# Enumerating linear systems on graphs

@article{Brauner2020EnumeratingLS, title={Enumerating linear systems on graphs}, author={Sarah Brauner and Forrest Glebe and David Perkinson}, journal={Mathematische Zeitschrift}, year={2020} }

The divisor theory of graphs views a finite connected graph $G$ as a discrete version of a Riemann surface. Divisors on $G$ are formal integral combinations of the vertices of $G$, and linear equivalence of divisors is determined by the discrete Laplacian operator for $G$. As in the case of Riemann surfaces, we are interested in the complete linear system $|D|$ of a divisor $D$---the collection of nonnegative divisors linearly equivalent to $D$. Unlike the case of Riemann surfaces, the complete…

## 2 Citations

Simplicial Dollar Game

- MathematicsThe Electronic Journal of Combinatorics
- 2022

The dollar game is a chip-firing game introduced by Baker as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices.…

## References

SHOWING 1-10 OF 34 REFERENCES

Linear systems on tropical curves

- Mathematics
- 2009

A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system…

Trees, parking functions, syzygies, and deformations of monomial ideals

- Mathematics
- 2003

For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial…

Invariants of finite groups and their applications to combinatorics

- Mathematics
- 1979

1 CONTENTS 1. Introduction 2. Molien's theorem 3. Cohen-Macaulay rings 4. Groups generated by pseudo-reflections 5. Three applications 6. Syzygies 7. The canonical module 8. Gorenstein rings 9.…

Chip-firing games, potential theory on graphs, and spanning trees

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 2013

Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation)

- Computer Science
- 2008

The book of Sturmfels is both an easy-to-read textbook for invariants and a challenging research monograph that introduces a new approach to the algorithmic side of invariant theory.

Chip firing on Dynkin diagrams and McKay quivers

- Mathematics
- 2016

This paper establishes new connections between the representation theory of finite groups and sandpile dynamics. Two classes of avalanche-finite matrices and their critical groups (integer cokernels)…

The lattice of integral flows and the lattice of integral cuts on a finite graph

- Mathematics
- 1997

Les flots entiers sur un graphe fini Γ constituent naturellement un reseau entier Λ 1 (Γ) dans l'espace euclidien Ker(Δ 1 ) des fonctions harmoniques a valeurs reelles sur l'ensemble des aretes de Γ.…

A Riemann–Roch theorem in tropical geometry

- Mathematics
- 2006

Recently, Baker and Norine have proven a Riemann–Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann–Roch theorem for divisors on (abstract) tropical…