Balaban 11-cage

Balaban 11-cage
The Balaban 11-cage
Named after	A. T. Balaban
Vertices	112
Edges	168
Radius	6
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 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 : $(x-3) x^{12} (x^2-6)^5 (x^2-2)^{12} (x^3-x^2-4 x+2)^2\cdot$ $\cdot(x^3+x^2-6 x-2) (x^4-x^3-6 x^2+4 x+4)^4 (x^5+x^4-8 x^3-6 x^2+12 x+4)^8$.

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

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.^[1]

1. ^ P. Eades, J. Marks, P. Mutzel, S. North. "Graph-Drawing Contest Report", TR98-16, December 1998, Mitsubishi Electric Research Laboratories.

References

1. Weisstein, Eric W., "Balaban 11-Cage", MathWorld.
2. Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships Among the Cages." Rev. Roumaine Math. 18, 1033-1043, 1973.
3. Weisstein, Eric W., "Cage Graph", MathWorld.
4. Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008)