Tutte matrix

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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 $V=\{1,2,\dots ,n\}$ {\displaystyle V=\{1,2,\dots ,n\}} then the Tutte matrix is an n-by-n matrix A with entries

A_{ij}={\begin{cases}x_{ij}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i<j\\-x_{ji}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i>j\0円\;\;\;\;{\mbox{otherwise}}\end{cases}}

{\displaystyle A_{ij}={\begin{cases}x_{ij}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i<j\\-x_{ji}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i>j\0円\;\;\;\;{\mbox{otherwise}}\end{cases}}}

where the x_ij are indeterminates. The determinant of this skew-symmetric matrix is then a polynomial (in the variables x_ij, 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

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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.
W.T. Tutte (April 1947). "The factorization of linear graphs" (PDF). J. London Math. Soc. 22 (2): 107–111. doi:10.1112/jlms/s1-22.2.107 . Retrieved 2008年06月15日.

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