Editing Complete bipartite graph (section)

{{short description|Bipartite graph where each node of 1st set is linked to all nodes of 2nd set}}
{{infobox graph
 | name = Complete bipartite graph
 | image = [[Image:Biclique_K_3_5.svg|170px]]
 | image_caption = A complete bipartite graph with {{math|1=''m'' = 5}} and {{math|1=''n'' = 3}}
 | automorphisms = <math>\left\{\begin{array}{ll}2 m! n! & n = m\\ m! n! & \text{otherwise}\end{array}\right.</math>
 | vertices = {{math|''n'' + ''m''}}
 | edges = {{mvar|mn}}
 | chromatic_number = 2
 | chromatic_index = {{math|max{''m'', ''n''} }}
 | radius = <math>\left\{\begin{array}{ll}1 & m = 1 \vee n = 1\\ 2 & \text{otherwise}\end{array}\right.</math>
 | diameter = <math>\left\{\begin{array}{ll}1 & m = n = 1\\ 2 & \text{otherwise}\end{array}\right.</math>
 | girth = <math>\left\{\begin{array}{ll}\infty & m = 1 \lor n = 1\\ 4 & \text{otherwise}\end{array}\right.</math>
 | spectrum = <math>\left\{0^{n + m - 2}, (\pm\sqrt{nm})^1\right\}</math>
 | notation = {{math|''K''{{sub|''m'',''n''}}}}
}}

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
 | last1=Bondy
 | first1=John Adrian
 | author-link1=John Adrian Bondy
 | last2=Murty
 | first2=U. S. R.
 | author-link2=U. S. R. Murty
 | page=[https://archive.org/details/graphtheorywitha0000bond/page/5 5]
 | title=Graph Theory with Applications
 | year=1976
 | publisher=North-Holland
 | isbn=0-444-19451-7
 | url=https://archive.org/details/graphtheorywitha0000bond/page/5
}}.</ref><ref name="d">{{Citation
 | last=Diestel | first=Reinhard | author-link = Reinhard Diestel
 | title=Graph Theory
 | publisher=[[Springer Science+Business Media|Springer]]
 | year=2005
 | edition=3rd
 | isbn=3-540-26182-6
}}. [http://diestel-graph-theory.com/ Electronic edition], page 17.</ref>

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|author-link=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 |contribution-url=https://books.google.com/books?id=vj1oAgAAQBAJ&pg=PA7 |isbn=978-0191630620}}.
</ref>