Grassmann-Berezin Calculus and Theorems of the Matrix-Tree Type

TL;DR

This paper extends classical matrix-tree theorems using Grassmann-Berezin calculus, providing new proofs and generalizations including a 'massive' case and a hyperpfaffian-tree theorem, bridging combinatorics and physics.

Contribution

It introduces novel generalizations of matrix-tree theorems utilizing Grassmann-Berezin calculus, including a new hyperpfaffian-tree theorem and a noninductive proof approach.

Findings

Extended the all minors matrix-tree theorem to the massive case.

Generalized the Pfaffian-tree theorem to a hyperpfaffian-cactus theorem.

Provided explicit, noninductive proofs using Grassmann-Berezin calculus.

Abstract

We prove two generalizations of the matrix-tree theorem. The first one, a result essentially due to Moon for which we provide a new proof, extends the ``all minors'' matrix-tree theorem to the ``massive'' case where no condition on row or column sums is imposed. The second generalization, which is new, extends the recently discovered Pfaffian-tree theorem of Masbaum and Vaintrob into a ``Hyperpfaffian-cactus'' theorem. Our methods are noninductive, explicit and make critical use of Grassmann-Berezin calculus that was developed for the needs of modern theoretical physics.