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| [[File:Paley13.svg|thumb|240px|The [[Paley graph]] of order 13, an example of a circulant graph.]]
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| [[File:Crown graphs.svg|thumb|400px|Crown graphs with six, eight, and ten vertices.]]
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| In [[graph theory]], a '''circulant graph''' is an [[undirected graph]] that has a [[cyclic group]] of [[graph automorphism|symmetries]] that includes a symmetry [[vertex-transitive graph|taking any vertex to any other vertex]].
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| ==Equivalent definitions==
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| Circulant graphs can be described in several equivalent ways:<ref name="v04">{{citation|first=V.|last=Vilfred|contribution=On circulant graphs|title=Graph Theory and its Applications (Anna University, Chennai, March 14–16, 2001)|publisher=Alpha Science|editor1-first=R.|editor1-last=Balakrishnan|editor2-first=G.|editor2-last=Sethuraman|editor3-first=Robin J.|editor3-last=Wilson|year=2004|url=http://books.google.com/books?id=wG-08Lv8E-0C&pg=PA34|pages=34–36}}.</ref>
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| *The [[automorphism group]] of the graph includes a [[cyclic group|cyclic]] [[subgroup]] that [[group action|acts transitively]] on the graph's vertices.
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| *The graph has an [[adjacency matrix]] that is a [[circulant matrix]].
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| *The {{mvar|n}} vertices of the graph can be numbered from 0 to {{math|''n'' − 1}} in such a way that, if some two vertices numbered {{mvar|x}} and {{mvar|y}} are adjacent, then every two vertices numbered {{mvar|z}} and {{math|(''z'' − ''x'' + ''y'') mod ''n''}} are adjacent.
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| *The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing.
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| *The graph is a [[Cayley graph]] of a [[cyclic group]].<ref>{{citation
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| | last = Alspach | first = Brian
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| | contribution = Isomorphism and Cayley graphs on abelian groups
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| | location = Dordrecht
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| | mr = 1468786
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| | pages = 1–22
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| | publisher = Kluwer Acad. Publ.
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| | series = NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.
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| | title = Graph symmetry (Montreal, PQ, 1996)
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| | url = http://books.google.com/books?id=-tIaXdII8egC&pg=PA1
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| | volume = 497
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| | year = 1997}}.</ref>
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| ==Examples==
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| Every [[cycle graph]] is a circulant graph, as is every [[crown graph]].
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| The [[Paley graph]]s of order {{mvar|n}} (where {{mvar|n}} is a [[prime number]] congruent to {{nowrap|1 modulo 4}}) is a graph in which the vertices are the numbers from 0 to {{math|''n'' − 1}} and two vertices are adjacent if their difference is a [[quadratic residue]] modulo {{mvar|n}}. Since the presence or absence of an edge depends only on the difference modulo {{mvar|n}} of two vertex numbers, any Paley graph is a circulant graph.
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| Every [[Möbius ladder]] is a circulant graph, as is every [[complete graph]]. A [[complete bipartite graph]] is a circulant graph if it has the same number of vertices on both sides of its bipartition.
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| If two numbers {{mvar|m}} and {{mvar|n}} are [[relatively prime]], then the {{math|''m'' × ''n''}} [[rook's graph]] (a graph that has a vertex for each square of an {{math|''m'' × ''n''}} chessboard and an edge for each two squares that a chess rook can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group {{math|''C<sub>mn</sub>'' = ''C<sub>m</sub>''×''C<sub>n</sub>''}}. More generally, in this case, the [[tensor product of graphs]] between any {{mvar|m}}- and {{mvar|n}}-vertex circulants is itself a circulant.<ref name="v04"/>
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| Many of the known [[lower bound]]s on [[Ramsey number]]s come from examples of circulant graphs that have small [[maximum clique]]s and small [[maximum independent set]]s.<ref>[http://www.combinatorics.org/Surveys/ds1/sur.pdf Small Ramsey Numbers], Stanisław P. Radziszowski, ''Electronic J. Combinatorics'', dynamic survey updated 2009.</ref>
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| == A Specific Example ==
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| The circulant graph <math> C_n^{s_1,\ldots,s_k} </math> with jumps <math> s_1, \ldots, s_k </math> is defined as the graph with <math> n </math> nodes labeled <math>0, 1, \ldots, n-1</math> where each node ''i'' is adjacent to 2''k'' nodes <math>i \pm s_1, \ldots, i \pm s_k \mod n</math>.
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| * The graph <math>C_n^{s_1, \ldots, s_k}</math> is connected if and only if <math>\gcd(n, s_1, \ldots, s_k) = 1</math>.
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| * If <math> 1 \leq s_1 < \cdots < s_k </math> are fixed integers then the number of [[spanning tree]]s <math>t(C_n^{s_1,\ldots,s_k})=na_n^2</math> where <math>a_n</math> satisfies a [[recurrence relation]] of order <math>2^{s_k-1}</math>.
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| ** In particular, <math>t(C_n^{1,2}) = nF_n^2 </math> where <math>F_n</math> is the ''n''-th [[Fibonacci number]].
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| ==Self-complementary circulants==
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| A [[self-complementary graph]] is a graph in which replacing every edge by a non-edge and vice versa produces an [[graph isomorphism|isomorphic]] graph.
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| For instance, a five-vertex cycle graph is self-complementary, and is also a circulant graph. More generally every [[Paley graph]] is a self-complementary circulant graph.<ref name="s62">{{Cite journal
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| | last = Sachs | first = Horst | authorlink = Horst Sachs
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| | mr = 0151953
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| | journal = Publicationes Mathematicae Debrecen
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| | pages = 270–288
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| | title = Über selbstkomplementäre Graphen
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| | volume = 9
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| | year = 1962}}.</ref> [[Horst Sachs]] showed that, if a number {{mvar|n}} has the property that every prime factor of {{mvar|n}} is congruent to {{nowrap|1 modulo 4}}, then there exists a self-complementary circulant with {{mvar|n}} vertices. He conjectured that this condition is also necessary: that no other values of {{mvar|n}} allow a self-complementary circulant to exist.<ref name="v04"/><ref name="s62"/> The conjecture was proven some 40 years later, by Vilfred.<ref name="v04"/>
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| ==Ádám's conjecture==
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| Define a ''circulant numbering'' of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to {{math|''n'' − 1}} in such a way that, if some two vertices numbered {{mvar|x}} and {{mvar|y}} are adjacent, then every two vertices numbered {{mvar|z}} and {{math|(''z'' − ''x'' + ''y'') mod ''n''}} are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix.
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| Let {{mvar|a}} be an integer that is [[relatively prime]] to {{mvar|n}}, and let {{mvar|b}} be any integer. Then the [[linear function]] that takes a number {{mvar|x}} to {{math|''ax'' + ''b''}} transforms a circulant numbering to another circulant numbering. András Ádám conjectured that these linear maps are the only ways of renumbering a circulant graph while preserving the circulant property: that is, if {{mvar|G}} and {{mvar|H}} are isomorphic circulant graphs, with different numberings, then there is a linear map that transforms the numbering for {{mvar|G}} into the numbering for {{mvar|H}}. However, Ádám's conjecture is now known to be false. A counterexample is given by graphs {{mvar|G}} and {{mvar|H}} with 16 vertices each; a vertex {{mvar|x}} in {{mvar|G}} is connected to the six neighbors {{math|''x'' ± 1}}, {{math|''x'' ± 2}}, and {{math|''x'' ± 7}} (modulo 16), while in {{mvar|H}} the six neighbors are {{math|''x'' ± 2}}, {{math|''x'' ± 3}}, and {{math|''x'' ± 5}} (modulo 16). These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.<ref name="v04"/>
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{mathworld|title=Circulant Graph|urlname=CirculantGraph}}
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| [[Category:Graph families]]
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| [[Category:Regular graphs]]
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