Foster cage

Foster cage
Named after	Ronald Martin Foster
Vertices	30
Edges	75
Radius	3
Diameter	3
Girth	5
Automorphisms	30
Chromatic number	4
Chromatic index	5
Properties	Cage

In the mathematical field of graph theory, the Foster cage is a 5-regular undirected graph with 30 vertices and 75 edges.^[1][2] It is one of the four (5,5)-cage graphs, the others being the Meringer graph, the Robertson–Wegner graph, and the Wong graph.

Like the unrelated Foster graph, it is named after R. M. Foster.

It has chromatic number 4, diameter 3, and is 5-vertex-connected.

Algebraic properties

The characteristic polynomial of the Foster cage is

${\displaystyle (x-5)(x+1)(x^{2}-5)^{2}(x^{2}+2x-4)^{2}(x-2)^{4}(x^{4}+2x^{3}-6x^{2}-7x+11)^{4}.}$

References

1. Weisstein, Eric W. "Foster Cage". MathWorld.
2. Meringer, Markus (1999), "Fast generation of regular graphs and construction of cages", Journal of Graph Theory, 30 (2): 137–146, doi:10.1002/(SICI)1097-0118(199902)30:2<137::AID-JGT7>3.0.CO;2-G, MR 1665972.