scholarly journals $k$-shellable simplicial complexes and graphs

2018 ◽  
Vol 122 (2) ◽  
pp. 161
Author(s):  
Rahim Rahmati-Asghar
Keyword(s):  
Simplicial Complex ◽  
Large Class ◽  
Simplicial Complexes ◽  
Face Ring ◽  
Stanley Conjecture ◽  
The Face ◽  
Shellable Simplicial Complex

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying an expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture.Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.

scholarly journals Face numbers of pseudomanifolds with isolated singularities

2012 ◽  
Vol 110 (2) ◽  
pp. 198 ◽  
Author(s):  
Isabella Novik ◽  
Ed Swartz
Keyword(s):  
Simplicial Complex ◽  
Upper Bound ◽  
Hilbert Function ◽  
Simplicial Complexes ◽  
Topological Invariant ◽  
Two Dimensional ◽  
Isolated Singularities ◽  
Face Ring ◽  
The Face

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower and upper bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the $f$-vector can be completely characterized are described. We also show that the Hilbert function of a generic Artinian reduction of the face ring of a simplicial complex $\Delta$ with isolated singularities minus the $h$-vector of $\Delta$ is a PL-topological invariant.

scholarly journals On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m

2018 ◽  
Vol 16 (1) ◽  
pp. 250-259
Author(s):  
Zahid Raza ◽  
Agha Kashif ◽  
Imran Anwar
Keyword(s):  
Simplicial Complex ◽  
Hilbert Series ◽  
Current Work ◽  
Algebraic Characterization ◽  
Face Ring ◽  
The Face

AbstractIn this paper, some algebraic and combinatorial characterizations of the spanning simplicial complex Δs(𝓙n,m) of the Jahangir’s graph 𝓙n,m are explored. We show that Δs(𝓙n,m) is pure, present the formula for f-vectors associated to it and hence deduce a recipe for computing the Hilbert series of the Face ring k[Δs(𝓙n,m)]. Finally, we show that the face ring of Δs(𝓙n,m) is Cohen-Macaulay and give some open scopes of the current work.

scholarly journals Geometric Realization of $\gamma$-Vectors of Subdivided Cross Polytopes

10.37236/9301 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Natalie Aisbett ◽  
Vadim Volodin
Keyword(s):  
Simplicial Complex ◽  
Geometric Realization ◽  
Simplicial Complexes ◽  
The Face ◽  
The Cross

For any flag simplicial complex $\Theta$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\Delta(\Theta)$ whose $f$-vector is the $\gamma$-vector of $\Theta$. This proves that the $\gamma$-vector of any such simplicial complex is the face vector of a flag simplicial complex, partially solving a conjecture by Nevo and Petersen. As a corollary we obtain that such simplicial complexes satisfy the Frankl-Füredi-Kalai inequalities.

scholarly journals Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex

2021 ◽  
pp. 207-224
Author(s):  
Arindam Banerjee ◽  
Kriti Goel ◽  
J. Verma
Keyword(s):  
Simplicial Complex ◽  
Rees Algebra ◽  
Homogeneous Ideal ◽  
Content Type ◽  
Face Ring ◽  
The Face ◽  
S Period

Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}({\mathfrak {n}}) be the Rees algebra of the maximal homogeneous ideal n {\mathfrak {n}} of R . R. We show that the generalized Hilbert-Kunz function H K ( s ) = ℓ ( R ( n ) / ( n , n t ) [ s ] ) HK(s)=\ell ({\mathcal {R}}({\mathfrak {n}})/({\mathfrak {n}}, {\mathfrak {n}} t)^{[s]}) is given by a polynomial for all large s . s. We calculate it in many examples and also provide a Macaulay2 code for computing H K ( s ) . HK(s).

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

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].

scholarly journals On Ptolemaic metric simplicial complexes

2010 ◽  
Vol 149 (1) ◽  
pp. 93-104 ◽  
Author(s):  
S. M. BUCKLEY ◽  
D. J. WRAITH ◽  
J. McDOUGALL
Keyword(s):  
Euclidean Space ◽  
Simplicial Complex ◽  
Large Class ◽  
Metric Spaces ◽  
Simplicial Complexes ◽  
Mild Conditions

AbstractWe show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy's inequality is closely related to inversions of metric spaces. For a large class of metric simplicial complexes, we characterize those which are isometric to Euclidean space in terms of metric inversions.

scholarly journals Face Ring Multiplicity via CM-Connectivity Sequences

2009 ◽  
Vol 61 (4) ◽  
pp. 888-903 ◽  
Author(s):  
Isabella Novik ◽  
Ed Swartz
Keyword(s):  
Lower Bound ◽  
Simplicial Complexes ◽  
Main Ingredient ◽  
Face Ring ◽  
The Face ◽  
New Interpretation ◽  
Dimension One

Abstract.The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen–Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all (d−1)-dimensional d-Cohen– Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring via the Cohen–Macaulay connectivity of the skeletons of .

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

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