Matrices | Matching (graph theory) | Algebraic graph theory
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 then the Tutte matrix is an n × n matrix A with entries 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. (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. (Wikipedia).
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