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Balaban 10-Cage

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Balaban10Cage

The Balaban 10-cage is one of the three (3,10)-cage graphs (Read and Wilson 1998, p. 272). The Balaban (3,10)-cage was the first known example of a 10-cage (Balaban 1973, Pisanski et al. 2001). Embeddings of all three possible (3,10)-cages (the others being the Harries graph and Harries-Wong graph) are given by Pisanski et al. (2001). Several embeddings are illustrated above (e.g., Pisanski and Randić 2000).

It is implemented in the Wolfram Language as GraphData["Balaban10Cage"].

Balaban10CageLCF

It is a Hamiltonian graph and has 91440 Hamiltonian cycles. It has 1003 distinct LCF notations, with four of length two (illustrated above) and 999 of length 1.

This graph has graph diameter 6, girth 10, graph radius 6, chromatic number 2, edge connectivity 3, vertex connectivity, edge chromatic number 3, and is Hamiltonian and bipartite but not planar. It has automorphism group order 80 (Pisanski et al. 2001). Its graph spectrum is given by

(-3)^1(-sqrt(6))^2(-sqrt(3+sqrt(6)))^8(-sqrt(5))^4(-2)^1(-sqrt(2))^2(-1)^8(-sqrt(3-sqrt(6)))^80^2(sqrt(3-sqrt(6)))^81^8(sqrt(2))^22^1(sqrt(5))^4(sqrt(3+sqrt(6)))^8(sqrt(6))^23^1.
Balaban10CageMatrices

The plots above show the adjacency, incidence, and distance matrices of the graph.

See also

Balaban 11-Cage, Cage Graph, Harries Graph, Harries-Wong Graph

Portions of this entry contributed by Ed Pegg, Jr. (author's link)

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References

Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships Among the Cages." Rev. Roumaine Math. 18, 1033-1043, 1973.Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. http://citeseer.ist.psu.edu/448980.html.Pisanski, T. and Randić, M. "Bridges between Geometry and Graph Theory." In Geometry at Work: A Collection of Papers Showing Applications of Geometry (Ed. C. A. Gorini). Washington, DC: Math. Assoc. Amer., pp. 174-194, 2000.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998.Royle, G. "Cubic Cages." http://school.maths.uwa.edu.au/~gordon/remote/cages/.Wong, P. K. "Cages--A Survey." J. Graph Th. 6, 1-22, 1982.

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Balaban 10-Cage

Cite this as:

Pegg, Ed Jr. and Weisstein, Eric W. "Balaban 10-Cage." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Balaban10-Cage.html

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