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| {{otheruses4|the 3-regular graph|the graph associated with a Coxeter group|Coxeter diagram}}
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| {{infobox graph
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| | name = Coxeter graph
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| | image = [[Image:Coxeter graph.svg|250px]]
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| | image_caption = The Coxeter graph
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| | namesake =
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| | vertices = 28
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| | edges = 42
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| | automorphisms= 336 ([[Projective linear group|PGL]]<sub>2</sub>(7))
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| | girth = 7
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| | diameter = 4
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| | radius = 4
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| | chromatic_number = 3
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| | chromatic_index = 3
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| | properties = [[Symmetric graph|Symmetric]]<br>[[distance-regular graph|Distance-regular]]<br>[[distance-transitive graph|Distance-transitive]]<br>[[Cubic graph|Cubic]]<br>[[Hypohamiltonian graph|Hypohamiltonian]]
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| }}
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| In the [[mathematics|mathematical]] field of [[graph theory]], the '''Coxeter graph''' is a 3-[[regular graph]] with 28 vertices and 42 edges.<ref>{{MathWorld|urlname=CoxeterGraph|title=Coxeter Graph}}</ref> 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 Coxeter graph is one of the 13 such graphs.
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| ==Properties==
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| The Coxeter graph has [[chromatic number]] 3, [[chromatic index]] 3, radius 4, diameter 4 and [[girth (graph theory)|girth]] 7. It is also a 3-[[k-vertex-connected graph|vertex-connected graph]] and a 3-[[k-edge-connected graph|edge-connected graph]].
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| The Coxeter graph is [[hypohamiltonian graph|hypohamiltonian]] : it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It has [[Crossing number (graph theory)|rectilinear crossing number]] 11, and is the smallest cubic graph with that crossing number currently known, but an 11-crossing, 26-vertex graph may exist {{OEIS|id=A110507}}.
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| The Coxeter graph may be constructed from the smaller distance-regular [[Heawood graph]] by constructing a vertex for each 6-cycle in the Heawood graph and an edge for each disjoint pair of 6-cycles.<ref>{{citation|first=Italo J.|last=Dejter|title=From the Coxeter graph to the Klein graph|journal=Journal of Graph Theory|year=2011|doi=10.1002/jgt.20597|arxiv=1002.1960}}.</ref>
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| ==Algebraic properties==
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| The automorphism group of the Coxeter graph is a group of order 336.<ref>Royle, G. [http://www.csse.uwa.edu.au/~gordon/foster/F028A.html F028A data]</ref> It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Coxeter 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 Coxeter graph, referenced as F28A, is the only cubic symmetric graph on 28 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 Coxeter graph is also uniquely determined by its [[graph spectrum]], the set of graph eigenvalues of its [[adjacency matrix]].<ref>E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189-202, 2003</ref>
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| As a finite connected vertex-transitive graph that contains no [[Hamiltonian cycle]], the Coxeter graph is a counterexample to a variant of the [[Lovász conjecture]], but the canonical formulation of the conjecture asks for an Hamiltonian path and is verified by the Coxeter graph.
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| Only five examples of vertex-transitive graph with no Hamiltonian cycles are known : the [[complete graph]] ''K''<sub>2</sub>, the [[Petersen graph]], the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.<ref>Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."]</ref>
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| The [[characteristic polynomial]] of the Coxeter graph is <math>(x-3) (x-2)^8 (x+1)^7 (x^2+2 x-1)^6</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
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| ==Gallery== | |
| <gallery>
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| Image:Edge excised Coxeter graph.svg|The graph obtained by any edge excision from the Coxeter is Hamilton-connected.
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| Image:coxeter_graph_3COL.svg|The [[chromatic number]] of the Coxeter graph is 3.
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| Image:Coxeter graph 11C.svg|The [[Crossing number (graph theory)|rectilinear crossing number]] of the Coxeter graph is 11.
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| </gallery>
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| == References ==
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| {{reflist}}
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| [[Category:Individual graphs]]
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| [[Category:Regular graphs]]
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