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arXiv · 2309.05827

Digraph Branchings and Matrix Determinants

Sayani Ghosh·Bradley S. Meyer
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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.

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Sayani Ghosh, Bradley S. Meyer. 2023-09-11. Digraph Branchings and Matrix Determinants. https://arxiv.org/abs/2309.05827

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