Coplanarity of rooted spanning-tree vectors

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 $\ddot{I}{„}_x^{\varnothing }$, $\ddot{I}{„}_x^{+}$, and $\ddot{I}{„}_x^{-}$ 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 $(\ddot{I}{„}_x^{\varnothing },\ddot{I}{„}_x^{+},\ddot{I}{„}_x^{-})$ 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.

21 pages, 3 figures