Harmonic coordinates - Revision history

en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-24T12:56:04Z

References: replace mr template with mr parameter in CS1 templates; using AWB

← Older revision		Revision as of 14:56, 24 September 2014
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	~~{{infobox graph~~		She is known by the title of Myrtle Shryock. Years in the past we moved to Puerto Rico and my family members loves it. My day occupation is a librarian. To gather coins is what his family members and him appreciate.<br><br>Also visit my web-site - [http://www.pornextras.info/user/G49Z http://www.pornextras.info/user/G49Z]
	~~\| name = Pappus graph~~
	~~\| image = [[Image:Pappus graph LS.svg\|250px]]~~
	~~\| image_caption = The Pappus graph.~~
	~~\| namesake = [[Pappus of Alexandria]]~~
	~~\| vertices = 18~~
	~~\| edges = 27~~
	~~\| automorphisms=216~~
	~~\| girth = 6~~
	~~\| radius = 4~~
	~~\| diameter = 4~~
	~~\| chromatic_number = 2~~
	~~\| chromatic_index = 3~~
	\| properties = [[Bipartite graph\|Bipartite]]<br>[[Symmetric graph\|Symmetric]]<br>[[Distance-transitive graph\|Distance-transitive]]<br>[[Distance-regular graph\|Distance-regular]]<br>[[Cubic graph\|Cubic]]<br>[[Hamiltonian graph\|Hamiltonian]]
	}}

	~~In the [[mathematics\|mathematical]] field of [[graph theory]], the '''Pappus graph'''~~ is ~~a [[Bipartite graph\|bipartite]] 3-[[regular graph\|regular]] [[undirected graph]] with 18 vertices and 27 edges, formed as~~ the ~~[[Levi graph]] of the [[Pappus configuration]].<ref>{{MathWorld\|urlname=PappusGraph\|~~title~~=Pappus Graph}}</ref> It is named after [[Pappus~~ of ~~Alexandria]], an ancient [[Greek mathematics\|Greek mathematician]] who is believed to have discovered the "hexagon theorem" describing the Pappus configuration~~. ~~All~~ the ~~[[cubic graph\|cubic]] [[distance-regular graph]]s are known; the Pappus graph is one of the 13 such graphs.<ref>Brouwer, A. E.; Cohen, A. M.;~~ and ~~Neumaier, A~~. ~~Distance-Regular Graphs. New York: Springer-Verlag, 1989.</ref>~~

	~~The Pappus graph has [[Crossing number (graph theory)\|rectilinear crossing number]] 5, and~~ is ~~the smallest cubic graph with that crossing number {{OEIS\|id=A110507}}~~. ~~It has [[Girth (graph theory)\|girth]] 6, diameter 4, radius 4, [[chromatic number]] 2, [[chromatic index]] 3 and~~ is ~~both 3-[[k-vertex-connected graph\|vertex-connected]]~~ and ~~3-[[k-edge-connected graph\|edge-connected]]~~.

	~~The Pappus graph has a [[chromatic polynomial]] equal to:~~ <~~math~~>~~(x-1)x(x^{16}-26x^{15}+325x^{14}-2600x^{13}+14950x^{12}-65762x^{11}+~~<~~/math~~>~~<math>229852x^{10}~~-~~653966x^9+1537363x^8~~-~~3008720x^7+4904386x^6-</math><math>6609926x^5+7238770x^4-6236975x^3+3989074x^2-1690406x+356509)</math>.~~

	~~The name "Pappus graph" has also been used to refer to a related nine-vertex graph,<ref>{{citation~~
	~~\| last = Kagno \| first = I. N.~~
	~~\| title = Desargues' and Pappus' graphs and their groups~~
	~~\| journal = American Journal of Mathematics~~
	~~\| year = 1947~~
	~~\| volume = 69~~
	~~\| issue = 4~~
	~~\| pages = 859–863~~
	~~\| doi = 10.2307/2371806~~
	~~\| jstor = 2371806~~
	\| publisher = The Johns Hopkins University Press}}</ref> with a vertex for each point of the Pappus configuration and an edge for every pair of points on the same line; this nine-vertex graph is 6-regular, and is the [[complement graph]] of the union of three disjoint [[triangle graph]]s. The first Pappus graph can be embedded in the torus to form a [[regular map (graph theory)\|regular map]] with nine hexagonal faces; the second, to form a regular map with 18 triangular faces.

	~~==Algebraic properties==~~
	The automorphism group of the Pappus graph is a group of order 216. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Pappus 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 Pappus graph, referenced as F018A, is the only cubic symmetric graph on 18 vertices.<ref>Royle, G. [http://www.~~cs.uwa.edu~~.au/~~~gordon~~/~~remote~~/~~foster~~/~~#census "Cubic Symmetric Graphs (The Foster Census)."]</ref><ref>[[Marston Conder\|Conder, M~~.~~]] and Dobcsányi, P~~. ~~"Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.<~~/~~ref>~~

	~~The [[characteristic polynomial]] of the Pappus graph is <math>(x-3) x^4 (x+3) (x^2-3)^6<~~/~~math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.~~

	~~==Gallery==~~
	~~<gallery>~~
	~~Image:Pappus graph colored.svg\|Pappus graph coloured to highlight various cycles.~~
	~~Image:Pappus graph 3color edge.svg\|The [[chromatic index]] of the Pappus graph is 3.~~
	~~Image:Pappus graph 2COL.svg\|The [[chromatic number]] of the Pappus graph is 2.~~
	~~</gallery>~~

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

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

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2996688

2013-03-16T04:50:13Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2996688

← Older revision		Revision as of 06:50, 16 March 2013
Line 1:		Line 1:
	The ~~author~~'~~s name~~ is ~~Christy~~. ~~Her family life in Ohio but her husband desires them~~ to ~~move~~. ~~It'~~s ~~not a common factor but what she likes performing~~ is ~~to perform domino but she doesn't have~~ the ~~time recently~~. ~~Credit authorising~~ is ~~how she makes~~ [~~http~~://~~Test~~.~~jeka~~-nn.ru/~~node~~/~~129 best psychic readings~~] a ~~residing~~.<~~br><br~~>~~Also visit my blog post; real~~ [http://www.~~seekavideo~~.~~com~~/~~playlist~~/~~2199~~/~~video~~/ ~~phone psychic readings~~] ([~~http://www~~.~~january~~-~~yjm~~.~~com~~/xe/~~index~~.~~php?mid~~=~~video&document_srl~~=~~158289 http~~:~~//www~~.~~january-yjm~~.~~com/xe/~~index.~~php?mid~~=~~video&document_srl~~=~~158289~~])		{{infobox graph
			\| name = Pappus graph
			\| image = [[Image:Pappus graph LS.svg\|250px]]
			\| image_caption = The Pappus graph.
			\| namesake = [[Pappus of Alexandria]]
			\| vertices = 18
			\| edges = 27
			\| automorphisms=216
			\| girth = 6
			\| radius = 4
			\| diameter = 4
			\| chromatic_number = 2
			\| chromatic_index = 3
			\| properties = [[Bipartite graph\|Bipartite]]<br>[[Symmetric graph\|Symmetric]]<br>[[Distance-transitive graph\|Distance-transitive]]<br>[[Distance-regular graph\|Distance-regular]]<br>[[Cubic graph\|Cubic]]<br>[[Hamiltonian graph\|Hamiltonian]]
			}}

			In the [[mathematics\|mathematical]] field of [[graph theory]], the '''Pappus graph''' is a [[Bipartite graph\|bipartite]] 3-[[regular graph\|regular]] [[undirected graph]] with 18 vertices and 27 edges, formed as the [[Levi graph]] of the [[Pappus configuration]].<ref>{{MathWorld\|urlname=PappusGraph\|title=Pappus Graph}}</ref> It is named after [[Pappus of Alexandria]], an ancient [[Greek mathematics\|Greek mathematician]] who is believed to have discovered the "hexagon theorem" describing the Pappus configuration. All the [[cubic graph\|cubic]] [[distance-regular graph]]s are known; the Pappus graph is one of the 13 such graphs.<ref>Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.</ref>

			The Pappus graph has [[Crossing number (graph theory)\|rectilinear crossing number]] 5, and is the smallest cubic graph with that crossing number {{OEIS\|id=A110507}}. It has [[Girth (graph theory)\|girth]] 6, diameter 4, radius 4, [[chromatic number]] 2, [[chromatic index]] 3 and is both 3-[[k-vertex-connected graph\|vertex-connected]] and 3-[[k-edge-connected graph\|edge-connected]].

			The Pappus graph has a [[chromatic polynomial]] equal to: <math>(x-1)x(x^{16}-26x^{15}+325x^{14}-2600x^{13}+14950x^{12}-65762x^{11}+</math><math>229852x^{10}-653966x^9+1537363x^8-3008720x^7+4904386x^6-</math><math>6609926x^5+7238770x^4-6236975x^3+3989074x^2-1690406x+356509)</math>.

			The name "Pappus graph" has also been used to refer to a related nine-vertex graph,<ref>{{citation
			\| last = Kagno \| first = I. N.
			\| title = Desargues' and Pappus' graphs and their groups
			\| journal = American Journal of Mathematics
			\| year = 1947
			\| volume = 69
			\| issue = 4
			\| pages = 859–863
			\| doi = 10.2307/2371806
			\| jstor = 2371806
			\| publisher = The Johns Hopkins University Press}}</ref> with a vertex for each point of the Pappus configuration and an edge for every pair of points on the same line; this nine-vertex graph is 6-regular, and is the [[complement graph]] of the union of three disjoint [[triangle graph]]s. The first Pappus graph can be embedded in the torus to form a [[regular map (graph theory)\|regular map]] with nine hexagonal faces; the second, to form a regular map with 18 triangular faces.

			==Algebraic properties==
			The automorphism group of the Pappus graph is a group of order 216. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Pappus 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 Pappus graph, referenced as F018A, is the only cubic symmetric graph on 18 vertices.<ref>Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."]</ref><ref>[[Marston Conder\|Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.</ref>

			The [[characteristic polynomial]] of the Pappus graph is <math>(x-3) x^4 (x+3) (x^2-3)^6</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

			==Gallery==
			<gallery>
			Image:Pappus graph colored.svg\|Pappus graph coloured to highlight various cycles.
			Image:Pappus graph 3color edge.svg\|The [[chromatic index]] of the Pappus graph is 3.
			Image:Pappus graph 2COL.svg\|The [[chromatic number]] of the Pappus graph is 2.
			</gallery>

			== References ==
			{{reflist}}

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

en>Rjwilmsi: /* References */Journal cites, added 1 DOI, using AWB (8079)

2012-06-18T10:25:31Z

References: Journal cites, added 1 DOI, using AWB (8079)

New page

The author's name is Christy. Her family life in Ohio but her husband desires them to move. It's not a common factor but what she likes performing is to perform domino but she doesn't have the time recently. Credit authorising is how she makes [http://Test.jeka-nn.ru/node/129 best psychic readings] a residing.<br><br>Also visit my blog post; real [http://www.seekavideo.com/playlist/2199/video/ phone psychic readings] ([http://www.january-yjm.com/xe/index.php?mid=video&document_srl=158289 http://www.january-yjm.com/xe/index.php?mid=video&document_srl=158289])

← Older revision		Revision as of 14:56, 24 September 2014
Line 1:		Line 1:
	~~{{infobox graph~~		She is known by the title of Myrtle Shryock. Years in the past we moved to Puerto Rico and my family members loves it. My day occupation is a librarian. To gather coins is what his family members and him appreciate.<br><br>Also visit my web-site - [http://www.pornextras.info/user/G49Z http://www.pornextras.info/user/G49Z]
	~~\| name = Pappus graph~~
	~~\| image = [[Image:Pappus graph LS.svg\|250px]]~~
	~~\| image_caption = The Pappus graph.~~
	~~\| namesake = [[Pappus of Alexandria]]~~
	~~\| vertices = 18~~
	~~\| edges = 27~~
	~~\| automorphisms=216~~
	~~\| girth = 6~~
	~~\| radius = 4~~
	~~\| diameter = 4~~
	~~\| chromatic_number = 2~~
	~~\| chromatic_index = 3~~
	\| properties = [[Bipartite graph\|Bipartite]]<br>[[Symmetric graph\|Symmetric]]<br>[[Distance-transitive graph\|Distance-transitive]]<br>[[Distance-regular graph\|Distance-regular]]<br>[[Cubic graph\|Cubic]]<br>[[Hamiltonian graph\|Hamiltonian]]
	}}

	~~In the [[mathematics\|mathematical]] field of [[graph theory]], the '''Pappus graph'''~~ is ~~a [[Bipartite graph\|bipartite]] 3-[[regular graph\|regular]] [[undirected graph]] with 18 vertices and 27 edges, formed as~~ the ~~[[Levi graph]] of the [[Pappus configuration]].<ref>{{MathWorld\|urlname=PappusGraph\|~~title~~=Pappus Graph}}</ref> It is named after [[Pappus~~ of ~~Alexandria]], an ancient [[Greek mathematics\|Greek mathematician]] who is believed to have discovered the "hexagon theorem" describing the Pappus configuration~~. ~~All~~ the ~~[[cubic graph\|cubic]] [[distance-regular graph]]s are known; the Pappus graph is one of the 13 such graphs.<ref>Brouwer, A. E.; Cohen, A. M.;~~ and ~~Neumaier, A~~. ~~Distance-Regular Graphs. New York: Springer-Verlag, 1989.</ref>~~

	~~The Pappus graph has [[Crossing number (graph theory)\|rectilinear crossing number]] 5, and~~ is ~~the smallest cubic graph with that crossing number {{OEIS\|id=A110507}}~~. ~~It has [[Girth (graph theory)\|girth]] 6, diameter 4, radius 4, [[chromatic number]] 2, [[chromatic index]] 3 and~~ is ~~both 3-[[k-vertex-connected graph\|vertex-connected]]~~ and ~~3-[[k-edge-connected graph\|edge-connected]]~~.

	~~The Pappus graph has a [[chromatic polynomial]] equal to:~~ <~~math~~>~~(x-1)x(x^{16}-26x^{15}+325x^{14}-2600x^{13}+14950x^{12}-65762x^{11}+~~<~~/math~~>~~<math>229852x^{10}~~-~~653966x^9+1537363x^8~~-~~3008720x^7+4904386x^6-</math><math>6609926x^5+7238770x^4-6236975x^3+3989074x^2-1690406x+356509)</math>.~~

	~~The name "Pappus graph" has also been used to refer to a related nine-vertex graph,<ref>{{citation~~
	~~\| last = Kagno \| first = I. N.~~
	~~\| title = Desargues' and Pappus' graphs and their groups~~
	~~\| journal = American Journal of Mathematics~~
	~~\| year = 1947~~
	~~\| volume = 69~~
	~~\| issue = 4~~
	~~\| pages = 859–863~~
	~~\| doi = 10.2307/2371806~~
	~~\| jstor = 2371806~~
	\| publisher = The Johns Hopkins University Press}}</ref> with a vertex for each point of the Pappus configuration and an edge for every pair of points on the same line; this nine-vertex graph is 6-regular, and is the [[complement graph]] of the union of three disjoint [[triangle graph]]s. The first Pappus graph can be embedded in the torus to form a [[regular map (graph theory)\|regular map]] with nine hexagonal faces; the second, to form a regular map with 18 triangular faces.

	~~==Algebraic properties==~~
	The automorphism group of the Pappus graph is a group of order 216. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Pappus 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 Pappus graph, referenced as F018A, is the only cubic symmetric graph on 18 vertices.<ref>Royle, G. [http://www.~~cs.uwa.edu~~.au/~~~gordon~~/~~remote~~/~~foster~~/~~#census "Cubic Symmetric Graphs (The Foster Census)."]</ref><ref>[[Marston Conder\|Conder, M~~.~~]] and Dobcsányi, P~~. ~~"Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.<~~/~~ref>~~

	~~The [[characteristic polynomial]] of the Pappus graph is <math>(x-3) x^4 (x+3) (x^2-3)^6<~~/~~math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.~~

	~~==Gallery==~~
	~~<gallery>~~
	~~Image:Pappus graph colored.svg\|Pappus graph coloured to highlight various cycles.~~
	~~Image:Pappus graph 3color edge.svg\|The [[chromatic index]] of the Pappus graph is 3.~~
	~~Image:Pappus graph 2COL.svg\|The [[chromatic number]] of the Pappus graph is 2.~~
	~~</gallery>~~

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

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