# Tutte matrix

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In graph theory, the Tutte matrix ${\displaystyle A}$ 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 2n elements then the Tutte matrix is a 2n × 2n 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 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. (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.