The Balaban 10-cage is one of the three cage graphs (Read and Wilson
1998, p. 272). The Balaban et al. 2001). Embeddings of all three possible 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" ].
It is a Hamiltonian graph and has 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
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 )
Explore with Wolfram|Alpha
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 Read, R. C.
and Wilson, R. J. An
Atlas of Graphs. 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. Referenced
on Wolfram|Alpha Balaban 10-Cage
Cite this as:
Pegg, Ed Jr. and Weisstein, Eric W. "Balaban 10-Cage." From MathWorld https://mathworld.wolfram.com/Balaban10-Cage.html
Subject classifications More... Less...
-cage graphs (Read and Wilson
1998, p. 272). The Balaban 
-cage was the first known example of a 10-cage (Balaban
1973, Pisanski et al. 2001). Embeddings of all three possible 
-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).
Hamiltonian cycles. It has 1003 distinct LCF
notations, with four of length two (illustrated above) and 999 of length 1.
