On Directed Graphs with the Same Sum over Arborescence Weights

TL;DR

This paper explores conditions under which different directed graphs with identical vertices have equal sums of arborescence weights, linking these findings to the Matrix-Tree Theorem and matrix determinant factorization.

Contribution

It introduces a novel graphical perspective connecting arborescence sums in digraphs to matrix determinants, extending classical theorems.

Findings

Identifies classes of digraphs with equal arborescence sums

Provides a graphical approach to matrix determinant factorization

Relates arborescence sums to the Matrix-Tree Theorem

Abstract

We show that certain digraphs with the same vertex set but different arc sets have the same sum over the weights of all arborescences with a given root vertex. We relate our results to the Matrix-Tree Theorem and show how they provide a graphical approach for factoring matrix determinants.