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| [[Image:Cayley graph of F2.svg|right|thumb|The Cayley graph of the [[free group]] on two generators ''a'' and ''b'']]
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| {{Graph families defined by their automorphisms}}
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| In [[mathematics]], a '''Cayley graph''', also known as a '''Cayley colour graph''', '''Cayley diagram''', '''group diagram''', or '''colour group'''<ref name = CGT>{{cite book|title=Combinatorial Group Theory|author=[[Wilhelm Magnus]], Abraham Karrass, [[Baumslag–Solitar group|Donald Solitar]] |year=1976|publisher=Dover Publications, Inc}}</ref> is a [[graph theory|graph]] that encodes the abstract structure of a [[group (mathematics)|group]]. Its definition is suggested by [[Cayley's theorem]] (named after [[Arthur Cayley]]) and uses a specified, usually finite, [[generating set of a group|set of generators]] for the group. It is a central tool in [[combinatorial group theory|combinatorial]] and [[geometric group theory]].
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| == Definition ==
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| Suppose that <math>G</math> is a [[group (mathematics)|group]] and <math>S</math> is a [[generating set of a group|generating set]]. The Cayley graph <math>\Gamma=\Gamma(G,S)</math> is a [[Graph coloring|colored]] [[directed graph]] constructed as follows: <ref>{{cite journal|first1= Arthur |last1=Cayley|journal= Amer. J. Math.|year=1878|volume=1|issue=2|pages=174–176|jstor=2369306|title=Desiderata and suggestions: No. 2. The Theory of groups: graphical representation|url=http://www.jstor.org/stable/2369306|publisher=The Johns Hopkins University Press}}</ref>
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| * Each element <math>g</math> of <math>G</math> is assigned a vertex: the vertex set <math>V(\Gamma)</math> of <math>\Gamma</math> is identified with <math>G.</math>
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| * Each generator <math>s</math> of <math>S</math> is assigned a color <math>c_s</math>.
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| * For any <math>g\in G, s\in S,</math> the vertices corresponding to the elements <math>g</math> and <math>gs</math> are joined by a directed edge of colour <math>c_s.</math> Thus the edge set <math>E(\Gamma)</math> consists of pairs of the form <math>(g, gs),</math> with <math>s\in S</math> providing the color.
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| In geometric group theory, the set <math>S</math> is usually assumed to be finite, [[Symmetric set|symmetric]] (i.e. <math>S=S^{-1}</math>) and not containing the identity element of the group. In this case, the uncolored Cayley graph is an ordinary [[graph (mathematics)|graph]]: its edges are not oriented and it does not contain loops (single-element cycles).
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| == Examples ==
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| * Suppose that <math>G=\mathbb{Z} \!</math> is the infinite cyclic group and the set ''S'' consists of the standard generator 1 and its inverse (−1 in the additive notation) then the Cayley graph is an infinite path.
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| * Similarly, if <math>G=\mathbb{Z}_n</math> is the finite [[cyclic group]] of order ''n'' and the set ''S'' consists of two elements, the standard generator of ''G'' and its inverse, then the Cayley graph is the [[cycle graph|cycle]] <math>C_n</math>.
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| * The Cayley graph of the [[direct product of groups]] (with the [[cartesian product]] of generating sets as a generating set) is the [[cartesian product of graphs|cartesian product]] of the corresponding Cayley graphs.<ref>{{citation
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| | last = Theron | first = Daniel Peter
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| | mr = 2636729
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| | page = 46
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| | publisher = University of Wisconsin, Madison
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| | series = Ph.D. thesis
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| | title = An extension of the concept of graphically regular representations
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| | year = 1988}}.</ref> Thus the Cayley graph of the abelian group <math>\mathbb{Z}^2</math> with the set of generators consisting of four elements <math>(\pm 1,0),(0,\pm 1)</math> is the infinite [[grid graph|grid]] on the plane <math>\mathbb{R}^2</math>, while for the direct product <math>\mathbb{Z}_n \times \mathbb{Z}_m</math> with similar generators the Cayley graph is the <math>n\times m</math> finite grid on a [[torus]].
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| [[Image:Cayley Graph of Dihedral Group D4.svg|220px|left|thumb|Cayley graph of the dihedral group Dih<sub>4</sub> on two generators ''a'' and ''b'']]
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| [[File:Cayley Graph of Dihedral Group D4 (generators b,c).svg|170px|right|thumb|On two generators of Dih<sub>4</sub>, which are both self-inverse]]
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| * A Cayley graph of the [[dihedral group]] ''D''<sub>4</sub> on two generators ''a'' and ''b'' is depicted to the left. Red arrows represent left-multiplication by element ''a''. Since element ''b'' is [[Cayley table|self-inverse]], the blue lines which represent left-multiplication by element ''b'' are undirected. Therefore the graph is mixed: it has eight vertices, eight arrows, and four edges. The [[Cayley table]] of the group ''D''<sub>4</sub> can be derived from the [[presentation of a group|group presentation]]
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| :: <math> \langle a, b | a^4 = b^2 = e, a b = b a^3 \rangle. \, </math>
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| A different Cayley graph of Dih<sub>4</sub> is shown on the right. ''b'' is still the horizontal reflection and represented by blue lines; ''c'' is a diagonal reflection and represented by green lines. As both reflections are self-inverse the Cayley graph on the right is completely undirected. This graph corresponds to the presentation
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| :: <math> \langle b, c | b^2 = c^2 = e, bcbc = cbcb \rangle. \, </math>
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| * The Cayley graph of the [[free group]] on two generators ''a'', ''b'' corresponding to the set ''S'' = {''a'', ''b'', ''a''<sup>−1</sup>, ''b''<sup>−1</sup>} is depicted at the top of the article, and ''e'' represents the [[identity element]]. Travelling along an edge to the right represents right multiplication by ''a'', while travelling along an edge upward corresponds to the multiplication by ''b''. Since the free group has no [[Presentation of a group|relations]], the Cayley graph has no [[Cycle (graph theory)|cycles]]. This Cayley graph is a key ingredient in the proof of the [[Banach–Tarski paradox]].
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| [[Image:HeisenbergCayleyGraph.png|thumb|240px|right|right|Part of a Cayley graph of the Heisenberg group. (The coloring is only for visual aid.)]]
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| * A Cayley graph of the [[discrete Heisenberg group]] <math>\left\{ \begin{pmatrix}
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| 1 & x & z\\
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| 0 & 1 & y\\
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| 0 & 0 & 1\\
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| \end{pmatrix},\ x,y,z \in \mathbb{Z}\right\} </math>
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| is depicted to the right. The generators used in the picture are the three matrices ''X, Y, Z'' given by the three permutations of 1, 0, 0 for the entries ''x, y, z''. They satisfy the relations
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| <math> Z^{}_{}=XYX^{-1}Y^{-1},\ XZ=ZX,\ YZ=ZY </math>, which can also be read off from the picture. This is a [[nonabelian group|non-commutative]] infinite group, and despite being three-dimensional in some sense, the Cayley graph has four-dimensional [[Growth rate (group theory)|volume growth]].
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| == Characterization ==
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| The group <math>G</math> [[group action|acts]] on itself by the left multiplication (see [[Cayley's theorem]]). This action may be viewed as the action of <math>G</math> on its Cayley graph. Explicitly, an element <math>h\in G</math> maps a vertex <math>g\in V(\Gamma)</math> to the vertex <math>hg\in V(\Gamma)</math>. The set of edges of the Cayley graph is preserved by this action: the edge <math>(g,gs)</math> is transformed into the edge <math>(hg,hgs)</math>. The left multiplication action of any group on itself is [[simply transitive]], in particular, the Cayley graph is [[vertex-transitive graph|vertex transitive]]. This leads to the following characterization of Cayley graphs:
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| : Sabidussi Theorem: ''A graph <math>\Gamma</math> is a Cayley graph of a group <math>G</math> if and only if it admits a simply transitive action of <math>G</math> by [[graph automorphism]]s (i.e. preserving the set of edges)''.<ref>{{cite journal|first1= Gert |last1=Sabidussi|authorlink=Gert Sabidussi|journal=Proceedings of the American Mathematical Society|year=1958|number=5|pages=800–804|title=On a Class of Fixed-Point-Free Graphs}}</ref>
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| To recover the group <math>G</math> and the generating set <math>S</math> from the Cayley graph <math>\Gamma=\Gamma(G,S)</math>, select a vertex <math>v_1\in V(\Gamma)</math> and label it by the identity element of the group. Then label each vertex <math>v</math> of <math>\Gamma</math> by the unique element of <math>G</math> that transforms <math>v_1</math> into <math>v.</math> The set <math>S</math> of generators of <math>G</math> that yields <math>\Gamma</math> as the Cayley graph is the set of labels of the vertices adjacent to the selected vertex. The generating set is finite (this is a common assumption for Cayley graphs) if and only if the graph is locally finite (i.e. each vertex is adjacent to finitely many edges).
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| == Elementary properties ==
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| * If a member <math>s</math> of the generating set is its own inverse, <math>s=s^{-1}</math>, then it is generally represented by an undirected edge.
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| * The Cayley graph <math>\Gamma(G,S)</math> depends in an essential way on the choice of the set <math>S</math> of generators. For example, if the generating set <math>S</math> has <math>k</math> elements then each vertex of the Cayley graph has <math>k</math> incoming and <math>k</math> outgoing directed edges. In the case of a symmetric generating set <math>S</math> with <math>r</math> elements, the Cayley graph is a [[regular graph]] of degree <math>r.</math>
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| * [[Path (graph theory)|Cycles]] (or ''closed walks'') in the Cayley graph indicate [[Presentation of a group|relations]] between the elements of <math>S.</math> In the more elaborate construction of the [[Cayley complex]] of a group, closed paths corresponding to relations are "filled in" by [[polygon]]s. This means that the problem of constructing the Cayley graph of a given presentation <math>\mathcal{P}</math> is equivalent to solving the [[Word problem for groups|Word Problem]] for <math>\mathcal{P}</math>.<ref name = CGT/>
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| * If <math>f: G'\to G</math> is a [[surjective]] [[group homomorphism]] and the images of the elements of the generating set <math>S'</math> for <math>G'</math> are distinct, then it induces a covering of graphs
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| :: <math> \bar{f}: \Gamma(G',S')\to \Gamma(G,S),\quad</math> where <math>S=f(S').</math>
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| : In particular, if a group <math>G</math> has <math>k</math> generators, all of order different from 2, and the set <math>S</math> consists of these generators together with their inverses, then the Cayley graph <math>\Gamma(G,S)</math> is covered by the infinite regular [[tree (graph theory)|tree]] of degree <math>2k</math> corresponding to the [[free group]] on the same set of generators.
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| * A graph <math>\Gamma(G,S)</math> can be constructed even if the set <math>S</math> does not generate the group <math>G.</math> However, it is [[connectivity (graph theory)|disconnected]] and is not considered to be a Cayley graph. In this case, each connected component of the graph represents a coset of the subgroup generated by <math>S</math>.
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| * For any finite Cayley graph, considered as undirected, the [[Connectivity (graph theory)|vertex connectivity]] is at least equal to 2/3 of the [[Degree (graph theory)|degree]] of the graph. If the generating set is minimal (removal of any element and, if present, its inverse from the generating set leaves a set which is not generating), the vertex connectivity is equal to the degree. The [[Connectivity (graph theory)|edge connectivity]] is in all cases equal to the degree.<ref>{{cite book|title=Technical Report TR-94-10|authorlink=L. Babai|author=Babai, L.|year=1996|publisher=University of Chicago}}[http://www.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps]</ref>
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| == Schreier coset graph ==
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| {{main|Schreier coset graph}}
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| If one, instead, takes the vertices to be right cosets of a fixed subgroup <math>H</math>, one obtains a related construction, the [[Schreier coset graph]], which is at the basis of [[coset enumeration]] or the [[Todd–Coxeter process]].
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| == Connection to group theory ==
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| Insights into the structure of the group can be obtained by studying the [[adjacency matrix]] of the graph and in particular applying the theorems of [[spectral graph theory]].
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| === Geometric group theory ===
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| For infinite groups, the [[Coarse structure|coarse geometry]] of the Cayley graph is fundamental to [[geometric group theory]]. For a [[finitely generated group]], this is independent of choice of finite set of generators, hence an intrinsic property of the group. This is only interesting for infinite groups: every finite group is coarsely equivalent to a point (or the trivial group), since one can choose as finite set of generators the entire group.
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| Formally, for a given choice of generators, one has the [[word metric]] (the natural distance on the Cayley graph), which determines a [[metric space]]. The coarse equivalence class of this space is an invariant of the group.
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| == History ==
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| The Cayley Graph was first considered for finite groups by [[Arthur Cayley]] in 1878.<ref>Cayley, A. (1878). The theory of groups: Graphical representation. Amer. J. Math. 1, 174–176. In his Collected Mathematical Papers 10: 403–405.</ref> [[Max Dehn]] in his unpublished lectures on group theory from 1909-10 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the [[word problem]] for the [[fundamental group]] of [[surface]]s with genus ≥ 2, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point.<ref>Dehn, M. (1987). Papers on Group Theory and Topology. New York: Springer-Verlag. Translated from the German and with introductions and an appendix by John Stillwell, and with an appendix by Otto Schreier.</ref>
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| == Bethe lattice ==
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| {{Main|Bethe lattice}}
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| The '''[[Bethe lattice]]''' or '''Cayley tree,''' is the Cayley graph of the free group on ''n'' generators. A presentation of a group ''G'' by ''n'' generators corresponds to a surjective map from the free group on ''n'' generators to the group ''G,'' and at the level of Cayley graphs to a map from the Cayley tree to the Cayley graph. This can also be interpreted (in [[algebraic topology]]) as the [[universal cover]] of the Cayley graph, which is not in general [[simply connected]].
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| == See also ==
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| * [[Vertex-transitive graph]]
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| * [[Generating set of a group]]
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| * [[Lovász conjecture]]
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| * [[Cube-connected cycles]]
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| * [[Algebraic graph theory]]
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| ==Notes==
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| {{reflist}}
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| == External links ==
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| * [http://www.weddslist.com/groups/cayley-plat/index.html Cayley diagrams]
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| * {{mathworld | urlname = CayleyGraph | title = Cayley graph }}
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| {{DEFAULTSORT:Cayley Graph}}
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| [[Category:Group theory]]
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| [[Category:Permutation groups]]
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| [[Category:Graph families]]
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| [[Category:Application-specific graphs]]
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| [[Category:Geometric group theory]]
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| [[Category:Algebraic graph theory]]
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