The edge ideals of the join of some vertex weighted oriented graphs

TL;DR

This paper investigates the algebraic properties of edge ideals formed from the join of weighted oriented graphs, providing explicit formulas and bounds for depth and regularity of their powers, advancing understanding in combinatorial commutative algebra.

Contribution

It offers new explicit formulas and bounds for the depth and regularity of powers of edge ideals of joined weighted oriented graphs, extending previous results in the field.

Findings

Explicit formulas for depth and regularity of edge ideals of joined graphs with oriented edges.

Upper bounds for the regularity of symbolic powers of such edge ideals.

Examples illustrating the bounds can be strict, highlighting the complexity of these algebraic invariants.

Abstract

In this paper, we describe primary decomposition of the edge ideal of the join of some graphs in terms of that information of the edge ideal of every weighted oriented graph. Meanwhile, we also study depth and regularity of symbolic powers and ordinary powers of such an edge ideal. We explicitly compute depth and regularity of ordinary powers of the edge ideal of the join of two graphs consisting of isolated vertices, and also provide upper bounds of regularity of symbolic powers of such an edge ideal. For the edge ideal of the join of two graphs with at least an oriented edge for per graph, we give the exact formulas for their depth and regularity, and also provide the upper bounds of regularity of ordinary powers of such an edge ideal. Some examples show that these upper bounds can be obtained, but may be strict.