Arborescences of Covering Graphs

TL;DR

This paper explores the relationship between arborescences in covering graphs and introduces a new explicit formula for their ratio using voltage Laplacian matrices, extending classical results with novel proofs.

Contribution

It derives a new explicit formula for arborescence ratios in covering graphs using voltage Laplacian matrices and provides a novel proof of Chaiken's results.

Findings

Derived a formula relating arborescences of covering graphs to voltage Laplacian determinants.

Established a new proof of Chaiken's results via deletion-contraction.

Connected arborescences with vector fields on graphs.

Abstract

An arborescence of a directed graph $\Gamma$ 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 $A_v(\Gamma)$ representing the sum of the weights of all such arborescences. The arborescences of a graph and the arborescences of a covering graph $\tilde{\Gamma}$ are closely related. Using voltage graphs as means to construct arbitrary regular covers, we derive a novel explicit formula for the ratio of $A_v(\Gamma)$ to the sum of arborescences in the lift $A_{\tilde{v}}(\tilde{\Gamma})$ 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 $\Gamma$ are reviewed, and we provide a new proof of Chaiken's results via a deletion-contraction argument.