On the Arithmetical Rank of the Edge Ideals of Some Graphs

In this paper, we compute the projective dimension of the edge ideals of graphs consisting of some cycles and lines which are joint in a common vertex. Moreover, we show that for such graphs, the arithmetical rank equals the projective dimension. As an application, we can compute the arithmetical rank for some homogenous monomial ideals.

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Algebraic Properties of Edge Ideals of Some Vertex-Weighted Oriented Cyclic Graphs

Algebra Colloquium ◽

10.1142/s1005386721000201 ◽

2021 ◽

Vol 28 (02) ◽

pp. 253-268

Author(s):

Hong Wang ◽

Guangjun Zhu ◽

Li Xu ◽

Jiaqi Zhang

Keyword(s):

Projective Dimension ◽

Common Vertex ◽

Edge Ideals ◽

Algebraic Properties ◽

Cyclic Graphs

We provide some exact formulas for the projective dimension and regularity of edge ideals associated to some vertex-weighted oriented cyclic graphs with a common vertex or edge. These formulas are functions in the weight of the vertices, and the numbers of edges and cycles. Some examples show that these formulas are related to direction selection and the assumption that [Formula: see text] for any vertex [Formula: see text] cannot be dropped.

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ARITHMETICAL RANK OF THE CYCLIC AND BICYCLIC GRAPHS

Journal of Algebra and Its Applications ◽

10.1142/s0219498811005634 ◽

2012 ◽

Vol 11 (02) ◽

pp. 1250039 ◽

Cited By ~ 10

Author(s):

MARGHERITA BARILE ◽

DARIUSH KIANI ◽

FATEMEH MOHAMMADI ◽

SIAMAK YASSEMI

Keyword(s):

Quotient Ring ◽

Projective Dimension ◽

Edge Ideals ◽

Arithmetical Rank ◽

Bicyclic Graphs

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex, the arithmetical rank equals the projective dimension of the corresponding quotient ring.

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Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

Asian-European Journal of Mathematics ◽

10.1142/s1793557117500619 ◽

2017 ◽

Vol 10 (03) ◽

pp. 1750061

Author(s):

Somayeh Moradi

Keyword(s):

Simplicial Complex ◽

Upper Bound ◽

Betti Numbers ◽

Recursive Formula ◽

Projective Dimension ◽

Simplicial Complexes ◽

Monomial Ideals ◽

Edge Ideals ◽

Graded Betti Numbers ◽

Shellable Simplicial Complex

In this paper, we study the regularity and the projective dimension of the Stanley–Reisner ring of a [Formula: see text]-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for [Formula: see text]-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable simplicial complex [Formula: see text], a formula for the regularity of the Stanley–Reisner ring of [Formula: see text] is presented. Finally, for a chordal clutter [Formula: see text], an upper bound for [Formula: see text] is given in terms of the regularities of edge ideals of some chordal clutters which are minors of [Formula: see text].

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The arithmetical rank of the edge ideals of cactus graphs

Communications in Algebra ◽

10.1080/00927872.2017.1310873 ◽

2017 ◽

Vol 45 (12) ◽

pp. 5407-5419

Author(s):

Margherita Barile ◽

Antonio Macchia

Keyword(s):

Edge Ideals ◽

Arithmetical Rank ◽

Cactus Graphs

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Arithmetical rank of Gorenstein squarefree monomial ideals of height three

Journal of Algebra ◽

10.1016/j.jalgebra.2014.09.005 ◽

2015 ◽

Vol 422 ◽

pp. 11-32 ◽

Cited By ~ 2

Author(s):

Kyouko Kimura ◽

Naoki Terai

Keyword(s):

Monomial Ideals ◽

Arithmetical Rank

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Regularity and projective dimension of edge ideals

Communications in Algebra ◽

10.1080/00927872.2018.1459638 ◽

2018 ◽

Vol 46 (11) ◽

pp. 4830-4843

Author(s):

Mengyao Sun

Keyword(s):

Projective Dimension ◽

Edge Ideals

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The stable property of projective dimension

Journal of Algebra and Its Applications ◽

10.1142/s0219498819502244 ◽

2019 ◽

Vol 18 (12) ◽

pp. 1950224

Author(s):

Somayeh Bandari ◽

Raheleh Jafari

Keyword(s):

Projective Dimension ◽

Monomial Ideals ◽

Prime Ideals ◽

Stable Property ◽

Polymatroidal Ideals

We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen–Macaulay property. Indeed, we study the class of monomial ideals [Formula: see text], whose projective dimension is stable under monomial localizations at monomial prime ideals [Formula: see text], with [Formula: see text]. We study the relations between this property and other sorts of Cohen–Macaulayness. Finally, we characterize some classes of polymatroidal ideals with stable projective dimension.

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