Projective dimension and regularity of edge ideals of some weighted oriented graphs

Abstract

In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to three special types of vertex-weighted oriented m-partite graphs. These formulas are functions of the weight and number of vertices. We also give some examples to show that these formulas are related to direction selection and the weight of vertices.

Betti Numbers of Weighted Oriented Graphs

“…In order to obtain more general results, we investigate the Betti numbers and provide recursive formulas for these invariants. With this approach, we successfully recover several results from [19] and suggest an explanation for the nature of regularity formulas given in [5].…”

“…Edge ideals of weighted oriented graphs are fairly new objects in the combinatorial commutative algebra community and there have been a few papers investigating algebraic properties and invariants of these objects (see [5,12,14,18,19]). There has been an extensive literature on the edge ideals of (unweighted, unoriented) graphs, and one of the reasons for such fruitful outcomes is due to the squarefree nature of edge ideals of graphs.…”

“…Current literature on algebraic invariants of edge ideals of weighted oriented graphs is obtained by focusing on specific classes of graphs with a predetermined orientation or weight assumptions. For instance, in [19], authors investigate the regularity and the projective dimension of edge ideals of weighted cycles and rooted forests with the assumption that all non-source vertices are of non-simple weight and graphs are naturally oriented (i.e., all edges are oriented in the same direction). A more general case, class of weighted naturally oriented paths and cycles under any weight distribution is studied in [5].…”

“…A more general case, class of weighted naturally oriented paths and cycles under any weight distribution is studied in [5]. All the formulas provided in [5,19] heavily depend on the orientation and weight distributions. In order to obtain more general results, we investigate the Betti numbers and provide recursive formulas for these invariants.…”

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

“…In this paper, we consider the edge ideal I(D) of a weighted oriented graph D. In this case, there exist integers b and t 0 such that reg (I t ) = (w + 1)t + b for all t ≥ t 0 , where w = max {w(x) | x ∈ V (D)}. We also derive some exact formulas for projective dimension and regularity of powers of the edge ideals of some weighted oriented graphs (see, for example, [34,35,36,37,38]). To the best of our knowledge, there is a few results about projective dimension and regularity of powers of the edge ideal for a weighted oriented graph.…”