# Picard group of a graph

From Egres Open

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.

## Remarks

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]}.

## References

- ↑ 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 - ↑ M. Baker, S. Norine,
*Riemann--Roch and Abel--Jacobi theory on a finite graph*, Advances in Mathematics (2007) DOI link, ArXiv Link - ↑ 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