Regularity in weighted oriented graphs

TL;DR

This paper investigates how the regularity of edge ideals in weighted oriented graphs behaves under certain modifications and explores the relationship between the regularity of these ideals and their underlying graphs, especially for paths and cycles.

Contribution

It establishes conditions under which the regularity remains unchanged after adding edges and relates the regularity of weighted oriented graphs to their underlying graphs for specific structures.

Findings

Regularity remains invariant after adding certain edges in weighted oriented graphs.

Relationship between regularity of weighted oriented paths/cycles and their underlying graphs.

Regularity of edge ideals for specific classes of weighted oriented graphs.

Abstract

Let $D$ be a weighted oriented graph with the underlying graph $G$ and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G$ respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph remains same even after adding certain kind of new edges to it. We also establish the relationship between the regularity of edge ideal of weighted oriented path and cycle with the regularity of edge ideal of their underlying graph when vertices of $V^+$ are sinks.