# Jeremy Usatine

## Thesis

Arithmetical Graphs, Riemann-Roch Structure for Lattices, and the Frobenius Number Problem

If $R = (r_1, \dots, r_n) \in \mathbb{Z}_{>0}^n$ with $\gcd(r_1, \dots, r_n)=1$, calculating the Frobenius number of $R$ is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of $R$. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the construction for Lorenzini's Riemann-Roch structure and how it relates to the Riemann-Roch theorem for finite graphs shown by Matthew Baker and Serguei Norine.
We then focus on the connection between the Frobenius number and arithmetical graphs. Using the Laplacian of an arithmetical graph and a formulation of chip-firing on the vertices of an arithmetical graph, we show results that can be used to find arithmetical graphs whose g-numbers correspond to the Frobenius number of $R$. We describe how this can be used to quickly calculate upper bounds for the Frobenius number of $R$.