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).

Baker, M. and and Norine, S. "Riemann-Roch and Abel-Jacobi Theory on a Finite Graph." Adv. Math.215, 766-788, 2007.Baker,
M. and and Norine, S. "Harmonic Morphisms and Hyperelliptic Graphs." Int.
Math Res. Not.1, 2914-2955, 2009.Echavarria, M.; Everett,
M.; Huang, R.; Jacoby, L.; Morrison, R.; Weber, B. "On the Scramble Number of
Graphs." 29 Mar 2021. https://arxiv.org/abs/2103.15253.Gijswijt,
D.; Smit, H.; and van der Wegen, M. "Computing Graph Gonality Is Hard."
Disc. Appl. Math.287, 134-149, 2020.Harp, M.; Jackson,
E.; Jensen, D.; and Speeter, N. "A New Lower Bound on Graph Gonality."
1 Jun 2020. https://arxiv.org/abs/2006.01020.Morrison,
R. and Tolley, L. "Computing Higher Graph Gonality Is Hard." 6 Aug 2022.
https://arxiv.org/abs/2208.03573.