# Minimal free resolutions of lattice ideals of digraphs

## TL;DR

This paper constructs minimal free resolutions for lattice ideals derived from finite, strongly connected, weighted directed graphs, extending previous work on undirected graphs and providing explicit algorithms for syzygy computation.

## Contribution

It introduces a method to compute minimal free resolutions of lattice ideals for directed graphs, refining existing algorithms and characterizing minimality in strongly complete digraphs.

## Key findings

- Provides a free resolution for lattice ideals of directed graphs.
- Characterizes when the resolution is minimal (strongly complete digraphs).
- Extends undirected graph results to directed, weighted graphs.

## Abstract

Based upon a previous work of Manjunath and Sturmfels for a finite, complete, undirected graph, and a refined algorithm by Er\"ocal, Motsak, Schreyer and Steenpa{\ss} for computing syzygies, we display a free resolution of the lattice ideal associated to a finite, strongly connected, weighted, directed graph. Moreover, the resolution is minimal precisely when the digraph is strongly complete.