Picard group of a graph

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The Picard group of a graph is the free Abelian group of the degree zero divisors factorized by linear equivalence:

[math] Pic^0(G)=Div^0(G) / Im(L) [/math]

Here, [math]L[/math] denotes the Laplacian matrix of the graph.


The Picard group of a graph is also known under the names Jacobian, and Sandpile group. There are various ways to define it, see for example [1], [2], [3].


  1. R. Bacher; P. de La Harpe; T. Nagnibeda The lattice of integral flows and the lattice of integral cuts on a finite graph, Bulletin de la Société Mathématique de France (1997) DOI link, Author Link
  2. M. Baker, S. Norine, Riemann--Roch and Abel--Jacobi theory on a finite graph, Advances in Mathematics (2007) DOI link, ArXiv Link
  3. A. E. Holroyd, L. Levine, K. Mészáros, Y. Peres, J. Propp, D. B. Wilson Chip-Firing and Rotor-Routing on Directed Graphs, Progress in Probability (2008) DOI link, Author Link