Nullity Invariance for Pivot and the Interlace Polynomial

We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof.

💡 Research Summary

The paper investigates the effect of the principal pivot transform (PPT) on the nullity (dimension of the nullspace) of principal submatrices of a square matrix and shows that this effect is governed by the symmetric difference operator on index sets. After recalling the definition of PPT—an operation that, given a square matrix A and a subset S of its row/column indices with A