Tutte matrix

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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 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 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. (It should be noted that 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.