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On Directed Graphs with the Same Sum over Arborescence Weights

Sayani Ghosh, Bradley S. Meyer

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.

On Directed Graphs with the Same Sum over Arborescence Weights

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.
Paper Structure (4 sections, 3 theorems, 6 equations, 6 figures)

This paper contains 4 sections, 3 theorems, 6 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Relation to the Matrix-Tree Theorem
  4. Graphical Calculation of a Matrix Determinant

Key Result

Theorem 2.1

Consider a directed graph $\Gamma_v$ with vertex set $V$ and arc set ${\cal A}$ and with root vertex $v \in V$. Consider further that $\{a, b, c\} \in V$ and that ${\cal A} = D \cup e$, where $D$ is a set of arcs and $e$ is an arc with weight $w(e)$ such that $s(e) = a$ and $t(e) = b$. Suppose anoth

Figures (6)

  • Figure 1: The matrix digraph $\Gamma_0$ corresponding to matrix $A$ in Eq. (\ref{['eq:A_ex']}).
  • Figure 2: The matrix digraph $\Gamma_0$ after moving arcs.
  • Figure 3: The matrix digraph $\Gamma_0$ corresponding to matrix $M$ in Eq. (\ref{['eq:M']}).
  • Figure 4: The digraphs resulting from sequential rooting of $\Gamma_0$ in Fig. \ref{['fig:matrix-digraph']}.
  • Figure 5: The digraphs of Fig. \ref{['fig:explicit']} after the isolation procedure.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 2.1: Moving-Arc Theorem
  • proof
  • Remark 2.1
  • Theorem 2.2: Combining-Arcs Theorem
  • proof
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 2 more