Edmonds matrix

In graph theory, the Edmonds matrix $$A$$ of a balanced bipartite graph $$G = (U, V, E)$$ with sets of vertices $$U = \{u_1, u_2, \dots, u_n \}$$ and $$V = \{v_1, v_2, \dots , v_n\}$$ is defined by


 * $$ A_{ij} = \left\{ \begin{array}{ll}

x_{ij} & (u_i, v_j) \in E \\ 0 & (u_i, v_j) \notin E \end{array}\right.$$

where the xij are indeterminates. One application of the Edmonds matrix of a bipartite graph is that the graph admits a perfect matching if and only if the polynomial det(Aij) in the xij is not identically zero. Furthermore, the number of perfect matchings is equal to the number of monomials in the polynomial det(A), and is also equal to the permanent of $$A$$. In addition, rank of $$A$$ is equal to the maximum matching size of $$G$$.

The Edmonds matrix is named after Jack Edmonds. The Tutte matrix is a generalisation to non-bipartite graphs.