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| {{About|connected, 2-regular graphs}}
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| {{Redirect|Triangle graph|data graphs plotted across three variables|Ternary plot}}
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| {{infobox graph
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| | name = Cycle graph
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| | image = [[Image:Undirected 6 cycle.svg|160px]]
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| | image_caption = A cycle graph of length 6
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| | vertices = ''n''
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| | edges = ''n''
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| | automorphisms = 2''n'' (''D<sub>n</sub>'')
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| | chromatic_number = 3 if ''n'' is odd<br/>2 if ''n'' is even
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| | chromatic_index = 3 if ''n'' is odd<br/>2 if ''n'' is even
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| | girth = ''n''
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| | spectrum = {2 cos(2 ''k'' π / ''n''); ''k''=1, ... ,''n''}<ref>[http://www.win.tue.nl/~aeb/2WF02/easyspectra.pdf Some simple graph spectra]. win.tue.nl</ref>
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| | notation = <math>C_n</math>
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| | properties = [[Regular graph|2-regular]]<br>[[Vertex-transitive graph|Vertex-transitive]]<br>[[Edge-transitive graph|Edge-transitive]]<br>[[Unit distance graph|Unit distance]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Eulerian graph|Eulerian]]
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| }}
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| In [[graph theory]], a '''cycle graph''' or '''circular graph''' is a [[graph (mathematics)|graph]] that consists of a single [[Cycle (graph theory)|cycle]], or in other words, some number of vertices connected in a closed chain. The cycle graph with ''n'' vertices is called ''C<sub>n</sub>''. The number of vertices in ''C<sub>n</sub>'' equals the number of [[Edge (graph theory)|edge]]s, and every vertex has [[degree (graph theory)|degree]] 2; that is, every vertex has exactly two edges incident with it.
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| ==Terminology==
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| There are many [[synonym]]s for "cycle graph". These include '''simple cycle graph''' and '''cyclic graph''', although the latter term is less often used, because it can also refer to graphs which are merely not [[directed acyclic graph|acyclic]]. Among graph theorists, '''cycle''', '''polygon''', or '''''n''-gon''' are also often used. A cycle with an even number of vertices is called an '''even cycle'''; a cycle with an odd number of vertices is called an '''odd cycle'''.
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| ==Properties==
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| A cycle graph is:
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| * [[Connected graph|Connected]]
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| * [[regular graph|2-regular]]
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| * [[Eulerian graph|Eulerian]]
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| * [[Hamiltonian graph|Hamiltonian]]
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| * [[Bipartite graph|2-vertex colorable]], if and only if it has an even number of vertices. More generally, a graph is bipartite [[if and only if]] it has no odd cycles ([[Dénes Kőnig|Kőnig]], 1936).
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| * [[k-edge colorable|2-edge colorable]], if and only if it has an even number of vertices
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| * 3-vertex colorable and 3-edge colorable, for any number of vertices
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| * A [[unit distance graph]]
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| In addition:
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| *As cycle graphs can be [[graph drawing|drawn]] as [[regular polygon]]s, the [[automorphism group|symmetries]] of an ''n''-cycle are the same as those of a regular polygon with ''n'' sides, the [[dihedral group]] of order 2''n''. In particular, there exist symmetries taking any vertex to any other vertex, and any edge to any other edge, so the ''n''-cycle is a [[symmetric graph]].
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| ==Directed cycle graph==
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| [[Image:DC8.png|frame|right|A directed cycle graph of length 8]]
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| A '''directed cycle graph''' is a directed version of a cycle graph, with all the edges being oriented in the same direction.
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| In a [[directed graph]], a set of edges which contains at least one edge (or ''arc'') from each directed cycle is called a [[feedback arc set]]. Similarly, a set of vertices containing at least one vertex from each directed cycle is called a [[feedback vertex set]].
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| A directed cycle graph has uniform in-degree 1 and uniform out-degree 1.
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| Directed cycle graphs are [[Cayley graph]]s for [[cyclic group]]s (see e.g. Trevisan).
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| ==See also==
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| {{commons category|Cycle graphs}}
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| * [[Complete bipartite graph]]
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| * [[Path graph]]
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| * [[Complete graph]]
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| * [[Null graph]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{MathWorld |urlname=CycleGraph |title=Cycle Graph}} (discussion of both 2-regular cycle graphs and the group-theoretic concept of [[cycle diagram]]s)
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| *[[Luca Trevisan]], [http://in-theory.blogspot.com/2006/12/characters-and-expansion.html Characters and Expansion].
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| [[Category:Parametric families of graphs]]
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| [[Category:Regular graphs]]
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