that is not a [[Moore graph]].


First discovered by Sachs but unpublished,<ref>Kárteszi, F. "Piani finit ciclici come risoluzioni di un certo problemo di minimo." Boll. Un. Mat. Ital. 15, 522-528, 1960</ref> the graph is named after McGee who published the result in 1960.<ref>McGee, W. F. "A Minimal Cubic Graph of Girth Seven." Canad. Math. Bull. 3, 149-152, 1960</ref> Then, the McGee graph was the proven the unique (3,7)-cage by Tutte in 1966.<ref>Tutte, W. T. Connectivity in Graphs. Toronto, Ontario: University of Toronto Press, 1966</ref><ref>Wong, P. K. "Cages--A Survey." J. Graph Th. 6, 1-22, 1982</ref><ref>Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, p. 209, 1989</ref>
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The smallest cubic graphs with crossing numbers 1–8 are known {{OEIS|id=A110507}}. The smallest 8-crossing graph is the McGee graph. There exists 5 non-isomorphic cubic graphs of order 24 with crossing number 8.<ref>Pegg, E. T. and Exoo, G. "Crossing Number Graphs." Mathematica J. 11, 2009</ref> One of them is the [[generalized Petersen graph]] {{nobr|''G''(12,5)}}, also known as the [[Nauru graph]].<ref>{{MathWorld|urlname=GraphCrossingNumber|title=Graph Crossing Number}}</ref>
 
The McGeeGraph has radius 4, diameter 4, [[chromatic number]] 3 and [[chromatic index]] 3. It is also a 3-[[k-vertex-connected graph|vertex-connected]] and a 3-[[k-edge-connected graph|edge-connected]] graph.
 
==Algebraic properties==
The [[characteristic polynomial]] of the McGeeGraph graph is : <math>x^3(x-3)(x-2)^3(x+1)^2(x+2)(x^2+x-4)(x^3+x^2-4x-2)^4</math>.  
 
The automorphism group of the McGee graph is of order 32 and doesn't acts transitively upon its vertices:  there are two vertex orbits of lengths 8 and 16. The McGee is the smallest cubic cage that is not a [[vertex-transitive graph]].<ref>Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.</ref>
 
==Gallery==
<gallery>
Image:McGee graph crossing number.svg|The [[Crossing number (graph theory)|crossing number]] of the McGee graph is&nbsp;8.
Image:McGee graph 3COL.svg |The [[chromatic number]] of the McGee graph is&nbsp;3.
Image:McGee graph  3color edge.svg|The [[chromatic index]] of the McGee graph is&nbsp;3.
Image:Acyclic_coloring.svg|The [[Acyclic coloring|acyclic chromatic number]] of the McGee graph is&nbsp;3.
Image:McGee graph.svg|Alternative drawing of the McGee graph.
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Latest revision as of 12:32, 5 May 2014


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