scholarly journals On the Betti Numbers of some Classes of Binomial Edge Ideals

10.37236/3689 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sohail Zafar ◽  
Zohaib Zahid
Keyword(s):  
Lower Bounds ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Induced Subgraphs ◽  
Edge Ideals ◽  
Arbitrary Graphs

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs depending on induced subgraphs.

On the Betti numbers of edge ideals of skew Ferrers graphs

2019 ◽  
Vol 30 (01) ◽  
pp. 125-139
Author(s):  
Do Trong Hoang
Keyword(s):  
Explicit Formula ◽  
Betti Number ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Ferrers Graph ◽  
Ferrers Graphs ◽  
Extremal Betti Number

We prove that [Formula: see text] for any staircase skew Ferrers graph [Formula: see text], where [Formula: see text] and [Formula: see text]. As a consequence, Ene et al. conjecture is confirmed to hold true for the Betti numbers in the last column of the Betti table in a particular case. An explicit formula for the unique extremal Betti number of the binomial edge ideal of some closed graphs is also given.

scholarly journals On the Extremal Betti Numbers of Binomial Edge Ideals of Block Graphs

10.37236/7689 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Jürgen Herzog ◽  
Giancarlo Rinaldo
Keyword(s):  
Betti Number ◽  
Betti Numbers ◽  
Block Graph ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Block Graphs ◽  
Extremal Betti Number

We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.

scholarly journals Depth and Stanley depth of the edge ideals of the powers of paths and cycles

2019 ◽  
Vol 27 (3) ◽  
pp. 113-135
Author(s):  
Zahid Iqbal ◽  
Muhammad Ishaq
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Quotient Ring ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

AbstractLet k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

scholarly journals Betti Numbers of Weighted Oriented Graphs

10.37236/9887 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Beata Casiday ◽  
Selvi Kara
Keyword(s):  
Betti Number ◽  
Betti Numbers ◽  
Oriented Graph ◽  
Projective Dimension ◽  
Edge Ideal ◽  
Oriented Graphs ◽  
Edge Ideals ◽  
Mapping Cone ◽  
Recursive Formulas ◽  
Extremal Betti Number

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that $\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.

scholarly journals Stanley depth of edge ideals

2012 ◽  
Vol 49 (4) ◽  
pp. 501-508 ◽  
Author(s):  
Muhammad Ishaq ◽  
Muhammad Qureshi
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

We give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of a s-uniform complete bipartite hypergraph.

scholarly journals Comparing powers of edge ideals

2019 ◽  
Vol 18 (10) ◽  
pp. 1950184 ◽  
Author(s):  
Mike Janssen ◽  
Thomas Kamp ◽  
Jason Vander Woude
Keyword(s):  
Symbolic Power ◽  
Edge Ideal ◽  
Homogeneous Ideal ◽  
Edge Ideals ◽  
Ideal Formula ◽  
Number Of Generators ◽  
Recent Interest ◽  
Odd Cycle ◽  
Ordinary Power

Given a nontrivial homogeneous ideal [Formula: see text], a problem of great recent interest has been the comparison of the [Formula: see text]th ordinary power of [Formula: see text] and the [Formula: see text]th symbolic power [Formula: see text]. This comparison has been undertaken directly via an exploration of which exponents [Formula: see text] and [Formula: see text] guarantee the subset containment [Formula: see text] and asymptotically via a computation of the resurgence [Formula: see text], a number for which any [Formula: see text] guarantees [Formula: see text]. Recently, a third quantity, the symbolic defect, was introduced; as [Formula: see text], the symbolic defect is the minimal number of generators required to add to [Formula: see text] in order to get [Formula: see text]. We consider these various means of comparison when [Formula: see text] is the edge ideal of certain graphs by describing an ideal [Formula: see text] for which [Formula: see text]. When [Formula: see text] is the edge ideal of an odd cycle, our description of the structure of [Formula: see text] yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

scholarly journals Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals

2010 ◽  
Vol 106 (1) ◽  
pp. 88 ◽  
Author(s):  
Luis A. Dupont ◽  
Rafael H. Villarreal
Keyword(s):  
Monomial Ideal ◽  
Lattice Points ◽  
Edge Ideal ◽  
Comparability Graph ◽  
Decomposition Property ◽  
Edge Ideals ◽  
Finite Poset ◽  
Integer Decomposition Property ◽  
Torsion Free ◽  
Min Cut

The normality of a monomial ideal is expressed in terms of lattice points of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property characterizes normality. Let $G$ be the comparability graph of a finite poset. If $\mathrm{cl}(G)$ is the clutter of maximal cliques of $G$, we prove that $\mathrm{cl}(G)$ satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. Then we prove that edge ideals of complete admissible uniform clutters are normally torsion free.

On Betti numbers of edge ideals of crown graphs

2018 ◽  
Vol 60 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Shahnawaz Ahmad Rather ◽  
Pavinder Singh
Keyword(s):  
Betti Numbers ◽  
Edge Ideals

ON BROADCASTING TIME

1999 ◽  
Vol 09 (01) ◽  
pp. 3-8
Author(s):  
VASSILIOS V. DIMAKOPOULOS ◽  
NIKITAS J. DIMOPOULOS
Keyword(s):  
Lower Bounds ◽  
Information Dissemination ◽  
Applied Mathematics ◽  
The Other ◽  
Arbitrary Graphs

Broadcasting is an information dissemination problem in which a particular node in a network must transmit an item of information to all the other nodes. In this work we present new lower bounds for the time needed to complete this process in arbitrary graphs. In particular we generalize a result of P. Fraigniaud and E. Lazard [Discrete Applied Mathematics, 53 (1994) 79–133] which states that if in a graph there are at least two vertices in distance equal to the diameter from the originator, then broadcasting requires time at least equal to the diameter plus one.

scholarly journals Lower bounds for Betti numbers of special extensions

2005 ◽  
Vol 201 (1-3) ◽  
pp. 328-339
Author(s):  
Melvin Hochster ◽  
Benjamin Richert
Keyword(s):  
Lower Bounds ◽  
Betti Numbers
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