No abstract

An arborescence of a directed graph Γ is a spanning tree directed toward a particular vertex v. The arborescences of a graph rooted at a particular vertex may be encoded as a polynomial Av(Γ) representing the sum of the weights of all such arborescences. The arborescences of a graph and the arborescences of a covering graph Γ are closely related. Using voltage graphs as means to construct arbitrary regular covers, we derive a novel explicit formula for the ratio of Av(Γ) to the sum of arborescences in the lift A ṽ ( Γ) in terms of the determinant of Chaiken's voltage Laplacian matrix, a generalization of the Laplacian matrix. Chaiken's results on the relationship between the voltage Laplacian and vector fields on Γ are reviewed, and we provide a new proof of Chaiken's results via a deletion-contraction argument.and to examine cases where this ratio is especially computationally nice. The ratio Aṽ( Γ)Av(Γ) first arose in Galashin and Pylyavskyy's study of R-systems [GP19]. The R-system is a discrete dynamical system on a edge-weighted strongly connected simple directed graph Γ = (V, E, wt) whose state vector X = (X v ) v∈V evolves to its next state X ′ = (X ′ v ) v∈V according to the following relation:This system is homogeneous in both X and X ′ , so we consider solutions in projective space. Galashin and Pylyavskyy determined all solutions X ′ of this equation as a function of X:Theorem 1.1.[GP19] The system given by equation (1) has solutionThis solution is unique up to scalar multiplication, yielding a unique solution to the R-system in P V .