Coplanarity of rooted spanning-tree vectors
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929432186322944 |
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| author | Polettini, Matteo Harunari, Pedro E. Cengio, Sara Dal Lecomte, Vivien |
| author_facet | Polettini, Matteo Harunari, Pedro E. Cengio, Sara Dal Lecomte, Vivien |
| contents | Employing a recent technology of tree surgery we prove a ``deletion-constriction'' formula for products of rooted spanning trees on weighted directed graphs that generalizes deletion-contraction on undirected graphs. The formula implies that, letting $τ_x^\varnothing$, $τ_x^+$, and $τ_x^-$ be the rooted spanning tree polynomials obtained respectively by removing an edge in both directions or by forcing the tree to pass through either direction of that edge, the vectors $(τ_x^\varnothing, τ_x^+, τ_x^-)$ are coplanar for all roots $x$. We deploy the result to give an alternative derivation of a recently found mutual linearity of stationary currents of Markov chains. We generalize deletion-constriction and current linearity among two edges, and conjecture that similar results may hold for arbitrary subsets of edges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16093 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coplanarity of rooted spanning-tree vectors Polettini, Matteo Harunari, Pedro E. Cengio, Sara Dal Lecomte, Vivien Combinatorics Statistical Mechanics Mathematical Physics Employing a recent technology of tree surgery we prove a ``deletion-constriction'' formula for products of rooted spanning trees on weighted directed graphs that generalizes deletion-contraction on undirected graphs. The formula implies that, letting $τ_x^\varnothing$, $τ_x^+$, and $τ_x^-$ be the rooted spanning tree polynomials obtained respectively by removing an edge in both directions or by forcing the tree to pass through either direction of that edge, the vectors $(τ_x^\varnothing, τ_x^+, τ_x^-)$ are coplanar for all roots $x$. We deploy the result to give an alternative derivation of a recently found mutual linearity of stationary currents of Markov chains. We generalize deletion-constriction and current linearity among two edges, and conjecture that similar results may hold for arbitrary subsets of edges. |
| title | Coplanarity of rooted spanning-tree vectors |
| topic | Combinatorics Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2407.16093 |