Balaban 11-cage

Balaban 11-cage
The Balaban 11-cage
Named after	Alexandru T. Balaban
Vertices	112
Edges	168
Radius	6
Diameter	8
Girth	11
Automorphisms	64
Chromatic number	3
Chromatic index	3
Properties	Cubic Cage Hamiltonian
Table of graphs and parameters

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 Alexandru T. Balaban.

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

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.

It has independence number 52, 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.

The characteristic polynomial of the Balaban 11-cage is:

$(x-3)x^{12}(x^{2}-6)^{5}(x^{2}-2)^{12}(x^{3}-x^{2}-4x+2)^{2}\cdot$

$\cdot (x^{3}+x^{2}-6x-2)(x^{4}-x^{3}-6x^{2}+4x+4)^{4}\cdot$

$\cdot (x^{5}+x^{4}-8x^{3}-6x^{2}+12x+4)^{8}$.

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

Gallery

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  The chromatic number of the Balaban 11-cage is 3.
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  The chromatic index of the Balaban 11-cage is 3.
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  Alternative drawing of the Balaban 11-cage.

References

• Heal, Maher (2016), "A Quadratic Programming Formulation to Find the Maximum Independent Set of Any Graph", The 2016 International Conference on Computational Science and Computational Intelligence, Las Vegas: IEEE Computer Society