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arXiv · 1912.01060

Arborescences of Covering Graphs

Sunita Chepuri·CJ Dowd·Andy Hardt·Gregory Michel·Sylvester W. Zhang·Valerie Zhang
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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.

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Sunita Chepuri, CJ Dowd, Andy Hardt, Gregory Michel, Sylvester W. Zhang, Valerie Zhang. 2019-12-02. Arborescences of Covering Graphs. https://arxiv.org/abs/1912.01060

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