# Linear equivalence of graph divisors

$x$ is linearly equivalent to $y$ (common notation: $x\sim y$) if there exists an integer vector $z\in \mathbb{Z}^{V(G)}$ such that $x = y + L_G z$.

Here, $G$ is a graph or a digraph; $x$ and $y$ are integer vectors indexed by the vertices of $G$ (i.e. distributions of a chip-firing game or divisors on G); and $L_G$ denote the Laplacian matrix of $G$.

## Remarks

• $x\sim y$ is an equivalence relation.
• Linear equivalence of graph divisors was defined by Baker and Norine[1], motivated by the divisor theory of Riemann surfaces.
• If we consider $x$ and $y$ as chip-distributions, $x\sim y$ is equivalent to the following: $y$ is reachable from $x$ by a sequence offirings which are not necessarily legal.

### References

1. M. Baker, S. Norine, Riemann--Roch and Abel--Jacobi theory on a finite graph, DOI link, ArXiv Link