scholarly journals A Construction of Sequentially Cohen–Macaulay Graphs

2021 ◽  
Vol 28 (03) ◽  
pp. 399-414
Author(s):  
Aming Liu ◽  
Tongsuo Wu
Keyword(s):  
Simplicial Complex ◽  
Betti Numbers ◽  
Simple Graph ◽  
Ideal Formula ◽  
Vertex Decomposable ◽  
Linear Quotients ◽  
Facet Ideal

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

scholarly journals On Generalization of Cycles and Chordality to Clutters from an Algebraic Viewpoint

2017 ◽  
Vol 24 (04) ◽  
pp. 611-624 ◽  
Author(s):  
Ashkan Nikseresht ◽  
Rashid Zaare-Nahandi
Keyword(s):  
Simplicial Complex ◽  
Commutative Algebra ◽  
Point Of View ◽  
Algebraic Point ◽  
Linear Resolution ◽  
Vertex Decomposable Simplicial Complex ◽  
Vertex Decomposable ◽  
The Relationship ◽  
Linear Quotients

In this paper, we study the notion of chordality and cycles in clutters from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. We mainly consider the generalization of chordality proposed by Bigdeli et al. in 2017 and the concept of cycles introduced by Cannon and Faridi in 2013, and study their interrelations and algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of clutters. Also, we show that if [Formula: see text] is a clutter such that 〈[Formula: see text]〉 is a vertex decomposable simplicial complex or I([Formula: see text]) is squarefree stable, then [Formula: see text] is chordal.

scholarly journals On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

2016 ◽  
Vol 118 (1) ◽  
pp. 43 ◽  
Author(s):  
Somayeh Moradi ◽  
Fahimeh Khosh-Ahang
Keyword(s):  
Simplicial Complex ◽  
Vertex Cover ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Projective Dimension ◽  
Graded Betti Numbers ◽  
Alexander Dual ◽  
Special Cases ◽  
Vertex Decomposable Simplicial Complex ◽  
Vertex Decomposable

In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $\Delta$ is vertex decomposable if and only if $I_{\Delta^{\vee}}$ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of $R/I_{\Delta}$, when $\Delta$ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph $G$, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when $G$ is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution.

Spanning Simplicial Complex of Wheel Graph Wn

2019 ◽  
Vol 26 (02) ◽  
pp. 309-320 ◽  
Author(s):  
A. Zahid ◽  
M.U. Saleem ◽  
A. Kashif ◽  
M. Khan ◽  
M.A. Meraj ◽  
...  
Keyword(s):  
Simplicial Complex ◽  
Hilbert Series ◽  
Associated Primes ◽  
Face Ring ◽  
Combinatorial Properties ◽  
Ideal Formula ◽  
Wheel Graph ◽  
Vertex Set ◽  
Facet Ideal

In this paper, we explore the spanning simplicial complex of wheel graph Wn on vertex set [n]. Combinatorial properties of the spanning simplicial complex of wheel graph are discussed, which are then used to compute the f-vector and Hilbert series of face ring k[Δs(Wn)] for the spanning simplicial complex Δs(Wn). Moreover, the associated primes of the facet ideal [Formula: see text] are also computed.

scholarly journals Symbolic powers of vertex cover ideals

2020 ◽  
Vol 30 (06) ◽  
pp. 1167-1183
Author(s):  
S. Selvaraja
Keyword(s):  
Polynomial Ring ◽  
Vertex Cover ◽  
Simple Graph ◽  
Symbolic Powers ◽  
Vertex Decomposable ◽  
Decomposable Graphs ◽  
Linear Quotients

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote its vertex cover ideal in a polynomial ring over a field [Formula: see text]. In this paper, we show that all symbolic powers of vertex cover ideals of certain vertex-decomposable graphs have linear quotients. Using these results, we give various conditions on a subset [Formula: see text] of the vertices of [Formula: see text] so that all symbolic powers of vertex cover ideals of [Formula: see text], obtained from [Formula: see text] by adding a whisker to each vertex in [Formula: see text], have linear quotients. For instance, if [Formula: see text] is a vertex cover of [Formula: see text], then all symbolic powers of [Formula: see text] have linear quotients. Moreover, we compute the Castelnuovo–Mumford regularity of symbolic powers of certain vertex cover ideals.

scholarly journals The facet ideal of a simplicial complex

2002 ◽  
Vol 109 (2) ◽  
pp. 159-174 ◽  
Author(s):  
Sara Faridi
Keyword(s):  
Simplicial Complex ◽  
Facet Ideal

scholarly journals Simplicial Complexes of Whisker Type

10.37236/4894 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mina Bigdeli ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Antonio Macchia
Keyword(s):  
Simplicial Complex ◽  
Short Proof ◽  
Monomial Ideal ◽  
Simplicial Complexes ◽  
Linear Resolution ◽  
Cw Complex ◽  
Alexander Dual ◽  
Vertex Decomposable ◽  
The Ideal

Let $I\subset K[x_1,\ldots,x_n]$ be  a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that  $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.

scholarly journals On the Facet Ideal of an Expanded Simplicial Complex

2018 ◽  
Vol 44 (3) ◽  
pp. 719-727
Author(s):  
S. Moradi ◽  
R. Rahmati-Asghar
Keyword(s):  
Simplicial Complex ◽  
Facet Ideal

scholarly journals Random simplicial complexes, duality and the critical dimension

2019 ◽  
pp. 1-31
Author(s):  
Michael Farber ◽  
Lewis Mead ◽  
Tahl Nowik
Keyword(s):  
Simplicial Complex ◽  
Knot Theory ◽  
Betti Numbers ◽  
Critical Dimension ◽  
Simplicial Complexes ◽  
Alexander Duality ◽  
Random Simplicial Complexes

In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.

scholarly journals Componentwise linear ideals

1999 ◽  
Vol 153 ◽  
pp. 141-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Homogeneous Polynomials ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Alexander Dual ◽  
Componentwise Linear Ideals ◽  
The Ideal

AbstractA componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

scholarly journals Betti numbers of toric ideals of graphs: A case study

2019 ◽  
Vol 18 (12) ◽  
pp. 1950226
Author(s):  
Federico Galetto ◽  
Johannes Hofscheier ◽  
Graham Keiper ◽  
Craig Kohne ◽  
Adam Van Tuyl ◽  
...  
Keyword(s):  
Bipartite Graph ◽  
Hilbert Series ◽  
Complete Bipartite Graph ◽  
Betti Numbers ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Initial Ideal ◽  
Graded Betti Numbers ◽  
Linear Quotients

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and [Formula: see text]-vector for all the toric ideals of graphs in this family.

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