Into the Woods: A graph theoretic proof of the Dodgson/Muir identity

TL;DR

This paper presents a graph theoretic proof of the Dodgson/Muir Identity by establishing a forest identity through the matrix tree theorem and an edge-swapping involution, extending previous algorithms.

Contribution

It introduces a new forest identity equivalent to the Dodgson/Muir Identity, generalizing the Red Hot Potato algorithm for determinant identities.

Findings

Proves the generalized Forest Identity using graph theory.

Establishes the Dodgson/Muir Identity via an edge-swapping involution.

Extends the Red Hot Potato algorithm to a broader class of determinant identities.

Abstract

The Dodgson/Muir Identity is an identity on determinants of matrix minors that generalizes the Dodgson Identity. Using the matrix tree theorem, we create an equivalent forest identity on ordered sets of $k$ forests. We then prove the generalized Forest Identity, and by extension the Dodgson/Muir Identity, using an edge-swapping involution. This algorithm is a generalization of the Red Hot Potato algorithm, developed by the author in 2021 to prove the Dodgson Identity.