The gonality (also called divisorial gonality) 
of a (finite) graph 
is the minimum degree of a rank 1 divisor on that graph. It
can be thought of as the minimum number of chips that can be placed on that graph
such that a debt of 1 can be eliminated via "chip-firing moves" over all
possible debt placements. Graph gonality was introduced by Baker and Norine (2007,
2009) and is analogous to the theory of divisors on algebraic curves.
The gonality of a graph is one of several graph analogs of the gonality of an algebraic curve, which is the minimum degree of a rational map from the curve to the projective line (Echavarria 2021).
The gonality of a graph and satisfies
 |
where 
is the vertex
connectivity, 
is the edge connectivity, 
is the treewidth, and 
is the scramble
number of 
(Harp et al. 2020, Echavarria et al. 2021).
Trees have gonality of 1. Gonalities for a number of special classes of graphs are summarized by Echavarria et al. (2021, Examples 2.7 and 2.8).
The gonality of a graph is NP-hard to compute (Gijswijt et al. 2020, Echavarria 2021).
