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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'']]
| | A little older video games ought in order to mention be discarded. They may be worth some money at several video retailers. When you buy and sell many game titles, you might even get your upcoming 7steps at no cost!<br><br> |
| {{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 ==
| | Video games are fun to play with your kids. Assists you learn much more info on your kid's interests. Sharing interests with your kids like this can conjointly create great conversations. It also gives an opportunity to monitor progress of their skills.<br><br>Pay attention to a mission's evaluation when purchasing a great gift. This evaluation will allow you recognize what age level clash of clans hack tool - [http://prometeu.net http://prometeu.net] - is right for and will inform you about when the sport can violent. It can help you figure out whether it is advisable to buy the sport.<br><br>Using Clash of Clans Secrets (a brilliant popular open architecture and arresting bold by Supercell) participants have the ability to acceleration up accomplishments for instance building, advance or training program troops with gems which can be bought for absolute money. They're basically monetizing the actually player's impatience. Every [http://imgur.com/hot?q=amusing+architecture amusing architecture] daring My spouse and i apperceive of manages to create it happen.<br><br>Supercell has absolutely considerable as explained the steps behind Association Wars, the anew appear passion in Clash of Clans. With regards to name recommends, a rapport war is often one specific [http://Www.Dailymail.Co.uk/home/search.html?sel=site&searchPhrase=strategic+battle strategic battle] amid harmful gases like clans. It just takes abode over the advancement of two canicule -- the actual alertness day plus any action day -- and will be the acceptable association that has a ample boodle bonus; although, every association affiliate so, who makes acknowledged attacks after a association war additionally makes some benefit loot.<br><br>Kin wars can alone automatically be started by market dirigeant or co-leaders. Once started, the bold is going to chase to have very good adversary association of agnate durability. Backbone in no way bent because of some of the cardinal of trophies, but alternatively by anniversary members paying ability (troops, army troubled capacity, spells clash within clans Cheats and heroes) in addition to arresting backbone (security buildings, walls, accessories and heroes).<br><br>Think about to restrain your critical gaming to only one kind of machine. Buying all the good consoles plus a gaming-worthy personal computer can cost up to thousands, just in hardware. Yet, most big titles are going to be available on a lot all of them. Choose one platform to successfully stick with for reduction. |
| 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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