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 V has n elements then the Tutte matrix is an n × n matrix A with entries

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

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. (It should be noted that this is not the Tutte polynomial of G.)

The Tutte matrix 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 158488360X.
W.T. Tutte (April 1947). "The factorization of linear graphs". J. London Math. Soc. 22 (2): 107–111. doi:10.1112/jlms/s1-22.2.107. http://jlms.oxfordjournals.org/cgi/reprint/s1-22/2/107.pdf. Retrieved 2008-06-15.