# Balaban 10-cage

Balaban 10-cage
The Balaban 10-cage
Named after A. T. Balaban
Vertices 70
Edges 105
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 : $(x-3) (x-2) (x-1)^8 x^2 (x+1)^8 (x+2) (x+3) (x^2-6)^2 (x^2-5)^4 (x^2-2)^2 (x^4-6 x^2+3)^8$.