Theta function of a lattice - Revision history

en>K9re11: removed Category:Number theory using HotCat as there is already the more specific category ''theta functions''

2014-11-21T10:52:06Z

removed Category:Number theory using HotCat as there is already the more specific category ''theta functions''

← Older revision		Revision as of 12:52, 21 November 2014
Line 1:		Line 1:
	~~{{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~~
	~~\| image = [[Image:Dyck graph hamiltonian.svg\|250px]]~~
	~~\| image_caption = The Dyck graph~~
	~~\| namesake = W. Dyck~~
	~~\| vertices = 32~~
	~~\| edges = 48~~
	~~\| automorphisms= 192~~
	~~\| girth = 6~~
	~~\| radius = 5~~
	~~\| diameter = 5~~
	~~\| chromatic_number = 2~~
	~~\| chromatic_index = 3~~
	~~\| properties = [[Symmetric graph\|Symmetric]]<br>[[Cubic graph\|Cubic]]<br>[[Hamiltonian graph\| Hamiltonian]]<br>[[Bipartite graph\|Bipartite]]<br>[[Cayley graph]]~~
	}}

	~~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~~
	~~\| last = Dyck \| first = W. \| author-link = Walther von Dyck~~
	~~\| journal = Math. Ann.~~
	~~\| page = 473~~
	~~\| title = Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann'scher Flächen~~
	~~\| volume = 17~~
	~~\| year = 1881}}.</ref~~><~~ref~~>~~{{MathWorld\|urlname=DyckGraph\|title=Dyck Graph}}</ref>~~

	~~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~~.

	~~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.~~

	~~==Algebraic properties==~~
	~~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
	~~\| last1 = Conder \| first1 = M. \| author1~~-~~link = Marston Conder~~
	~~\| last2 = Dobcsányi \| first2 = P.~~
	~~\| journal = J. Combin. Math. Combin. Comput.~~
	~~\| pages = 41–63~~
	~~\| title = Trivalent symmetric graphs up to 768 vertices~~
	~~\| volume = 40~~
	~~\| year = 2002}}.</ref>~~

	~~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>.~~

	~~==Dyck map==~~
	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
	~~\| last = Dyck \| first = W. \| author~~-~~link = Walther von Dyck~~
	~~\| journal = Math. Ann.~~
	~~\| pages = 510–516~~
	~~\| title = Notiz über eine reguläre Riemannsche Fläche vom Geschlecht 3 und die zugehörige Normalkurve 4. Ordnung~~
	~~\| volume = 17~~
	~~\| year = 1880}}.</ref><ref>{{citation~~
	~~\| last = King \| first = R. B.~~
	~~\| doi = 10.1080/00268970410001728681~~
	~~\| issue = 11~~
	~~\| journal = Molecular physics~~
	~~\| pages = 1231–1241~~
	~~\| title = Toroidal polyhexes, tripartite graphs, and double groups~~
	~~\| volume = 102~~
	~~\| year = 2004}}.</ref>~~

	~~==Gallery==~~
	~~<gallery>~~
	~~Image:Dyck graph.svg\| Alternative drawing of the Dyck graph.~~
	~~Image:Dyck_graph_2COL.svg\|The [[chromatic number]] of the Dyck graph is 2.~~
	~~Image:Dyck graph 3color edge.svg\|The [[chromatic index]] of the Dyck graph is 3.~~
	~~</gallery>~~

	~~== References ==~~
	~~{{reflist}}~~

	~~[[Category:Individual graphs]]~~
	~~[[Category:Regular graphs]~~]

en>Alex Selby: /* Definition */ Disambiguated "Lattice"

2013-10-24T14:53:17Z

Definition: Disambiguated "Lattice"

← Older revision		Revision as of 16:53, 24 October 2013
Line 1:		Line 1:
	~~e - Shop Word - Press~~ is a ~~excellent cart for your on the web shopping organization~~. ~~Good luck on continue learning how to make a wordpress website~~. ~~Step~~-~~4 Testing:~~ It is ~~the foremost important of your Plugin development process~~. ~~Out of the various designs of photography identified these days~~, ~~sports photography is preferred most~~, ~~probably for the enjoyment~~ and ~~enjoyment associated with it~~. It is ~~found that most of the visitors only look for the results that are displayed on~~ the ~~first page~~ of the ~~search engines~~ and ~~so if you get~~ the ~~service from professional service providers then they strive for the first page ranking~~ of ~~your site and improve the online visibility~~. <br>~~<br>If you beloved this article and you also would like to acquire more info relating to~~ [http://~~shortener~~.us/~~wordpress_backup_2232559 wordpress backup plugin~~] ~~please visit~~ the ~~page~~. ~~Creating~~ a ~~website from scratch can be such a pain~~. ~~While direct advertising is limited~~ to ~~few spots in your site~~ and ~~tied~~ to ~~fixed monthly payment by~~ the ~~advertisers~~, ~~affiliate marketing can give you unlimited income~~ as ~~long as you can convert your traffic to sales~~. ~~Which is perfect for building a mobile site for business use~~. ~~They provide many such popular products which you can buy for your baby~~. ~~For a Wordpress website, you don't need a powerful web hosting account~~ to ~~host your site~~. <br><br>~~Saying that, despite~~ the ~~launch~~ of ~~Wordpress Express many months ago, there has still been no sign~~ of a ~~Wordpress video tutorial on offer UNTIL NOW~~. ~~After sending these details~~, ~~your Word~~ - ~~Press blog will be setup within a few days~~. ~~This platform can be customizedaccording to the requirements of the business~~. ~~In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably~~. ~~Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait~~. <br><br>~~The primary differences are in the plugins that I install, as all sites don't need all the normal plugins~~. ~~Cameras with a pentaprism (as in comparison to pentamirror) ensure that little mild is lost before it strikes your eye, however these often increase the cost of the digital camera considerably~~. ~~One of the great features of Wordpress is its ability to integrate SEO into your site~~. ~~It supports backup scheduling and allows you to either download the backup file or email it to you. Fortunately~~, ~~Word - Press Customization Service is available these days~~, ~~right from custom theme design, to plugin customization~~ and ~~modifying your website, you can take any bespoke service for your Word - Press development project~~. <br><br>~~More it extends numerous opportunities where your firm is at comfort and rest assured~~ of ~~no risks & errors~~. ~~Mahatma Gandhi is known as one~~ of the ~~most prominent personalities and symbols of peace, non-violence and freedom~~. ~~However, you must also manually approve or reject comments so that your website does not promote parasitic behavior~~. ~~If this is not possible you still have the choice~~ of the ~~default theme that~~ is ~~Word - Press~~ 3. ~~For your information, it is an open source web content management system.~~		{{infobox graph
			\| name = Dyck graph
			\| image = [[Image:Dyck graph hamiltonian.svg\|250px]]
			\| image_caption = The Dyck graph
			\| namesake = W. Dyck
			\| vertices = 32
			\| edges = 48
			\| automorphisms= 192
			\| girth = 6
			\| radius = 5
			\| diameter = 5
			\| chromatic_number = 2
			\| chromatic_index = 3
			\| properties = [[Symmetric graph\|Symmetric]]<br>[[Cubic graph\|Cubic]]<br>[[Hamiltonian graph\| Hamiltonian]]<br>[[Bipartite graph\|Bipartite]]<br>[[Cayley graph]]
			}}

			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
			\| last = Dyck \| first = W. \| author-link = Walther von Dyck
			\| journal = Math. Ann.
			\| page = 473
			\| title = Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann'scher Flächen
			\| volume = 17
			\| year = 1881}}.</ref><ref>{{MathWorld\|urlname=DyckGraph\|title=Dyck Graph}}</ref>

			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.

			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.

			==Algebraic properties==
			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
			\| last1 = Conder \| first1 = M. \| author1-link = Marston Conder
			\| last2 = Dobcsányi \| first2 = P.
			\| journal = J. Combin. Math. Combin. Comput.
			\| pages = 41–63
			\| title = Trivalent symmetric graphs up to 768 vertices
			\| volume = 40
			\| year = 2002}}.</ref>

			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>.

			==Dyck map==
			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
			\| last = Dyck \| first = W. \| author-link = Walther von Dyck
			\| journal = Math. Ann.
			\| pages = 510–516
			\| title = Notiz über eine reguläre Riemannsche Fläche vom Geschlecht 3 und die zugehörige Normalkurve 4. Ordnung
			\| volume = 17
			\| year = 1880}}.</ref><ref>{{citation
			\| last = King \| first = R. B.
			\| doi = 10.1080/00268970410001728681
			\| issue = 11
			\| journal = Molecular physics
			\| pages = 1231–1241
			\| title = Toroidal polyhexes, tripartite graphs, and double groups
			\| volume = 102
			\| year = 2004}}.</ref>

			==Gallery==
			<gallery>
			Image:Dyck graph.svg\| Alternative drawing of the Dyck graph.
			Image:Dyck_graph_2COL.svg\|The [[chromatic number]] of the Dyck graph is 2.
			Image:Dyck graph 3color edge.svg\|The [[chromatic index]] of the Dyck graph is 3.
			</gallery>

			== References ==
			{{reflist}}

			[[Category:Individual graphs]]
			[[Category:Regular graphs]]

en>Michael Hardy: link to Lattice (discrete subgroup) and some other small edits

2010-05-16T05:37:51Z

link to Lattice (discrete subgroup) and some other small edits

New page

e - Shop Word - Press is a excellent cart for your on the web shopping organization. Good luck on continue learning how to make a wordpress website. Step-4 Testing: It is the foremost important of your Plugin development process. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility. If you beloved this article and you also would like to acquire more info relating to [http://shortener.us/wordpress_backup_2232559 wordpress backup plugin] please visit the page. Creating a website from scratch can be such a pain. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. Which is perfect for building a mobile site for business use. They provide many such popular products which you can buy for your baby. For a Wordpress website, you don't need a powerful web hosting account to host your site. Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. After sending these details, your Word - Press blog will be setup within a few days. This platform can be customizedaccording to the requirements of the business. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. The primary differences are in the plugins that I install, as all sites don't need all the normal plugins. Cameras with a pentaprism (as in comparison to pentamirror) ensure that little mild is lost before it strikes your eye, however these often increase the cost of the digital camera considerably. One of the great features of Wordpress is its ability to integrate SEO into your site. It supports backup scheduling and allows you to either download the backup file or email it to you. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. If this is not possible you still have the choice of the default theme that is Word - Press 3. For your information, it is an open source web content management system.

← Older revision		Revision as of 12:52, 21 November 2014
Line 1:		Line 1:
	~~{{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~~
	~~\| image = [[Image:Dyck graph hamiltonian.svg\|250px]]~~
	~~\| image_caption = The Dyck graph~~
	~~\| namesake = W. Dyck~~
	~~\| vertices = 32~~
	~~\| edges = 48~~
	~~\| automorphisms= 192~~
	~~\| girth = 6~~
	~~\| radius = 5~~
	~~\| diameter = 5~~
	~~\| chromatic_number = 2~~
	~~\| chromatic_index = 3~~
	~~\| properties = [[Symmetric graph\|Symmetric]]<br>[[Cubic graph\|Cubic]]<br>[[Hamiltonian graph\| Hamiltonian]]<br>[[Bipartite graph\|Bipartite]]<br>[[Cayley graph]]~~
	}}

	~~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~~
	~~\| last = Dyck \| first = W. \| author-link = Walther von Dyck~~
	~~\| journal = Math. Ann.~~
	~~\| page = 473~~
	~~\| title = Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann'scher Flächen~~
	~~\| volume = 17~~
	~~\| year = 1881}}.</ref~~><~~ref~~>~~{{MathWorld\|urlname=DyckGraph\|title=Dyck Graph}}</ref>~~

	~~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~~.

	~~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.~~

	~~==Algebraic properties==~~
	~~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
	~~\| last1 = Conder \| first1 = M. \| author1~~-~~link = Marston Conder~~
	~~\| last2 = Dobcsányi \| first2 = P.~~
	~~\| journal = J. Combin. Math. Combin. Comput.~~
	~~\| pages = 41–63~~
	~~\| title = Trivalent symmetric graphs up to 768 vertices~~
	~~\| volume = 40~~
	~~\| year = 2002}}.</ref>~~

	~~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>.~~

	~~==Dyck map==~~
	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
	~~\| last = Dyck \| first = W. \| author~~-~~link = Walther von Dyck~~
	~~\| journal = Math. Ann.~~
	~~\| pages = 510–516~~
	~~\| title = Notiz über eine reguläre Riemannsche Fläche vom Geschlecht 3 und die zugehörige Normalkurve 4. Ordnung~~
	~~\| volume = 17~~
	~~\| year = 1880}}.</ref><ref>{{citation~~
	~~\| last = King \| first = R. B.~~
	~~\| doi = 10.1080/00268970410001728681~~
	~~\| issue = 11~~
	~~\| journal = Molecular physics~~
	~~\| pages = 1231–1241~~
	~~\| title = Toroidal polyhexes, tripartite graphs, and double groups~~
	~~\| volume = 102~~
	~~\| year = 2004}}.</ref>~~

	~~==Gallery==~~
	~~<gallery>~~
	~~Image:Dyck graph.svg\| Alternative drawing of the Dyck graph.~~
	~~Image:Dyck_graph_2COL.svg\|The [[chromatic number]] of the Dyck graph is 2.~~
	~~Image:Dyck graph 3color edge.svg\|The [[chromatic index]] of the Dyck graph is 3.~~
	~~</gallery>~~

	~~== References ==~~
	~~{{reflist}}~~

	~~[[Category:Individual graphs]]~~
	~~[[Category:Regular graphs]~~]