Abstract :
Let A = (Aij) be an n × n skew-symmetric matrix and G(A) be the associated graph; the vertices of G(A) are identified with the rows (and columns) of A, and the edges with the nonzero entries of A. A basic fact, used in Tutteʹs 1947 paper on matching, is: det A ≠ 0 if and only if G(A) has a perfect matching. This holds generically, i.e., when the nonzero entries Aij (i < j) are independent parameters. The present paper gives a weighted and nongeneric version of this relation. Let A(x) = (Aij(x)) be an n × n skew-symmetric polynomial matrix in x, and define δ(A) = degx det A(x) [the degree of the determinant of A(x)]. We attach degx Aij(x) to edge (i, j) of G(A) as a weight and define image [maximum weight of a perfect matching in G(A)]. Then image, with equality in the generic case. It is proven by a combinatorial argument that the gap between δ(A) and image, if any, can be resolved by an unimodular congruence transformation. That is, for a nonsingular A(x) there exists a unimodular U(x) such that Aʹ(x) = U(x)A(x)U(x)T satisfies image. The proof relies on the dual integrality of perfect matching polytopes known in polyhedral combinatorics.
