# Regularity of powers of edge ideals of vertex-weighted oriented unicyclic graphs

## TL;DR

This paper derives exact formulas for the regularity of powers of edge ideals in vertex-weighted oriented unicyclic graphs, highlighting the influence of vertex weights and edge directions on algebraic properties.

## Contribution

It introduces new formulas linking vertex weights and edge orientations to the regularity of edge ideal powers in unicyclic graphs.

## Key findings

- Formulas depend on vertex weights and edge directions.
- Regularity varies with the choice of orientation.
- Examples illustrate the relationship between weights, directions, and regularity.

## Abstract

In this paper we provide some exact formulas for the regularity of powers of edge ideals of vertex-weighted oriented cycles and vertex-weighted unicyclic graphs. These formulas are functions of the weight of vertices and the number of edges. We also give some examples to show that these formulas are related to direction selection and the weight of vertices.