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, 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.