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{{infobox graph
| name = Balaban 11-cage
| image = [[Image:Balaban 11-cage.svg|240px]]
| image_caption = The Balaban 11-cage
| namesake = A. T. Balaban
| vertices = 112
| edges = 168
| automorphisms = 64
| girth = 11
| radius = 6
| diameter = 8
| chromatic_number = 3
| chromatic_index = 3
| properties = [[Cubic graph|Cubic]]<br>[[Cage (graph theory)|Cage]]<br>[[Hamiltonian graph|Hamiltonian]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Balaban 11-cage''' or '''Balaban (3-11)-cage''' is a 3-[[regular graph]] with 112 vertices and 168 edges named after [[Alexandru Balaban|A. T. Balaban]].<ref>{{MathWorld|urlname=Balaban11-Cage|title=Balaban 11-Cage}}</ref>


The Balaban 11-cage is the unique [[cage graph|(3-11)-cage]]. It was discovered by Balaban in 1973.<ref>Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships Among the Cages." Rev. Roumaine Math. 18, 1033-1043, 1973.</ref> The uniqueness was proved by McKay and Myrvold in 2003.<ref>{{MathWorld|urlname=CageGraph|title=Cage Graph}}</ref>


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The Balaban 11-cage is a [[Hamiltonian graph]] and can be constructed by excision from the [[Tutte 12-cage]] by removing a small subtree and suppressing the resulting vertices of degree two.<ref name="SURV">Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008)</ref>
 
It has chromatic number 3, chromatic index 3, radius 6, diameter 8 and girth 11. It is also a 3-[[k-vertex-connected graph|vertex-connected graph]] and a 3-[[k-edge-connected graph|edge-connected graph]].
 
==Algebraic properties==
The [[characteristic polynomial]] of the Balaban 11-cage is :
<math>(x-3) x^{12} (x^2-6)^5 (x^2-2)^{12} (x^3-x^2-4 x+2)^2</math> <math>(x^3+x^2-6 x-2) (x^4-x^3-6 x^2+4 x+4)^4 (x^5+x^4-8 x^3-6 x^2+12 x+4)^8</math>.
 
The automorphism group of the Balaban 11-cage is of order 64.<ref name="SURV"/>
 
==Gallery==
<gallery>
Image:balaban_11-cage_3COL.svg|The [[chromatic number]] of the Balaban 11-cage is 3.
Image:balaban_11-cage_3color_edge.svg|The [[chromatic index]] of the Balaban 11-cage is 3.
Image: balaban_11-cage_alternative_drawing.svg|Alternative drawing of the Balaban 11-cage.<ref>[[Peter Eades|P. Eades]], J. Marks, P. Mutzel, S. North. "Graph-Drawing Contest Report", TR98-16, December 1998, Mitsubishi Electric Research Laboratories.</ref>
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Revision as of 21:49, 12 April 2013

Template:Infobox graph In the mathematical field of graph theory, the Balaban 11-cage or Balaban (3-11)-cage is a 3-regular graph with 112 vertices and 168 edges named after A. T. Balaban.[1]

The Balaban 11-cage is the unique (3-11)-cage. It was discovered by Balaban in 1973.[2] The uniqueness was proved by McKay and Myrvold in 2003.[3]

The Balaban 11-cage is a Hamiltonian graph and can be constructed by excision from the Tutte 12-cage by removing a small subtree and suppressing the resulting vertices of degree two.[4]

It has chromatic number 3, chromatic index 3, radius 6, diameter 8 and girth 11. It is also a 3-vertex-connected graph and a 3-edge-connected graph.

Algebraic properties

The characteristic polynomial of the Balaban 11-cage is : .

The automorphism group of the Balaban 11-cage is of order 64.[4]

Gallery

References

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  1. ↑

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  2. ↑ Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships Among the Cages." Rev. Roumaine Math. 18, 1033-1043, 1973.
  3. ↑

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  4. ↑ 4.0 4.1 Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008)
  5. ↑ P. Eades, J. Marks, P. Mutzel, S. North. "Graph-Drawing Contest Report", TR98-16, December 1998, Mitsubishi Electric Research Laboratories.
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