Digraph Branchings and Matrix Determinants

TL;DR

This paper extends the matrix-tree theorem to directed graphs with non-zero column sums, relating matrix determinants to arborescences and forests, and applies these results to system evolution and determinant computation strategies.

Contribution

It introduces a generalized matrix-tree theorem for directed graphs with non-zero column sums and derives a matrix-forest theorem, expanding the theoretical framework for analyzing matrix determinants.

Findings

Extended the matrix-tree theorem to directed graphs with non-zero column sums.

Proved a matrix-forest theorem relating minors to arborescences and forests.

Applied the theorems to system evolution calculations and determinant computation methods.

Abstract

We present a version of the matrix-tree theorem, which relates the determinant of a matrix to sums of weights of arborescences of its directed graph representation. Our treatment allows for non-zero column sums in the parent matrix by adding a root vertex to the usually considered matrix directed graph. We use our result to prove a version of the matrix-forest, or all-minors, theorem, which relates minors of the matrix to forests of arborescences of the matrix digraph. We apply the theorems to calculations of the time-evolution of a system with discrete states and then consider two strategies using these theorems to compute determinants.