| Balaban 10-cage | |
| Namesake: | Alexandru T. Balaban |
| Vertices: | 70 |
| Edges: | 105 |
| Automorphisms: | 80 |
| Girth: | 10 |
| Diameter: | 6 |
| Radius: | 6 |
| Chromatic Number: | 2 |
| Chromatic Index: | 3 |
| Book Thickness: | 3 |
| Queue Number: | 2 |
| Genus: | 9 |
| Properties: | Cubic Cage Hamiltonian |
| Book Thickness: | 3 |
| Queue Number: | 2 |
In the mathematical field of graph theory, the Balaban 10-cage or Balaban -cage is a 3-regular graph with 70 vertices and 105 edges named after Alexandru T. Balaban. Published in 1972,[1] It was the first 10-cage discovered but it is not unique.[2]
The complete list of 10-cages and the proof of minimality was given by Mary R. O'Keefe and Pak Ken Wong.[3] There exist 3 distinct -cages, the other two being the Harries graph and the Harries–Wong graph.[4] 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 3-edge-connected. The book thickness is 3 and the queue number is 2.[5]
The characteristic polynomial of the Balaban 10-cage is
(x-3)(x-2)(x-1)8x2(x+1)8(x+2)(x+3) ⋅
⋅ (x2-6)2(x2-5)4(x2-2)2(x4-6x2+3)8.
Molecular graph
Balaban 11-cage