Taylor Expansion Proof of the Matrix Tree Theorem - Part II

TL;DR

This paper provides a concise proof of the All Minors Matrix Tree Theorem using Taylor series expansions, simplifying the understanding of how determinants relate to forests in matrices with zero-sum columns.

Contribution

It introduces a novel, succinct proof of the theorem by leveraging Taylor series, offering a new perspective on the relationship between determinants and forests.

Findings

Short proof of the All Minors Matrix Tree Theorem

Demonstrates the use of Taylor expansions in matrix combinatorics

Simplifies understanding of determinants in zero-column-sum matrices

Abstract

The All Minors Matrix Tree Theorem states that the determinant of any submatrix of a matrix whose columns sum to zero can be computed as a sum over certain oriented forests. We offer a particularly short proof of this result, which amounts to comparing Taylor series expansions.