Balaban 10 cage - Alchetron, The Free Social Encyclopedia

Named after A. T. Balaban Edges 105 Diameter 6		Vertices 70 Radius 6 Girth 10

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. Published in 1972, It was the first (3,10)-cage discovered but is not unique.

The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong. There exists 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 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 .

References

Balaban 10-cage Wikipedia

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