Divisors on graphs, Connected flags, and Syzygies

Fatemeh Mohammadi (1) , Farbod Shokrieh (2)

1 Fachbereich Mathematik und Informatik [Marburg] [Dept. of Math and Computer Science]
2 Georgia Institute of Technology [Atlanta]

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of $I_G$ and its initial ideal (with respect to a natural term order) coincide and they correspond to the number of ``connected flags'' in $G$. Moreover, the Betti numbers are independent of the characteristic of the base field.

Keywords: Graph; divisors; chip-firing; Gröbner bases; Betti numbers; connected flags.

Domains: Discrete Mathematics [cs.DM]

Fatemeh Mohammadi, Farbod Shokrieh. Divisors on graphs, Connected flags, and Syzygies. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.885-896, ⟨10.46298/dmtcs.2351⟩. ⟨hal-01229678⟩