# ALGEBRAIC COMBINATORICS

Arborescences of covering graphs
Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 319-346.

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 ${A}_{v}\left(\Gamma \right)$ representing the sum of the weights of all such arborescences. The arborescences of a graph and the arborescences of a covering graph $\stackrel{˜}{\Gamma }$ are closely related. Using voltage graphs to construct arbitrary regular covers, we derive a novel explicit formula for the ratio of ${A}_{v}\left(\Gamma \right)$ to the sum of arborescences in the lift ${A}_{\stackrel{˜}{v}}\left(\stackrel{˜}{\Gamma }\right)$ 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.212
Classification: 05C50,  05E18,  05C20,  05C05,  05C22
Keywords: Arborescence, covering graph, voltage graph.
Chepuri, Sunita 1; Dowd, CJ 2; Hardt, Andrew 3; Michel, Gregory 3; Zhang, Sylvester W. 3; Zhang, Valerie 2

1 University of Michigan Department of Mathematics 2074 East Hall 530 Church St. Ann Arbor MI 48109, USA
2 Harvard University Department of Mathematics Science Center Room 325 1 Oxford Street Cambridge MA 02138, USA
3 University of Minnesota School of Mathematics 127 Vincent Hall 206 Church St. SE Minneapolis MN 55414, USA
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Chepuri, Sunita; Dowd, CJ; Hardt, Andrew; Michel, Gregory; Zhang, Sylvester W.; Zhang, Valerie. Arborescences of covering graphs. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 319-346. doi : 10.5802/alco.212. https://alco.centre-mersenne.org/articles/10.5802/alco.212/

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