# Foster cage

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{{infobox graph
 | name = Foster cage
 | image = 170px
 | namesake = [Ronald Martin Foster](/source/R._M._Foster)
 | vertices = 30
 | edges = 75
 | automorphisms = 30
 | girth = 5
 | diameter = 3
 | radius = 3
 | chromatic_number = 4
 | chromatic_index = 5
 | properties = [Cage](/source/Cage_(graph_theory))
}}
In the [mathematical](/source/mathematics) field of [graph theory](/source/graph_theory), the '''Foster cage''' is a 5-[regular](/source/regular_graph) [undirected graph](/source/undirected_graph) with 30 vertices and 75 edges.<ref>{{MathWorld|urlname=FosterCage|title=Foster Cage}}</ref><ref>{{citation
 | last = Meringer | first = Markus
 | doi = 10.1002/(SICI)1097-0118(199902)30:2<137::AID-JGT7>3.0.CO;2-G
 | issue = 2
 | journal = [Journal of Graph Theory](/source/Journal_of_Graph_Theory)
 | mr = 1665972
 | pages = 137–146
 | title = Fast generation of regular graphs and construction of cages
 | volume = 30
 | year = 1999}}.</ref> It is one of the four [(5,5)-cage graphs](/source/cage_graph), the others being the [Meringer graph](/source/Meringer_graph), the [Robertson–Wegner graph](/source/Robertson%E2%80%93Wegner_graph), and the [Wong graph](/source/Wong_graph).

Like the unrelated [Foster graph](/source/Foster_graph), it is named after [R. M. Foster](/source/R._M._Foster).

It has [chromatic number](/source/chromatic_number) 4, diameter 3, and is 5-[vertex-connected](/source/k-vertex-connected_graph). 

==Algebraic properties==
The [characteristic polynomial](/source/characteristic_polynomial) of the Foster cage is

: <math>(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 ==
{{reflist}}

Category:Individual graphs
Category:Regular graphs

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Adapted from the Wikipedia article [Foster cage](https://en.wikipedia.org/wiki/Foster_cage) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Foster_cage?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.