Principal minors Pfaffian half-tree theorem

TL;DR

This paper proves a new half-tree theorem for Pfaffian principal minors of skew-symmetric matrices with zero column sum, linking perfect matchings and trees, and extends the matrix-tree theorem with explicit algorithms.

Contribution

It introduces a novel half-tree theorem for Pfaffian minors, providing explicit algorithms and generalizations of the matrix-tree theorem for skew-symmetric matrices.

Findings

Established a half-tree theorem for Pfaffian principal minors.

Developed an explicit algorithm to characterize involved half-trees.

Extended the matrix-tree theorem to skew-symmetric matrices with zero column sum.

Abstract

A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an explicit algorithm, we fully characterize half-trees involved. This question naturally arose in the context of statistical mechanics where we aimed at relating perfect matchings and trees on the same graph. As a consequence of the Pfaffian half-tree theorem, we obtain a refined version of the matrix-tree theorem in the case of skew-symmetric matrices, as well as a line-bundle version of this result.