# Tutte matrix

In graph theory, the Tutte matrix A of a graph G = (VE) 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 ${\displaystyle V=\{1,2,\dots ,n\}}$ then the Tutte matrix is an n × n matrix A with entries

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

where the xij are indeterminates. The determinant of this skew-symmetric matrix is then a polynomial (in the variables xiji < 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.
• 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 2008-06-15.