# Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests

## TL;DR

This paper derives exact formulas for the projective dimension and regularity of powers of edge ideals in vertex-weighted rooted forests, linking algebraic invariants to vertex weights and forest structure.

## Contribution

It provides novel explicit formulas for these invariants in weighted rooted forests, highlighting the influence of weights and structure on algebraic properties.

## Key findings

- Formulas depend on vertex weights and number of edges
- Examples illustrate the impact of direction choice and assumptions
- Results show specific conditions where formulas hold

## Abstract

In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also give some examples to show that these formulas are related to direction selection and the assumptions about "rooted" forest such that $w(x)\geq 2$ if $d(x)\neq 1$ cannot be dropped.