A graph divisor on a nonempty finite connected graph 
with no loops,
allowing multiple edges, is a formal integer linear combination
 |
The divisors form the free Abelian group on the vertices of 
. The degree of 
is 
.
It is effective when every 
.
The coefficients represent chips, with negative values representing debt. Firing a vertex sends one chip along each incident graph edge to its other endpoint. Two divisors are linearly equivalent when a sequence of firings and inverse firings transforms one into the other. Intermediate coefficients may be negative.
The rank 
is 
if no effective divisor is linearly
equivalent to 
.
Otherwise it is the largest nonnegative integer 
for which 
is linearly equivalent to an effective divisor for every
effective divisor 
of degree 
(Baker and Norine 2007). The gonality of 
is the least degree of a divisor with rank at least 1.
