Projective dimension and regularity of edge ideals of some vertex-weighted oriented unicyclic graphs

TL;DR

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

Contribution

It provides new explicit formulas linking graph weights, orientations, and algebraic invariants of edge ideals, with insights into the role of vertex weights and orientations.

Findings

Formulas depend on vertex weights and edge counts.

Direction choice affects the formulas and results.

Vertex weights of at least 2 are necessary for the formulas to hold.

Abstract

In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals of vertex-weighted oriented unicyclic graphs. These formulas are in function of the weight of the vertices, the numbers of edges. We also give some examples to show that these formulas are related to direction selection and the assumptions that $w(x)\geq 2$ for any vertex $x$ cannot be dropped.