Dimensionality of deletion–constriction spanning-tree vectors for n reversible edges

Establish that for any weighted directed graph and any set of n pinned reversible edges (present in both orientations with nonzero weights), the family {τ_x : x ∈ X} of 3^n-dimensional vectors whose entries are the conditioned rooted spanning-tree polynomials for all deletion–constriction configurations across those n edges spans an (n+1)-dimensional subspace of R^{3^n}.

Background

The paper develops a deletion–constriction framework for rooted spanning-tree polynomials on weighted directed graphs. For a single pinned edge, the authors prove a second-order deletion–constriction identity that implies coplanarity (2-dimensional span) of the associated 3-component spanning-tree vectors across all roots.

They extend the approach to two pinned reversible edges and argue, under technical assumptions, that the corresponding 9-component spanning-tree vectors span a 3-dimensional subspace, which enables mutual linearity of two Markov currents.

Motivated by these results for one and two edges, the authors formulate a general conjecture asserting that for n pinned reversible edges the 3^n-dimensional spanning-tree vectors (collecting all deletion–constriction conditions across the n edges) should span an (n+1)-dimensional space. This, if true, would underpin higher-order linearity relations among multiple Markov currents.