Symbolic powers in weighted oriented graphs

Mandal, Mousumi; Pradhan, Dipak Kumar

doi:10.1142/s0218196721500260

Cited by 11 publications

(4 citation statements)

References 17 publications

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“…The strong vertex covers of weighted oriented graphs always play an important role in the study of symbolic powers of their edge ideals. 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].…”

Section: Introductionmentioning

confidence: 99%

Symbolic powers of edge ideals of weighted oriented graphs

Mandal¹,

Pradhan²

2022

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%

Symbolic powers of edge ideals of weighted oriented graphs

Mandal¹,

Pradhan²

2022

Preprint

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“…The aims of this paper are to determine when I(D) 2 is equal to I(D) (2) in terms of D and to give a combinatorial classification for the equality "I(D) n = I(D) (n) for n ≥ 1". It is an open problem to classify the equality "I(D) n = I(D) n for n ≥ 1", for some of the advances to solve this problem see [14,15] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Normally torsion-free edge ideals of weighted oriented graphs

Grisalde¹,

Martinez-Bernal²,

Villarreal³

2021

Preprint

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Let I = I(D) be the edge ideal of a weighted oriented graph D, let G be the underlying graph of D, and let I (n) be the n-th symbolic power of I defined using the associated minimal primes of I. We prove that I 2 = I (2) if and only if every vertex of D with weight greater than 1 is a sink and G has no triangles. As a consequence, using a result of Mandal and Pradhan and the classification of normally torsion-free edge ideals of graphs, it follows that I (n) = I n for all n ≥ 1 if and only if every vertex of D with weight greater than 1 is a sink and G is bipartite. If I has no embedded primes, these two properties classify when I is normally torsion-free.

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“…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%