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| {{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
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| | location = New York
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| | mr = 1481157
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| | page = 137
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| | publisher = John Wiley & Sons Inc.
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| | series = Wiley-Interscience Series in Discrete Mathematics and Optimization
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| | title = Fractional graph theory
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| | year = 1997}}.</ref> or '''semiregular bipartite graph'''<ref>{{citation
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| | last1 = Dehmer | first1 = Matthias
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| | last2 = Emmert-Streib | first2 = Frank
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| | isbn = 9783527627998
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| | page = 149
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| | 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.
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| | |
| [[File:Rhombicdodecahedron.jpg|thumb|left|The graph of the [[rhombic dodecahedron]] is biregular.]]
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| | |
| ==Example==
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| Every [[complete bipartite graph]] <math>K_{a,b}</math> is <math>(b,a)</math>-biregular.<ref name="ls03"/>
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| The [[rhombic dodecahedron]] is another example; it is (3,4)-biregular.<ref>{{citation
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| | last = Réti | first = Tamás
| |
| | journal = MATCH Commun. Math. Comput. Chem.
| |
| | pages = 169–188
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| | 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
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| | year = 2012}}.</ref>
| |
| | |
| ==Vertex counts==
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| 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.
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| | |
| ==Symmetry==
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| Every [[regular graph|regular]] bipartite graph is also biregular.
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| 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
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| | pages = 20–21
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| | publisher = Cambridge University Press
| |
| | series = London Mathematical Society Student Texts
| |
| | title = Topics in Graph Automorphisms and Reconstruction
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| | url = http://books.google.com/books?id=hsymFm0E0uIC&pg=PA20
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| | year = 2003}}.</ref> In particular every edge-transitive graph is either regular or biregular.
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| | |
| ==Configurations==
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| 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>
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| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Graph theory]]
| |
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.
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.
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http://www.shlgrouptr.com/warm/?p=234
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http://www.shlgrouptr.com/warm/?p=421
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