1. Coplanarity of rooted spanning-tree vectors
- Author
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Polettini, Matteo, Harunari, Pedro E., Cengio, Sara Dal, and Lecomte, Vivien
- Subjects
Mathematics - Combinatorics ,Condensed Matter - Statistical Mechanics ,Mathematical Physics - Abstract
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 $\tau_x^\varnothing$, $\tau_x^+$, and $\tau_x^-$ 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 $(\tau_x^\varnothing, \tau_x^+, \tau_x^-)$ 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., Comment: 21 pages, 3 figures
- Published
- 2024