Tutte matrix
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is [math]\displaystyle{ V = \{1, 2, \dots , n\} }[/math] then the Tutte matrix is an n × n matrix A with entries
- [math]\displaystyle{ A_{ij} = \begin{cases} x_{ij}\;\;\mbox{if}\;(i,j) \in E \mbox{ and } i\lt j\\ -x_{ji}\;\;\mbox{if}\;(i,j) \in E \mbox{ and } i\gt j\\ 0\;\;\;\;\mbox{otherwise} \end{cases} }[/math]
where the xij are indeterminates. The determinant of this skew-symmetric matrix is then a polynomial (in the variables xij, i < j ): this coincides with the square of the pfaffian of the matrix A and is non-zero (as a polynomial) if and only if a perfect matching exists. (This polynomial is not the Tutte polynomial of G.)
The Tutte matrix is named after W. T. Tutte, and is a generalisation of the Edmonds matrix for a balanced bipartite graph.
References
- R. Motwani, P. Raghavan (1995). Randomized Algorithms. Cambridge University Press. p. 167.
- Allen B. Tucker (2004). Computer Science Handbook. CRC Press. p. 12.19. ISBN 1-58488-360-X.
- {{cite journal|url= http://jlms.oxfordjournals.org/cgi/reprint/s1-22/2/107.pdf%7Ctitle= The factorization of linear graphs
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