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| {{infobox graph
| | The name of the author is Nestor. Interviewing is how I make a living and it's some thing I truly appreciate. I presently live in Alabama. Bottle tops gathering is the only hobby his spouse doesn't approve of.<br><br>Check out my web site - [http://Louisianastrawberries.net/ActivityFeed/MyProfile/tabid/61/UserId/95947/Default.aspx louisianastrawberries.net] |
| | name = Foster graph
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| | image = [[Image:Foster graph.svg|240px]]
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| | image_caption = The Foster graph
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| | namesake = [[R. M. Foster|Ronald Martin Foster]]
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| | vertices = 90
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| | edges = 135
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| | chromatic_number = 2
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| | chromatic_index = 3
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| | automorphisms = 4320
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| | girth = 10
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| | diameter = 8
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| | radius = 8
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| | properties = [[Cubic graph|Cubic]]<br>[[Bipartite graph|Bipartite]]<br>[[symmetric graph|Symmetric]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Distance-transitive graph|Distance-transitive]]
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| }}
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| In the [[mathematics|mathematical]] field of [[graph theory]], the '''Foster graph''' is a [[Bipartite graph|bipartite]] 3-[[regular graph]] with 90 vertices and 135 edges.<ref>{{MathWorld|urlname=FosterGraph|title=Foster Graph}}</ref>
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| The Foster graph is [[Hamiltonian graph|Hamiltonian]] and has [[chromatic number]] 2, [[chromatic index]] 3, radius 8, diameter 8 and [[girth (graph theory)|girth]] 10. It is also a 3-[[k-vertex-connected graph|vertex-connected]] and 3-[[k-edge-connected graph|edge-connected]] graph.
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| All the [[cubic graph|cubic]] [[distance-regular graph]]s are known.<ref>Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.</ref> The Foster graph is one of the 13 such graphs. It is the unique [[Distance-transitive graph|distance-transitive]] graph with [[intersection array]] {3,2,2,2,2,1,1,1;1,1,1,1,2,2,2,3}.<ref>[http://www.win.tue.nl/~aeb/graphs/cubic_drg.html Cubic distance-regular graphs], A. Brouwer.</ref> It can be constructed as the [[incidence graph]] of the [[partial linear space]] which is the unique triple [[Covering space|cover]] with no 8-gons of the [[generalized quadrangle]] [[Tutte-Coxeter graph|''GQ''(2,2)]]. It is named after [[R. M. Foster]], whose ''[[Foster census]]'' of [[cubic graph|cubic]] [[symmetric graph]]s included this graph.
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| ==Algebraic properties==
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| The automorphism group of the Foster graph is a group of order 4320.<ref>Royle, G. [http://www.csse.uwa.edu.au/~gordon/foster/F090A.html F090A data]</ref> It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Foster graph is a [[symmetric graph]]. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the ''Foster census'', the Foster graph, referenced as F90A, is the only cubic symmetric graph on 90 vertices.<ref>[[Marston Conder|Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.</ref>
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| The [[characteristic polynomial]] of the Foster graph is equal to <math>(x-3) (x-2)^9 (x-1)^{18} x^{10} (x+1)^{18} (x+2)^9 (x+3) (x^2-6)^{12}</math>.
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| ==Gallery==
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| <gallery>
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| Image:Foster graph colored.svg|Foster graph colored to highlight various cycles.
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| Image:Foster graph 2COL.svg|The [[chromatic number]] of the Foster graph is 2.
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| Image:Foster_graph_3color_edge.svg|The [[chromatic index]] of the Foster graph is 3.
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| </gallery>
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| ==References==
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| {{reflist}}
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| *{{citation
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| | last1 = Biggs | first1 = N. L. | last2 = Boshier | first2 = A. G. | last3 = Shawe-Taylor | first3 = J.
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| | title = Cubic distance-regular graphs
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| | journal = Journal of the London Mathematical Society
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| | volume = 33 | issue = 3 | year = 1986 | pages = 385–394
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| | doi = 10.1112/jlms/s2-33.3.385
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| | mr = 0850954 }}.
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| *{{citation
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| | last1 = Van Dam | first1 = Edwin R. | last2 = Haemers | first2 = Willem H.
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| | title = Spectral characterizations of some distance-regular graphs
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| | journal = Journal of Algebraic Combinatorics
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| | volume = 15 | issue = 2 | year = 2002 | pages = 189–202
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| | mr = 1887234
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| | doi = 10.1023/A:1013847004932}}.
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| *{{citation
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| | last = Van Maldeghem | first = Hendrik
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| | title = Ten exceptional geometries from trivalent distance regular graphs
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| | journal = Annals of Combinatorics
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| | volume = 6 | issue = 2 | year = 2002 | pages = 209–228
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| | mr = 1955521
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| | doi = 10.1007/PL00012587}}.
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| [[Category:Individual graphs]]
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| [[Category:Regular graphs]]
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The name of the author is Nestor. Interviewing is how I make a living and it's some thing I truly appreciate. I presently live in Alabama. Bottle tops gathering is the only hobby his spouse doesn't approve of.
Check out my web site - louisianastrawberries.net