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 x_ij 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(A_ij) in the x_ij 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.

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

References

R. Motwani, P. Raghavan (1995). Randomized Algorithms. Cambridge University Press. p. 167. http://books.google.com/books/cambridge?id=QKVY4mDivBEC&pg=PR5&sig=8KZG5MvVdHKKRcLYdN91fGyIrBQ#PPA167,M1.
Allen B. Tucker (2004). Computer Science Handbook. CRC Press. p. 12.19. ISBN 158488360X.