The Balaban 10-cage is one of the three 
-cage graphs (Read and Wilson
1998, p. 272). The Balaban 
-cage was the first known example of a 10-cage
graph (Balaban 1973, Pisanski et al. 2004). Drawings of all three possible
-cages
(the others being the Harries graph and Harries-Wong
graph) are given by Pisanski et al. (2004). Several drawings 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 
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. 2004). Its graph spectrum is given by
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The plots above show the adjacency, incidence, and distance matrices of the graph.
