Coplanarity of rooted spanning-tree vectors

TL;DR

This paper introduces a new formula for rooted spanning trees on weighted directed graphs, showing that certain associated vectors are coplanar, and applies this to Markov chain stationary currents, suggesting broader generalizations.

Contribution

It generalizes deletion-contraction formulas to directed graphs and establishes coplanarity of rooted spanning tree vectors, providing new insights into Markov chain currents.

Findings

Vectors of rooted spanning trees are coplanar for all roots.

Derived an alternative proof for linearity of stationary currents.

Proposed conjectures for broader edge subset generalizations.

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.