Digraph Branchings and Matrix Determinants

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.

Figures (5)

• Figure 1: The matrix digraph corresponding to the matrix in Eq. (\ref{['eq:A']}).
• Figure 2: The sixteen branchings for the matrix digraph in Fig. \ref{['fig:3digraph']}.
• Figure 3: Arborescences in the calculation of the determinant of the matrix in Eq. (\ref{['eq:M_cof2']}).
• Figure 4: Matrix digraph for a tridiagonal matrix of dimension $n$.
• Figure 5: Matrix digraph $G_{i+1}$ derived from predecessor graph $G_i$.

Theorems & Definitions (26)

• Definition 1: Matrix Digraph
• Remark 1
• Definition 2: Incidence Matrix
• Definition 3: Weight Matrix
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof