Balaban 10-cage

Balaban 10-cage
The Balaban 10-cage
Named after	A. T. Balaban
Vertices	70
Edges	105
Radius	6
Diameter	6
Girth	10
Automorphisms	80
Chromatic number	2
Chromatic index	3
Properties	Cubic Cage Hamiltonian

In the mathematical field of graph theory, the Balaban 10-cage or Balaban (3,10)-cage is a 3-regular graph with 70 vertices and 105 edges named after A. T. Balaban.^[1] Published in 1972,^[2] It was the first (3,10)-cage discovered but is not unique.^[3]

The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong.^[4] There exists 3 distinct (3-10)-cages, the other two being the Harries graph and the Harries–Wong graph.^[5] Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.

The Balaban 10-cage has chromatic number 2, chromatic index 3, diameter 6, girth 10 and is hamiltonian. It is also a 3-vertex-connected graph and a 3-edge-connected graph.

The characteristic polynomial of the Balaban 10-cage is

${\displaystyle (x-3)(x-2)(x-1)^{8}x^{2}(x+1)^{8}(x+2)(x+3)(x^{2}-6)^{2}(x^{2}-5)^{4}(x^{2}-2)^{2}(x^{4}-6x^{2}+3)^{8}.}$

Gallery

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  The chromatic number of the Balaban 10-cage is 2.

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  The chromatic index of the Balaban 10-cage is 3.

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  Alternative drawing of the Balaban 10-cage.

See also

Molecular graph

References

1. Weisstein, Eric W. "Balaban 10-Cage". MathWorld.
2. A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1-5. 1972.
3. Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. .
4. M. O'Keefe and P.K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory Ser. B 29 (1980) 91–105.
5. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.