Taylor Expansion Proof of the Matrix Tree Theorem - Part I

TL;DR

This paper presents a concise proof of the Matrix-Tree Theorem using Taylor expansions, providing a novel mathematical approach to a classical graph theory result.

Contribution

It introduces a new proof technique for the Matrix-Tree Theorem based on Taylor expansion comparisons, simplifying the understanding of the theorem.

Findings

The proof offers a shorter, more elegant demonstration of the theorem.

Taylor expansion method can be applied to other combinatorial proofs.

The approach clarifies the relationship between Laplacian cofactors and spanning trees.

Abstract

The Matrix-Tree Theorem states that the number of spanning trees of a graph is given by the absolute value of any cofactor of the Laplacian matrix of the graph. We propose a very short proof of this result which amounts to comparing Taylor expansions.