Complete bipartite graph

In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.

Complete bipartite graph
A complete bipartite graph with m = 5 and n = 3
Vertices	n + m
Edges	mn
Radius	${\displaystyle \left\{{\begin{array}{ll}1&m=1\vee n=1\\2&{\text{otherwise}}\end{array}}\right.}$
Diameter	${\displaystyle \left\{{\begin{array}{ll}1&m=n=1\\2&{\text{otherwise}}\end{array}}\right.}$
Girth	${\displaystyle \left\{{\begin{array}{ll}\infty &m=1\lor n=1\\4&{\text{otherwise}}\end{array}}\right.}$
Automorphisms	${\displaystyle \left\{{\begin{array}{ll}2m!n!&n=m\\m!n!&{\text{otherwise}}\end{array}}\right.}$
Chromatic number	2
Chromatic index	max{m, n}
Spectrum	${\displaystyle \left\{0^{n+m-2},(\pm {\sqrt {nm}})^{1}\right\}}$
Notation	K{m,n}
Table of graphs and parameters

Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings 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. Llull himself had made similar drawings of complete graphs three centuries earlier.