An identity for matching and skew-symmetric determinant

Abstract

Let A = ( A ij ) 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 Tutte's 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 A ij ( i < j) are independent parameters. The present paper gives a weighted and nongeneric version of this relation. Let A( x) = ( A ij ( x)) be an n × n skew-symmetric polynomial matrix in x, and define δ( A) = deg x det A( x) [the degree of the determinant of A( x)]. We attach deg x A ij ( x) to edge ( i, j) of G( A) as a weight and define \\ ̂ gd(A) = 2× [maximum weight of a perfect matching in G( A)]. Then δ(A) ≤ \\ ̂ gd(A) , with equality in the generic case. It is proven by a combinatorial argument that the gap between δ( A) and \\ ̂ gd(A) , 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 A'( x) = U( x) A( x) U( x) T satisfies [δ(A) =] δ(A') = \\ ̂ gd(A') . The proof relies on the dual integrality of perfect matching polytopes known in polyhedral combinatorics.