Regularity and projective dimension of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs

TL;DR

This paper derives exact formulas and bounds for the regularity and projective dimension of powers of edge ideals in disjoint unions of weighted oriented gap-free bipartite graphs, highlighting the influence of direction selection.

Contribution

It provides new explicit formulas and bounds for regularity and projective dimension of edge ideals in a specific class of bipartite graphs, considering orientation effects.

Findings

Exact formulas for regularity of powers of edge ideals.

Precise formula for projective dimension of the edge ideal.

Bounds for the projective dimension of higher powers.

Abstract

In this paper we provide some precise formulas for regularity of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. For the projective dimension of such an edge ideal, we give its exact formula. Meanwhile, we also give the upper and lower bounds of projective dimension of higher power of such edge ideals. Some examples show that these formulas are related to direction selection.