Betti numbers of weighted oriented graphs

TL;DR

This paper studies the Betti numbers of edge ideals of weighted oriented graphs using algebraic and combinatorial methods, providing recursive formulas, characterizations, and explicit computations for specific classes.

Contribution

It introduces recursive formulas and characterizations for Betti numbers and regularity of edge ideals of weighted oriented graphs, expanding understanding of their algebraic invariants.

Findings

Recursive formulas for Betti numbers of certain classes

Identification of classes with unique extremal Betti number

Characterization of graphs with projective dimension equal to number of vertices

Abstract

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that $\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.