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| {{infobox graph
| | You'll find lots of people that use medicines now. This want to be prevented and folks ought to discover in the previous. You will discover a whole lot of men and women that expire for the reason that of unique medications, but other individuals never ever find out. They are destroying their lives each and every and every single time they take medications. Should you are a personnel, an individual or what ever your career is, the time can come which you might have a drug test. So how would you pass a drug test? You will find suggests so as for you personally to pass a drug test.<br><br> |
| | name = Complete bipartite graph
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| | image = [[Image:Biclique_K_3_5.svg|170px]]
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| | image_caption = A complete bipartite graph with ''m'' = 5 and ''n'' = 3
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| | automorphisms = <math>\left\{\begin{array}{ll}2 m! n! & n = m\\ m! n! & \text{otherwise}\end{array}\right.</math>
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| | vertices = ''n'' + ''m''
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| | edges = ''mn''
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| | chromatic_number = 2
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| | chromatic_index = max{''m'', ''n''}
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| | radius = <math>\left\{\begin{array}{ll}1 & m = 1 \vee n = 1\\ 2 & \text{otherwise}\end{array}\right.</math>
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| | diameter = <math>\left\{\begin{array}{ll}1 & m = n = 1\\ 2 & \text{otherwise}\end{array}\right.</math>
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| | girth = <math>\left\{\begin{array}{ll}\infty & m = 1 \vee n = 1\\ 4 & \text{otherwise}\end{array}\right.</math>
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| | spectrum = <math>\{0^{n + m - 2}, (\pm \sqrt{n m})^1\}</math>
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| | notation = <math>K_{m,n}</math>
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| }}
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| In the [[mathematical]] field of [[graph theory]], a '''complete bipartite graph''' or '''biclique''' is a special kind of [[bipartite graph]] where every [[vertex (graph theory)|vertex]] of the first set is connected to every vertex of the second set.<ref name="bm">{{citation
| | The First thing you should do is to cease employing medications. This may perhaps seem easy to say yet it can take some time to truly stop using it, particularly after you are certainly happy of it. So even it requires time, you actually ought to be identified. Devoid of dedication, you can not pass a drug test. Just consider of the results if you're continuously using drugs. To keep away from all of these kinds of challenges, make specific that you are prepared to do away with medications inside your regimen.<br><br>Then when you happen to be obvious from prescription drugs, the following issue you have to do is to beverage lots of water. This will likely allow you to complete and pee to take away the drug like weed. You can even ingest a gallon of normal water in the event you essentially want to remove medicines in your system for the short term. This approach just is not permanent, but it may preserve you to pass the drug test. There are actually many men and women which use drugs and continue requesting "how will you pass a drug test?" By far the most productive possible response is to stay away from it.<br><br>Substitution pee or synthetic test checks are just a couple of options as a way to pass a drug. Using the replacement strategy is often a uncomplicated way, nevertheless it requires time. You might have to have someone else's urine plus it ought to be covered totally. So as that this could be comprised, you also can place it within a freezer to have the pee final for a longer time. It just requires two days prior to the harmful bacteria pollute the urine. Once this happens during the test, there'll be suspicions and you may well go into problems whenever you get captured. So be certain to get it totally closed and secure. There are actually 2 kinds of a man-made trial, the powder and fluid type. The powder kind can be mixed to h2o along with the liquid form really should be preserved with all the appropriate temp.<br><br>Despite the fact that you will find lots of different approaches to pass a drug test, the very best approach is to quit making use of medications. For men and women that do not employ medicines, usually do not even feel about utilizing it only for practical experience. This can eliminate your life and it isn't an excellent factor to utilize it in the initial location. Then when another person requests you "how would you pass a drug test?", just explain to preserve clear of it. They then will undoubtedly be free from your anxieties to pass the drug test ([http://Www.Youtube.com/watch?v=6fC3FG8R8Cc click the up coming post]). You'll be free through the damage and you will hold a very good, healthy lifestyle. |
| | last1=Bondy | first1=John Adrian | authorlink1=John Adrian Bondy
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| | last2=Murty | first2=U. S. R. | authorlink2=U. S. R. Murty
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| | page = 5
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| | title=Graph Theory with Applications
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| | year=1976
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| | publisher=North-Holland
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| | isbn=0-444-19451-7}}.</ref><ref name="d">{{Citation
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| | last=Diestel | first=Reinhard
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| | title=Graph Theory
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| | publisher=[[Springer Science+Business Media|Springer]]
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| | year=2005
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| | edition=3rd
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| | isbn=3-540-26182-6
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| }}. [http://diestel-graph-theory.com/ Electronic edition], page 17.</ref>
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| Graph theory itself is typically dated as beginning with [[Leonhard Euler]]'s 1736 work on the [[Seven Bridges of Königsberg]]. However, [[graph drawing|drawing]]s of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of [[Ramon Llull]] edited by [[Athanasius Kircher]].<ref name="knuth"/><ref>{{citation|title=An Atlas of Graphs|first1=Ronald C.|last1=Read|first2=Robin J.|last2=Wilson|publisher=Clarendon Press|year=1998|isbn=9780198532897|page=ii}}.</ref> Llull himself had made similar drawings of [[complete graph]]s three centuries earlier.<ref name="knuth">{{citation|contribution=Two thousand years of combinatorics|first=Donald E.|last=Knuth|authorlink=Donald Knuth|pages=7–37|title=Combinatorics: Ancient and Modern|publisher=Oxford University Press|year=2013|editor1-first=Robin|editor1-last=Wilson|editor2-first=John J.|editor2-last=Watkins}}.
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| </ref>
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| == Definition ==
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| A '''complete bipartite graph''' is a graph whose vertices can be partitioned into two subsets ''V''<sub>1</sub> and ''V''<sub>2</sub> such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a [[bipartite graph]] (''V''<sub>1</sub>, ''V''<sub>2</sub>, ''E'') such that for every two vertices ''v''<sub>1</sub> ∈ ''V''<sub>1</sub> and ''v''<sub>2</sub> ∈ ''V''<sub>2</sub>, ''v''<sub>1</sub>''v''<sub>2</sub> is an edge in ''E''. A complete bipartite graph with partitions of size |''V''<sub>1</sub>|=''m'' and |''V''<sub>2</sub>|=''n'', is denoted ''K''<sub>m,n</sub>;<ref name="bm"/><ref name="d"/> every two graphs with the same notation are [[graph isomorphism|isomorphic]].
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| == Examples ==
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| [[Image:Star graphs.svg|thumb|500px|right|The star graphs ''S''<sub>3</sub>, ''S''<sub>4</sub>, ''S''<sub>5</sub> and ''S''<sub>6</sub>.]]
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| [[Image:Biclique_K_3_3.svg|thumb|150px|right|The utility graph ''K''<sub>3,3</sub>]]
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| * For any ''k'', ''K''<sub>1,''k''</sub> is called a [[Star (graph theory)|star]].<ref name="d"/> All complete bipartite graphs which are [[Tree (graph theory)|trees]] are stars.
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| * The graph ''K''<sub>1,3</sub> is called a [[Claw (graph theory)|claw]], and is used to define the [[claw-free graph]]s.<ref>{{citation
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| | last1 = Lovász | first1 = László | author1-link = László Lovász
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| | last2 = Plummer | first2 = Michael D. | author2-link = Michael D. Plummer
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| | isbn = 978-0-8218-4759-6
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| | mr = 2536865
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| | page = 109
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| | publisher = AMS Chelsea Publishing, Providence, RI
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| | title = Matching theory
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| | url = http://books.google.com/books?id=yW3WSVq8ygcC&pg=PA109
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| | year = 2009}}. Corrected reprint of the 1986 original.</ref>
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| * The graph ''K''<sub>3,3</sub> is called the [[Water, gas, and electricity|utility graph]]. This usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings; it is impossible to solve without crossings due to the [[planar graph|nonplanarity]] of ''K''<sub>3,3</sub>.<ref>{{citation|title=A Logical Approach to Discrete Math|first1=David|last1=Gries|author1-link=David Gries|first2=Fred B.|last2=Schneider|author2-link=Fred B. Schneider|publisher=Springer|year=1993|isbn=9780387941158|page=437|url=http://books.google.com/books?id=ZWTDQ6H6gsUC&pg=PA437}}.</ref>
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| == Properties ==
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| *Given a bipartite graph, testing whether it contains a complete bipartite subgraph ''K''<sub>''i'',''i''</sub> for a parameter ''i'' is an [[NP-complete]] problem.<ref>{{citation
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| | last1=Garey | first1=Michael R. | authorlink1=Michael R. Garey
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| | last2=Johnson | first2=David S. | authorlink2=David S. Johnson
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| | contribution = [GT24] Balanced complete bipartite subgraph
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| | page = 196
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| | year=1979
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| | title=[[Computers and Intractability: A Guide to the Theory of NP-Completeness]]
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| | publisher=[[W. H. Freeman|W. H. Freeman]]
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| | isbn=0-7167-1045-5
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| }}.</ref>
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| *A [[planar graph]] cannot contain ''K''<sub>3,3</sub> as a [[minor (graph theory)|minor]]; an [[outerplanar graph]] cannot contain ''K''<sub>3,2</sub> as a minor (These are not [[sufficient condition]]s for planarity and outerplanarity, but necessary). Conversely, every nonplanar graph contains either ''K''<sub>3,3</sub> or the [[complete graph]] ''K''<sub>5</sub> as a minor; this is [[Wagner's theorem]].<ref>Diestel, elect. ed. p. 105.</ref>
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| *Every complete bipartite graph. '''K'''<sub>''n'',''n''</sub> is a [[Moore graph]] and a (''n'',4)-[[cage (graph theory)|cage]].<ref>{{citation|title=Algebraic Graph Theory|first=Norman|last=Biggs|publisher=Cambridge University Press|year=1993|isbn=9780521458979|page=181|url=http://books.google.com/books?id=6TasRmIFOxQC&pg=PA181}}.</ref>
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| *The complete bipartite graphs '''K'''<sub>''n'',''n''</sub> and '''K'''<sub>''n'',''n''+1</sub> have the maximum possible number of edges among all [[triangle-free graph]]s with the same number of vertices; this is [[Mantel's theorem]]. Mantel's result was generalized to ''k''-partite graphs and graphs that avoid larger cliques as subgraphs in [[Turán's theorem]], and these two complete bipartite graphs are examples of [[Turán graph]]s, the extremal graphs for this more general problem.<ref>{{citation|title=Modern Graph Theory|volume=184|series=Graduate Texts in Mathematics|first=Béla|last=Bollobás|authorlink=Béla Bollobás|publisher=Springer|year=1998|isbn=9780387984889|page=104|url=http://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA104}}.</ref>
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| *The complete bipartite graph '''K'''<sub>''m'',''n''</sub> has a [[vertex covering number]] of '''min'''{''m'',''n''} and an [[edge covering number]] of '''max'''{''m'',''n''}.
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| *The complete bipartite graph '''K'''<sub>''m'',''n''</sub> has a [[maximum independent set]] of size '''max'''{''m'',''n''}.
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| *The [[adjacency matrix]] of a complete bipartite graph '''K'''<sub>''m'',''n''</sub> has eigenvalues √(''nm''), −√(''nm'') and 0; with multiplicity 1, 1 and ''n''+''m''−2 respectively.<ref>{{harvtxt|Bollobás|1998}}, p. 266.</ref>
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| *The [[laplacian matrix]] of a complete bipartite graph '''K'''<sub>''m'',''n''</sub> has eigenvalues ''n''+''m'', ''n'', ''m'', and 0; with multiplicity 1, ''m''−1, ''n''−1 and 1 respectively.
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| *A complete bipartite graph '''K'''<sub>''m'',''n''</sub> has ''m''<sup>''n''−1</sup> ''n''<sup>''m''−1</sup> [[spanning tree]]s.<ref>{{citation|title=Graphs, Networks and Algorithms|volume=5|series=Algorithms and Computation in Mathematic|first=Dieter|last=Jungnickel|publisher=Springer|year=2012|isbn=9783642322785|page=557|url=http://books.google.com/books?id=PrXxFHmchwcC&pg=PA557}}.</ref>
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| *A complete bipartite graph '''K'''<sub>''m'',''n''</sub> has a [[maximum matching]] of size '''min'''{''m'',''n''}.
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| *A complete bipartite graph '''K'''<sub>''n'',''n''</sub> has a proper [[edge coloring|''n''-edge-coloring]] corresponding to a [[Latin square]].<ref>{{citation|title=Graph Coloring Problems|volume=39|series=Wiley Series in Discrete Mathematics and Optimization|first1=Tommy R.|last1=Jensen|first2=Bjarne|last2=Toft|publisher=John Wiley & Sons|year=2011|isbn=9781118030745|page=16|url=http://books.google.com/books?id=leL0Y5N0bFoC&pg=PA16}}.</ref>
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| == See also ==
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| * [[Crown graph]], a graph formed by removing a [[perfect matching]] from a complete bipartite graph
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| ==References==
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| {{reflist|30em}}
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| [[Category:Parametric families of graphs]]
| |
You'll find lots of people that use medicines now. This want to be prevented and folks ought to discover in the previous. You will discover a whole lot of men and women that expire for the reason that of unique medications, but other individuals never ever find out. They are destroying their lives each and every and every single time they take medications. Should you are a personnel, an individual or what ever your career is, the time can come which you might have a drug test. So how would you pass a drug test? You will find suggests so as for you personally to pass a drug test.
The First thing you should do is to cease employing medications. This may perhaps seem easy to say yet it can take some time to truly stop using it, particularly after you are certainly happy of it. So even it requires time, you actually ought to be identified. Devoid of dedication, you can not pass a drug test. Just consider of the results if you're continuously using drugs. To keep away from all of these kinds of challenges, make specific that you are prepared to do away with medications inside your regimen.
Then when you happen to be obvious from prescription drugs, the following issue you have to do is to beverage lots of water. This will likely allow you to complete and pee to take away the drug like weed. You can even ingest a gallon of normal water in the event you essentially want to remove medicines in your system for the short term. This approach just is not permanent, but it may preserve you to pass the drug test. There are actually many men and women which use drugs and continue requesting "how will you pass a drug test?" By far the most productive possible response is to stay away from it.
Substitution pee or synthetic test checks are just a couple of options as a way to pass a drug. Using the replacement strategy is often a uncomplicated way, nevertheless it requires time. You might have to have someone else's urine plus it ought to be covered totally. So as that this could be comprised, you also can place it within a freezer to have the pee final for a longer time. It just requires two days prior to the harmful bacteria pollute the urine. Once this happens during the test, there'll be suspicions and you may well go into problems whenever you get captured. So be certain to get it totally closed and secure. There are actually 2 kinds of a man-made trial, the powder and fluid type. The powder kind can be mixed to h2o along with the liquid form really should be preserved with all the appropriate temp.
Despite the fact that you will find lots of different approaches to pass a drug test, the very best approach is to quit making use of medications. For men and women that do not employ medicines, usually do not even feel about utilizing it only for practical experience. This can eliminate your life and it isn't an excellent factor to utilize it in the initial location. Then when another person requests you "how would you pass a drug test?", just explain to preserve clear of it. They then will undoubtedly be free from your anxieties to pass the drug test (click the up coming post). You'll be free through the damage and you will hold a very good, healthy lifestyle.