Ordinal optimization: Difference between revisions

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In [[graph theory]], the '''Edmonds matrix''' <math>A</math> of a balanced [[bipartite graph]] <math>G(U, V, E)</math> with [[Set (mathematics)|set]]s of vertices <math>U = \{u_1, u_2, \dots , u_n \}</math> and <math>V = \{v_1, v_2, \dots , v_n\}</math> is defined by
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:<math> A_{ij} = \left\{ \begin{array}{ll}
  x_{ij} & (u_i, v_j) \in E \\
  0 & (u_i, v_j) \notin E
\end{array}\right.</math>
 
where the ''x''<sub>ij</sub> are indeterminates. One application of the Edmonds matrix of a bipartite graph is that the graph admits a [[perfect matching]] if and only if the polynomial det(''A''<sub>ij</sub>) in the ''x''<sub>ij</sub> is not identically zero. Furthermore, the number of perfect matchings is equal to the number of [[monomials]] in the polynomial det(''A''), and is also equal to the [[permanent]] of ''A''.
 
The Edmonds matrix is named after [[Jack Edmonds]]. The [[Tutte matrix]] is a generalisation to non-bipartite graphs.
 
==References==
*{{cite book|author=R. Motwani, P. Raghavan |title=Randomized Algorithms |url=http://books.google.com/books/cambridge?id=QKVY4mDivBEC&pg=PR5&sig=8KZG5MvVdHKKRcLYdN91fGyIrBQ#PPA167,M1 |publisher=Cambridge University Press|year=1995|page=167}}
*{{cite book|author=Allen B. Tucker|title=Computer Science Handbook|publisher=CRC Press|date=2004|isbn=1-58488-360-X|page=12.19}}
 
{{combin-stub}}
[[Category:Graph theory objects]]
[[Category:Matrices]]

Latest revision as of 07:26, 4 May 2014

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