Linear equivalence of graph divisors

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

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

Remarks

• [math]x\sim y[/math] 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 [math]x[/math] and [math]y[/math] as chip-distributions, [math]x\sim y[/math] is equivalent to the following: [math]y[/math] is reachable from [math]x[/math] by a sequence offirings which are not necessarily legal.