Foster cage

From HandWiki
Foster cage
Foster cage.svg
Named afterRonald Martin Foster
Vertices30
Edges75
Radius3
Diameter3
Girth5
Automorphisms30
Chromatic number4
Chromatic index5
PropertiesCage
Table of graphs and parameters

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

[math]\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. }[/math]

References

  1. Weisstein, Eric W.. "Foster Cage". http://mathworld.wolfram.com/FosterCage.html. 
  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 .