Editing Bipartite graph (section)

==Additional applications==
Bipartite graphs are extensively used in modern [[coding theory]], especially to decode [[Code word (communication)|codeword]]s received from the channel. [[Factor graph]]s and [[Tanner graph]]s are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors.<ref>{{citation
 | last = Moon | first = Todd K.
 | isbn = 9780471648000
 | page = 638
 | publisher = John Wiley & Sons
 | title = Error Correction Coding: Mathematical Methods and Algorithms
 | url = https://books.google.com/books?id=adxb8CRx5vQC&pg=PA638
 | year = 2005}}.</ref> A factor graph is a closely related [[belief network]] used for probabilistic decoding of [[LDPC]] and [[turbo codes]].<ref>{{harvtxt|Moon|2005}}, p. 686.</ref>

In computer science, a [[Petri net]] is a mathematical modeling tool used in analysis and simulations of concurrent systems. A system is modeled as a bipartite directed graph with two sets of nodes:  A set of "place" nodes that contain resources, and a set of "event" nodes which generate and/or consume resources.  There are additional constraints on the nodes and edges that constrain the behavior of the system.  Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.<ref>{{citation
 | last1 = Cassandras | first1 = Christos G.
 | last2 = Lafortune | first2 = Stephane
 | edition = 2nd
 | isbn = 9780387333328
 | page = 224
 | publisher = Springer
 | title = Introduction to Discrete Event Systems
 | url = https://books.google.com/books?id=AxguNHDtO7MC&pg=PA224
 | year = 2007}}.</ref>

In [[projective geometry]], [[Levi graph]]s are a form of bipartite graph used to model the incidences between points and lines in a [[configuration (geometry)|configuration]]. Corresponding to the geometric property of points and lines that every two lines meet in at most one point and every two points be connected with a single line, Levi graphs necessarily do not contain any cycles of length four, so their [[girth (graph theory)|girth]] must be six or more.<ref>{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | isbn = 9780821843086
 | page = 28
 | publisher = [[American Mathematical Society]]
 | series = [[Graduate Studies in Mathematics]]
 | title = Configurations of Points and Lines
 | url = https://books.google.com/books?id=mRw571GNa5UC&pg=PA28
 | volume = 103
 | year = 2009}}.</ref>