On Picard groups and Jacobians of directed graphs

TL;DR

This paper explores the structure of Picard groups and Jacobians in directed graphs, extending classical concepts from undirected graphs and computing these groups for various classes of directed graphs.

Contribution

It generalizes the computation of Picard groups and Jacobians to directed graphs and provides explicit calculations for several graph classes.

Findings

Picard groups of directed graphs can be computed using Smith normal form of the Laplacian.

Jacobians are the torsion subgroups of the Picard groups for directed graphs.

Explicit formulas are provided for directed trees, cycles, wheel, and multipartite graphs.

Abstract

The Picard group of an undirected graph is a finitely generated abelian group, and the Jacobian is the torsion subgroup of the Picard group. These groups can be computed by using the Smith normal form of the Laplacian matrix of the graph or by using chip-firing games associated with the graph. One may consider its generalization to directed graphs based on the Laplacian matrix. We compute Picard groups and Jacobians for several classes of directed trees, cycles, wheel, and multipartite graphs.