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| {{infobox graph
| | Hi there! :) My name is Johnie, I'm a student studying Arts and Sciences from Milston, Great Britain.<br><br>Also visit my web page ... [http://iteamsanz.com/2014/10/19/%d0%bf%d0%bb%d0%b0%d0%bd-%d0%ba%d0%b0%d0%ba-%d0%bf%d0%be%d1%85%d1%83%d0%b4%d0%b5%d1%82%d1%8c-%d0%be%d1%87%d0%b5%d0%bd%d1%8c-%d0%b1%d1%8b%d1%81%d1%82%d1%80%d0%be-%d0%b8-%d0%ba%d0%b0%d0%ba-%d0%bf%d0%be/ пойти в этот веб-сайте] |
| | name = Dyck graph
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| | image = [[Image:Dyck graph hamiltonian.svg|250px]]
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| | image_caption = The Dyck graph
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| | namesake = W. Dyck
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| | vertices = 32
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| | edges = 48
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| | automorphisms= 192
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| | girth = 6
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| | radius = 5
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| | diameter = 5
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| | chromatic_number = 2
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| | chromatic_index = 3
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| | properties = [[Symmetric graph|Symmetric]]<br>[[Cubic graph|Cubic]]<br>[[Hamiltonian graph| Hamiltonian]]<br>[[Bipartite graph|Bipartite]]<br>[[Cayley graph]]
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| }}
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| In the [[mathematics|mathematical]] field of [[graph theory]], the '''Dyck graph''' is a 3-[[regular graph]] with 32 vertices and 48 edges, named after [[Walther von Dyck]].<ref>{{citation
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| | last = Dyck | first = W. | author-link = Walther von Dyck
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| | journal = Math. Ann.
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| | page = 473
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| | title = Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann'scher Flächen
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| | volume = 17
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| | year = 1881}}.</ref><ref>{{MathWorld|urlname=DyckGraph|title=Dyck Graph}}</ref>
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| It is [[hamiltonian graph|Hamiltonian]] with 120 distinct Hamiltonian cycles. It has [[chromatic number]] 2, [[chromatic index]] 3, radius 5, diameter 5 and [[girth (graph theory)|girth]] 6. It is also a 3-[[k-vertex-connected graph|vertex-connected]] and a 3-[[k-edge-connected graph|edge-connected]] graph.
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| The Dyck graph is a [[toroidal graph]], and the dual of its symmetric toroidal embedding is the [[Shrikhande graph]], a strongly regular graph both symmetric and hamiltonian.
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| ==Algebraic properties==
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| The automorphism group of the Dyck graph is a group of order 192.<ref>Royle, G. [http://www.csse.uwa.edu.au/~gordon/foster/F032A.html F032A data]</ref> It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Dyck 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 Dyck graph, referenced as F32A, is the only cubic symmetric graph on 32 vertices.<ref>{{citation
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| | last1 = Conder | first1 = M. | author1-link = Marston Conder
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| | last2 = Dobcsányi | first2 = P.
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| | journal = J. Combin. Math. Combin. Comput.
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| | pages = 41–63
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| | title = Trivalent symmetric graphs up to 768 vertices
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| | volume = 40
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| | year = 2002}}.</ref>
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| The [[characteristic polynomial]] of the Dyck graph is equal to <math>(x-3) (x-1)^9 (x+1)^9 (x+3) (x^2-5)^6</math>.
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| ==Dyck map==
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| The Dyck graph is the [[skeleton (topology)|skeleton]] of a [[Regular map (graph theory)|symmetric tessellation]] of a surface of [[genus (topology)|genus]] three by twelve octagons, known as the '''Dyck map''' or '''Dyck tiling'''. The [[dual graph]] for this tiling is the [[complete bipartite graph|complete tripartite graph]] ''K''<sub>4,4,4</sub>.<ref>{{citation
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| | last = Dyck | first = W. | author-link = Walther von Dyck
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| | journal = Math. Ann.
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| | pages = 510–516
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| | title = Notiz über eine reguläre Riemannsche Fläche vom Geschlecht 3 und die zugehörige Normalkurve 4. Ordnung
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| | volume = 17
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| | year = 1880}}.</ref><ref>{{citation
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| | last = King | first = R. B.
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| | doi = 10.1080/00268970410001728681
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| | issue = 11
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| | journal = Molecular physics
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| | pages = 1231–1241
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| | title = Toroidal polyhexes, tripartite graphs, and double groups
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| | volume = 102
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| | year = 2004}}.</ref>
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| ==Gallery==
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| <gallery>
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| Image:Dyck graph.svg| Alternative drawing of the Dyck graph.
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| Image:Dyck_graph_2COL.svg|The [[chromatic number]] of the Dyck graph is 2.
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| Image:Dyck graph 3color edge.svg|The [[chromatic index]] of the Dyck graph is 3.
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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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Hi there! :) My name is Johnie, I'm a student studying Arts and Sciences from Milston, Great Britain.
Also visit my web page ... пойти в этот веб-сайте