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 τ x ∅ , τ x + , and τ x - be the rooted spanning-tree polynomials obtained, respectively, by removing both directed edges between two vertices, or by forcing the tree to pass through either edge, the vectors ( τ x ∅ , 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 for two pairs of edges and conjecture that similar results may hold for arbitrary subsets of edges.