Complete bipartite graph

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	${\displaystyle \left\{{\begin{array}{ll}\infty &m=1\vee n=1\\4&{\text{otherwise}}\end{array}}\right.}$
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	${\displaystyle \{0^{n+m-2},(\pm {\sqrt {nm}})^{1}\}}$
Notation	${\displaystyle K_{m,n}}$

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

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.

A complete bipartite graph is a graph whose vertices can be partitioned into two subsets V₁ and V₂ 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 (V₁, V₂, E) such that for every two vertices v₁ ∈ V₁ and v₂ ∈ V₂, v₁v₂ is an edge in E. A complete bipartite graph with partitions of size |V₁|=m and |V₂|=n, is denoted K_m,n; every two graphs with the same notation are isomorphic.