Digraph Branchings and Matrix Determinants
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912959607865344 |
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| author | Ghosh, Sayani Meyer, Bradley S. |
| author_facet | Ghosh, Sayani Meyer, Bradley S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05827 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Digraph Branchings and Matrix Determinants Ghosh, Sayani Meyer, Bradley S. Combinatorics Discrete Mathematics 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. |
| title | Digraph Branchings and Matrix Determinants |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2309.05827 |