Comparing symbolic powers of edge ideals of weighted oriented graphs

Mandal, Mousumi; Pradhan, Dipak Kumar

doi:10.1007/s10801-022-01118-1

Cited by 6 publications

(2 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [8], we see that, if the set of all vertices of a weighted oriented graph forms a strong vertex cover, all the ordinary and symbolic powers of its edge ideal coincide. Comparision of ordinary and symbolic powers has been done for several classes of weighted oriented graphs in [1], [2] and [9]. In all those papers, to compute the symbolic powers, the authors always find the minimal generators of the intersections of irreducible ideals associated to the strong vertex covers contained in a maximal strong vertex cover.…”

Section: Introductionmentioning

confidence: 99%

Symbolic powers of edge ideals of weighted oriented graphs

Mandal¹,

Pradhan²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Let D be a weighted oriented graph and I(D) be its edge ideal. We provide one method to find all the minimal generators of I ⊆C , where C is a maximal strong vertex cover of D and I ⊆C is the intersections of irreducible ideals associated to the strong vertex covers contained in C. If D ′ is an induced digraph of D, under certain condition on the strong vertex covers of D ′ and D, we show that I(D ′ ) (s) ≠ I(D ′ ) s for some s ≥ 2 implies I(D) (s) ≠ I(D) s . We characterize all the maximal strong vertex covers of D such that at most one edge is oriented into each of its vertex and w(x) ≥ 2 if deg D (x) ≥ 2 for all x ∈ V (D). If D is a weighted rooted tree with degree of root is 1 and w(x) ≥ 2 when deg D (x) ≥ 2 for all x ∈ V (D), we show that I(D) (s) = I(D) s for all s ≥ 2.

show abstract

Section: Introductionmentioning

confidence: 99%

Symbolic powers of edge ideals of weighted oriented graphs

Mandal¹,

Pradhan²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Mandal and Pradhan studied the symbolic powers of edge ideals of weighted oriented graphs. Recently, they showed that if D is any weighted oriented star graph or some specific weighted naturally oriented path, then I(D) s = I(D) (s) for all s ≥ 2, ( [8], [9]). In this paper, we study the ordinary and symbolic powers of edge ideals of some classes of oriented tree.…”

Section: Introductionmentioning

confidence: 99%