# Balaban 11-cage

Balaban 11-cage
The Balaban 11-cage
Named after A. T. Balaban
Vertices 112
Edges 168
Diameter 8
Girth 11
Automorphisms 64
Chromatic number 3
Chromatic index 3
Properties Cubic
Cage
Hamiltonian

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 independence number 52,[5] 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 : ${\displaystyle (x-3)x^{12}(x^{2}-6)^{5}(x^{2}-2)^{12}(x^{3}-x^{2}-4x+2)^{2}\cdot }$ ${\displaystyle \cdot (x^{3}+x^{2}-6x-2)(x^{4}-x^{3}-6x^{2}+4x+4)^{4}(x^{5}+x^{4}-8x^{3}-6x^{2}+12x+4)^{8}}$.

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

## References

1. ^
2. ^ Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships Among the Cages." Rev. Roumaine Math. 18, 1033-1043, 1973.
3. ^
4. ^ a b Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008)
5. ^ Maher Heal (2016)
6. ^ P. Eades, J. Marks, P. Mutzel, S. North. "Graph-Drawing Contest Report", TR98-16, December 1998, Mitsubishi Electric Research Laboratories.

## 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