Principal minors of tree distance matrices
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915665288364032 |
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| author | Richman, Harry Shokrieh, Farbod Wu, Chenxi |
| author_facet | Richman, Harry Shokrieh, Farbod Wu, Chenxi |
| contents | We prove that the principal minors of the distance matrix of a tree satisfy a combinatorial expression involving counts of rooted spanning forests of the underlying tree. This generalizes a result of Graham and Pollak, and refines a result of Graham and Lovász on the coefficients of the characteristic polynomial of the distance matrix. We also give such an expression for the case of trees with edge lengths. We use arguments motivated by potential theory on graphs. Our formulas can be expressed in terms of evaluations of Symanzik polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Principal minors of tree distance matrices Richman, Harry Shokrieh, Farbod Wu, Chenxi Combinatorics 05C50, 05C05, 05C12, 05C30, 31C15 We prove that the principal minors of the distance matrix of a tree satisfy a combinatorial expression involving counts of rooted spanning forests of the underlying tree. This generalizes a result of Graham and Pollak, and refines a result of Graham and Lovász on the coefficients of the characteristic polynomial of the distance matrix. We also give such an expression for the case of trees with edge lengths. We use arguments motivated by potential theory on graphs. Our formulas can be expressed in terms of evaluations of Symanzik polynomials. |
| title | Principal minors of tree distance matrices |
| topic | Combinatorics 05C50, 05C05, 05C12, 05C30, 31C15 |
| url | https://arxiv.org/abs/2411.11488 |