Balaban 10-cage

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 Alexandru T. Balaban. Published in 1972, It was the first 10-cage discovered but it is not unique.

The complete list of 10-cages and the proof of minimality was given by Mary R. O'Keefe and Pak Ken Wong. There exist 3 distinct $(3,10)$-cages, the other two being the Harries graph and the Harries–Wong graph. 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.

The characteristic polynomial of the Balaban 10-cage is
 * $$(x-3) (x-2) (x-1)^8 x^2 (x+1)^8 (x+2) (x+3) \cdot$$
 * $$\cdot(x^2-6)^2 (x^2-5)^4 (x^2-2)^2 (x^4-6 x^2+3)^8.$$