Algebraic invariants of weighted oriented graphs

TL;DR

This paper derives formulas for the Castelnuovo-Mumford regularity and computes the projective dimension of edge ideals associated with weighted oriented paths and cycles with unidirectional edges.

Contribution

It provides explicit formulas for algebraic invariants of edge ideals of specific weighted oriented graphs, extending understanding in combinatorial commutative algebra.

Findings

Formulas for Castelnuovo-Mumford regularity of these graphs.

Calculation of projective dimension for weighted oriented paths and cycles.

Extension of algebraic invariant computations to weighted oriented graph classes.

Abstract

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal in a polynomial ring $R$. We give the formula of Castelnuovo-Mumford regularity of $R/I(\mathcal{D})$ when $\mathcal{D}$ is a weighted oriented path or cycle such that edges of $\mathcal{D}$ are oriented in one direction. Additionally, we compute the projective dimension for this class of graphs.