Arborescences of Covering Graphs
Sunita Chepuri, CJ Dowd, Andy Hardt, Gregory Michel, Sylvester W., Zhang, and Valerie Zhang

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 is a spanning tree directed toward a particular vertex . The arborescences of a graph rooted at a particular vertex may be encoded as a polynomial representing the sum of the weights of all such arborescences. The arborescences of a graph and the arborescences of a covering graph are closely related. Using voltage graphs as means to construct arbitrary regular covers, we derive a novel explicit formula for the ratio of to the sum of arborescences in the lift 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 are reviewed, and we provide a new proof of Chaiken's results via a deletion-contraction argument.
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Taxonomy
TopicsMolecular Junctions and Nanostructures · Surface Chemistry and Catalysis · Graph theory and applications
