Binary matroid - Revision history

en>Rjwilmsi: /* Alternative characterizations */Added 1 doi to a journal cite using AWB (10222)

2014-06-08T13:11:11Z

Alternative characterizations: Added 1 doi to a journal cite using AWB (10222)

← Older revision		Revision as of 15:11, 8 June 2014
Line 1:		Line 1:
	~~{{Graph families defined~~ by ~~their automorphisms}}~~		EFI with maps: This is a little easier, but you need to reduce the air flow coming in by the amount of 'gaseous fuel' you're adding.<br>Ugh! I'm so mad that it cut my short, and notice I put winter in quotation marks because we don't get a real winter down here I'm in south Ga and it's 81 today, and the trees are blooming, and there's a light dusting of pollen on the car. Yes it's pretty and I do like spring, but I AM NOT READY FOR SUMMER! We totally overdo summer down here, and everyone makes a fuss about how cold it is in the when it's only 60 50 degrees outside.<br>http://www.shlgrouptr.com/warm/?p=14 <br /> http://www.shlgrouptr.com/warm/?p=234 <br /> http://www.shlgrouptr.com/warm/?p=263 <br /> http://www.shlgrouptr.com/warm/?p=413 <br /> http://www.shlgrouptr.com/warm/?p=421 <br /> <br>http://tw-kb.com/wiki/Five_Ways_Twitter_Destroyed_My_Bags_Outlet_Without_Me_Noticing<br>http://kemikgrefti.net/modules.php?name=Your_Account&op=userinfo&username=SF44<br><br>In the event you loved this informative article and you would want to receive more information relating to [http://www.restaurantcalcuta.com/outlet/ugg.asp Cheap Uggs Boots] kindly visit our web site.
	~~In [[graph theory\|graph-theoretic mathematics]], a '''biregular graph~~'''<~~ref~~>~~{{citation~~
	~~\| last1 = Scheinerman \| first1 = Edward R. \| authorlink = Ed Scheinerman~~
	~~\| last2 = Ullman \| first2 = Daniel H.~~
	~~\| isbn = 0-471-17864-0~~
	~~\| location = New York~~
	~~\| mr = 1481157~~
	~~\| page = 137~~
	~~\| publisher = John Wiley & Sons Inc.~~
	~~\| series = Wiley-Interscience Series~~ in ~~Discrete Mathematics and Optimization~~
	~~\| title = Fractional graph theory~~
	~~\| year = 1997}}.</ref> or~~ '''~~semiregular bipartite graph~~'''<~~ref~~>~~{{citation~~
	~~\| last1 = Dehmer \| first1 = Matthias~~
	~~\| last2 = Emmert-Streib \| first2 = Frank~~
	~~\| isbn = 9783527627998~~
	~~\| page = 149~~
	~~\| publisher = John Wiley & Sons~~
	~~\| title = Analysis of Complex Networks: From Biology to Linguistics~~
	~~\| url =~~ http://~~books~~.~~google~~.com/~~books~~?~~id=l9zURPH2GiUC&pg=PA149&lpg=PA149~~
	~~\| year~~ = ~~2009}}.~~</~~ref~~> ~~is a [[bipartite graph]] <math>G=(U,V,E)<~~/~~math> for which every two vertices on the same side of the given bipartition have the same [[Degree (graph theory)\|degree]] as each other~~. ~~If the degree of the vertices in <math>U<~~/~~math> is <math>x<~~/~~math> and the degree of the vertices in <math>V~~</~~math~~> ~~is <math>y<~~/~~math>, then the graph is said to be <math>(x,y)<~~/~~math>-biregular~~.

	~~[[File:Rhombicdodecahedron~~.~~jpg\|thumb\|left\|The graph of the [[rhombic dodecahedron]] is biregular.]]~~

	~~==Example==~~
	~~Every [[complete bipartite graph]] <math>K_{a,b}<~~/~~math> is <math>(b,a)<~~/~~math>-biregular.~~<~~ref name="ls03"~~/>
	~~The [[rhombic dodecahedron]] is another example; it is (3,4)-biregular.<ref>{{citation~~
	~~\| last = Réti \| first = Tamás~~
	~~\| journal = MATCH Commun. Math. Comput. Chem.~~
	~~\| pages = 169–188~~
	~~\| title = On the relationships between the first and second Zagreb indices~~
	~~\| url =~~ http://www.~~pmf~~.~~kg.ac.rs~~/~~match~~/~~electronic_versions~~/~~match68~~/n1/~~match68n1_169-188~~.~~pdf~~
	~~\| volume = 68~~
	~~\| year = 2012}}~~.</~~ref>~~

	~~==Vertex counts==~~
	~~An <math>(x,y)<~~/~~math>-biregular graph <math>G~~=~~(U,V,E)~~</~~math~~> ~~must satisfy the equation~~ <~~math~~>~~x\|U\|=y\|V\|</math>. This follows from a simple [[Double counting (proof technique)\|double counting argument]]~~: ~~the number of endpoints of edges in <math>U<~~/~~math> is <math>x\|U\|<~~/~~math>, the number of endpoints of edges in <math>V<~~/~~math> is <math>y\|V\|<~~/~~math>, and each edge contributes the same amount (one) to both numbers.~~

	~~==Symmetry==~~
	~~Every [[regular graph\|regular]] bipartite graph is also biregular.~~
	~~Every [[edge-transitive graph]] (disallowing graphs with [[isolated vertex\|isolated vertices]]) that is not also [[vertex-transitive graph\|vertex-transitive]] must be biregular.~~<~~ref name="ls03"~~>~~{{citation~~
	~~\| last1 = Lauri \| first1 = Josef~~
	~~\| last2 = Scapellato \| first2 = Raffaele~~
	~~\| isbn = 9780521529037~~
	~~\| pages = 20–21~~
	~~\| publisher = Cambridge University Press~~
	~~\| series = London Mathematical Society Student Texts~~
	~~\| title = Topics in Graph Automorphisms and Reconstruction~~
	~~\| url =~~ http://~~books~~.~~google~~.~~com/books~~?id=~~hsymFm0E0uIC~~&pg=~~PA20~~
	~~\| year~~ = ~~2003}}.~~<~~/ref~~> In ~~particular every edge-transitive graph is either regular or biregular.~~

	~~==Configurations==~~
	~~The [[Levi graph]]s of [[Configuration (geometry)\|geometric configuration]]s are biregular; a biregular graph is~~ the ~~Levi graph of an (abstract) configuration if~~ and ~~only if its~~ [~~[girth (graph theory)\|girth]] is at least six~~.~~<ref>{{citation~~
	~~\| last = Gropp \| first = Harald~~
	~~\| editor1-last = Colbourn \| editor1-first = Charles J~~.
	~~\| editor2-last = Dinitz \| editor2-first = Jeffrey H.~~
	~~\| contribution = VI.7 Configurations~~
	~~\| edition = Second~~
	~~\| pages = 353–355~~
	~~\| publisher = Chapman & Hall~~/~~CRC, Boca Raton, FL~~
	~~\| series = Discrete Mathematics and its Applications (Boca Raton)~~
	~~\| title = Handbook of combinatorial designs~~
	~~\| year = 2007}}~~.~~</ref>~~

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

	~~[[Category:Graph theory]~~]

en>John of Reading: /* Additional properties */Typo fixing, replaced: a independence oracle → an independence oracle using AWB (8686)

2013-03-01T12:07:24Z

Additional properties: Typo fixing, replaced: a independence oracle → an independence oracle using AWB (8686)

← Older revision		Revision as of 14:07, 1 March 2013
Line 1:		Line 1:
	~~Has learned that Fox st store some models are car sales~~, ~~the highest offer 30,000 yuan, and interested friends may wish to look.~~<br>~~The 5SB has a maximum pumping capacity of 7~~.~~5 gallons per minute and a maximum power rating of 1 horsepower~~. ~~The 7SB pumps a maximum of 10 gallons per minute~~ and ~~generates a maximum of 1 horsepower~~.<br>http://~~eastwest-tours~~.com/~~style/~~?p=29 <br /> ~~http:~~/~~/eastwest-tours~~.~~com~~/~~style~~/~~?p=85~~ <br /> ~~http:~~//~~eastwest~~-~~tours~~.~~com~~/~~style~~/?p=94 <~~br /~~> http://~~eastwest~~-~~tours~~.~~com~~/~~style~~/?p=12 <br /> ~~http:~~/~~/eastwest-tours~~.~~com~~/~~style~~/~~?p=197~~ <br /><br><br>In the ~~event you loved this post and you would like~~ to ~~receive more info regarding~~ [http://~~www~~.~~bendtrapclub~~.com/~~cheap~~/~~ugg~~.~~asp Cheap Uggs Boots~~] ~~assure visit~~ the ~~webpage~~.		{{Graph families defined by their automorphisms}}
			In [[graph theory\|graph-theoretic mathematics]], a '''biregular graph'''<ref>{{citation
			\| last1 = Scheinerman \| first1 = Edward R. \| authorlink = Ed Scheinerman
			\| last2 = Ullman \| first2 = Daniel H.
			\| isbn = 0-471-17864-0
			\| location = New York
			\| mr = 1481157
			\| page = 137
			\| publisher = John Wiley & Sons Inc.
			\| series = Wiley-Interscience Series in Discrete Mathematics and Optimization
			\| title = Fractional graph theory
			\| year = 1997}}.</ref> or '''semiregular bipartite graph'''<ref>{{citation
			\| last1 = Dehmer \| first1 = Matthias
			\| last2 = Emmert-Streib \| first2 = Frank
			\| isbn = 9783527627998
			\| page = 149
			\| publisher = John Wiley & Sons
			\| title = Analysis of Complex Networks: From Biology to Linguistics
			\| url = http://books.google.com/books?id=l9zURPH2GiUC&pg=PA149&lpg=PA149
			\| year = 2009}}.</ref> is a [[bipartite graph]] <math>G=(U,V,E)</math> for which every two vertices on the same side of the given bipartition have the same [[Degree (graph theory)\|degree]] as each other. If the degree of the vertices in <math>U</math> is <math>x</math> and the degree of the vertices in <math>V</math> is <math>y</math>, then the graph is said to be <math>(x,y)</math>-biregular.

			[[File:Rhombicdodecahedron.jpg\|thumb\|left\|The graph of the [[rhombic dodecahedron]] is biregular.]]

			==Example==
			Every [[complete bipartite graph]] <math>K_{a,b}</math> is <math>(b,a)</math>-biregular.<ref name="ls03"/>
			The [[rhombic dodecahedron]] is another example; it is (3,4)-biregular.<ref>{{citation
			\| last = Réti \| first = Tamás
			\| journal = MATCH Commun. Math. Comput. Chem.
			\| pages = 169–188
			\| title = On the relationships between the first and second Zagreb indices
			\| url = http://www.pmf.kg.ac.rs/match/electronic_versions/match68/n1/match68n1_169-188.pdf
			\| volume = 68
			\| year = 2012}}.</ref>

			==Vertex counts==
			An <math>(x,y)</math>-biregular graph <math>G=(U,V,E)</math> must satisfy the equation <math>x\|U\|=y\|V\|</math>. This follows from a simple [[Double counting (proof technique)\|double counting argument]]: the number of endpoints of edges in <math>U</math> is <math>x\|U\|</math>, the number of endpoints of edges in <math>V</math> is <math>y\|V\|</math>, and each edge contributes the same amount (one) to both numbers.

			==Symmetry==
			Every [[regular graph\|regular]] bipartite graph is also biregular.
			Every [[edge-transitive graph]] (disallowing graphs with [[isolated vertex\|isolated vertices]]) that is not also [[vertex-transitive graph\|vertex-transitive]] must be biregular.<ref name="ls03">{{citation
			\| last1 = Lauri \| first1 = Josef
			\| last2 = Scapellato \| first2 = Raffaele
			\| isbn = 9780521529037
			\| pages = 20–21
			\| publisher = Cambridge University Press
			\| series = London Mathematical Society Student Texts
			\| title = Topics in Graph Automorphisms and Reconstruction
			\| url = http://books.google.com/books?id=hsymFm0E0uIC&pg=PA20
			\| year = 2003}}.</ref> In particular every edge-transitive graph is either regular or biregular.

			==Configurations==
			The [[Levi graph]]s of [[Configuration (geometry)\|geometric configuration]]s are biregular; a biregular graph is the Levi graph of an (abstract) configuration if and only if its [[girth (graph theory)\|girth]] is at least six.<ref>{{citation
			\| last = Gropp \| first = Harald
			\| editor1-last = Colbourn \| editor1-first = Charles J.
			\| editor2-last = Dinitz \| editor2-first = Jeffrey H.
			\| contribution = VI.7 Configurations
			\| edition = Second
			\| pages = 353–355
			\| publisher = Chapman & Hall/CRC, Boca Raton, FL
			\| series = Discrete Mathematics and its Applications (Boca Raton)
			\| title = Handbook of combinatorial designs
			\| year = 2007}}.</ref>

			==References==
			{{reflist}}

			[[Category:Graph theory]]

en>David Eppstein: odd determinant isn't the right condition for sets smaller than a basis

2012-08-31T20:36:31Z

odd determinant isn't the right condition for sets smaller than a basis

New page

Has learned that Fox st store some models are car sales, the highest offer 30,000 yuan, and interested friends may wish to look. The 5SB has a maximum pumping capacity of 7.5 gallons per minute and a maximum power rating of 1 horsepower. The 7SB pumps a maximum of 10 gallons per minute and generates a maximum of 1 horsepower. http://eastwest-tours.com/style/?p=29 http://eastwest-tours.com/style/?p=85 http://eastwest-tours.com/style/?p=94 http://eastwest-tours.com/style/?p=12 http://eastwest-tours.com/style/?p=197 In the event you loved this post and you would like to receive more info regarding [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots] assure visit the webpage.

← Older revision		Revision as of 15:11, 8 June 2014
Line 1:		Line 1:
	~~{{Graph families defined~~ by ~~their automorphisms}}~~		EFI with maps: This is a little easier, but you need to reduce the air flow coming in by the amount of 'gaseous fuel' you're adding.<br>Ugh! I'm so mad that it cut my short, and notice I put winter in quotation marks because we don't get a real winter down here I'm in south Ga and it's 81 today, and the trees are blooming, and there's a light dusting of pollen on the car. Yes it's pretty and I do like spring, but I AM NOT READY FOR SUMMER! We totally overdo summer down here, and everyone makes a fuss about how cold it is in the when it's only 60 50 degrees outside.<br>http://www.shlgrouptr.com/warm/?p=14 <br /> http://www.shlgrouptr.com/warm/?p=234 <br /> http://www.shlgrouptr.com/warm/?p=263 <br /> http://www.shlgrouptr.com/warm/?p=413 <br /> http://www.shlgrouptr.com/warm/?p=421 <br /> <br>http://tw-kb.com/wiki/Five_Ways_Twitter_Destroyed_My_Bags_Outlet_Without_Me_Noticing<br>http://kemikgrefti.net/modules.php?name=Your_Account&op=userinfo&username=SF44<br><br>In the event you loved this informative article and you would want to receive more information relating to [http://www.restaurantcalcuta.com/outlet/ugg.asp Cheap Uggs Boots] kindly visit our web site.
	~~In [[graph theory\|graph-theoretic mathematics]], a '''biregular graph~~'''<~~ref~~>~~{{citation~~
	~~\| last1 = Scheinerman \| first1 = Edward R. \| authorlink = Ed Scheinerman~~
	~~\| last2 = Ullman \| first2 = Daniel H.~~
	~~\| isbn = 0-471-17864-0~~
	~~\| location = New York~~
	~~\| mr = 1481157~~
	~~\| page = 137~~
	~~\| publisher = John Wiley & Sons Inc.~~
	~~\| series = Wiley-Interscience Series~~ in ~~Discrete Mathematics and Optimization~~
	~~\| title = Fractional graph theory~~
	~~\| year = 1997}}.</ref> or~~ '''~~semiregular bipartite graph~~'''<~~ref~~>~~{{citation~~
	~~\| last1 = Dehmer \| first1 = Matthias~~
	~~\| last2 = Emmert-Streib \| first2 = Frank~~
	~~\| isbn = 9783527627998~~
	~~\| page = 149~~
	~~\| publisher = John Wiley & Sons~~
	~~\| title = Analysis of Complex Networks: From Biology to Linguistics~~
	~~\| url =~~ http://~~books~~.~~google~~.com/~~books~~?~~id=l9zURPH2GiUC&pg=PA149&lpg=PA149~~
	~~\| year~~ = ~~2009}}.~~</~~ref~~> ~~is a [[bipartite graph]] <math>G=(U,V,E)<~~/~~math> for which every two vertices on the same side of the given bipartition have the same [[Degree (graph theory)\|degree]] as each other~~. ~~If the degree of the vertices in <math>U<~~/~~math> is <math>x<~~/~~math> and the degree of the vertices in <math>V~~</~~math~~> ~~is <math>y<~~/~~math>, then the graph is said to be <math>(x,y)<~~/~~math>-biregular~~.

	~~[[File:Rhombicdodecahedron~~.~~jpg\|thumb\|left\|The graph of the [[rhombic dodecahedron]] is biregular.]]~~

	~~==Example==~~
	~~Every [[complete bipartite graph]] <math>K_{a,b}<~~/~~math> is <math>(b,a)<~~/~~math>-biregular.~~<~~ref name="ls03"~~/>
	~~The [[rhombic dodecahedron]] is another example; it is (3,4)-biregular.<ref>{{citation~~
	~~\| last = Réti \| first = Tamás~~
	~~\| journal = MATCH Commun. Math. Comput. Chem.~~
	~~\| pages = 169–188~~
	~~\| title = On the relationships between the first and second Zagreb indices~~
	~~\| url =~~ http://www.~~pmf~~.~~kg.ac.rs~~/~~match~~/~~electronic_versions~~/~~match68~~/n1/~~match68n1_169-188~~.~~pdf~~
	~~\| volume = 68~~
	~~\| year = 2012}}~~.</~~ref>~~

	~~==Vertex counts==~~
	~~An <math>(x,y)<~~/~~math>-biregular graph <math>G~~=~~(U,V,E)~~</~~math~~> ~~must satisfy the equation~~ <~~math~~>~~x\|U\|=y\|V\|</math>. This follows from a simple [[Double counting (proof technique)\|double counting argument]]~~: ~~the number of endpoints of edges in <math>U<~~/~~math> is <math>x\|U\|<~~/~~math>, the number of endpoints of edges in <math>V<~~/~~math> is <math>y\|V\|<~~/~~math>, and each edge contributes the same amount (one) to both numbers.~~

	~~==Symmetry==~~
	~~Every [[regular graph\|regular]] bipartite graph is also biregular.~~
	~~Every [[edge-transitive graph]] (disallowing graphs with [[isolated vertex\|isolated vertices]]) that is not also [[vertex-transitive graph\|vertex-transitive]] must be biregular.~~<~~ref name="ls03"~~>~~{{citation~~
	~~\| last1 = Lauri \| first1 = Josef~~
	~~\| last2 = Scapellato \| first2 = Raffaele~~
	~~\| isbn = 9780521529037~~
	~~\| pages = 20–21~~
	~~\| publisher = Cambridge University Press~~
	~~\| series = London Mathematical Society Student Texts~~
	~~\| title = Topics in Graph Automorphisms and Reconstruction~~
	~~\| url =~~ http://~~books~~.~~google~~.~~com/books~~?id=~~hsymFm0E0uIC~~&pg=~~PA20~~
	~~\| year~~ = ~~2003}}.~~<~~/ref~~> In ~~particular every edge-transitive graph is either regular or biregular.~~

	~~==Configurations==~~
	~~The [[Levi graph]]s of [[Configuration (geometry)\|geometric configuration]]s are biregular; a biregular graph is~~ the ~~Levi graph of an (abstract) configuration if~~ and ~~only if its~~ [~~[girth (graph theory)\|girth]] is at least six~~.~~<ref>{{citation~~
	~~\| last = Gropp \| first = Harald~~
	~~\| editor1-last = Colbourn \| editor1-first = Charles J~~.
	~~\| editor2-last = Dinitz \| editor2-first = Jeffrey H.~~
	~~\| contribution = VI.7 Configurations~~
	~~\| edition = Second~~
	~~\| pages = 353–355~~
	~~\| publisher = Chapman & Hall~~/~~CRC, Boca Raton, FL~~
	~~\| series = Discrete Mathematics and its Applications (Boca Raton)~~
	~~\| title = Handbook of combinatorial designs~~
	~~\| year = 2007}}~~.~~</ref>~~

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

	~~[[Category:Graph theory]~~]