Simplicial Complexes of Whisker Type

Simplicial Complexes of Whisker Type

Let I ⊂ K[x1, . . . , xn] be a zero-dimensional monomial ideal, and ∆(I) be the simplicial complex whose Stanley–Reisner ideal is the polarization of I. It follows from a result of Soleyman Jahan that ∆(I) is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that ∆(I) is even vertex decomposable. The ideal L(I), which is defined to be the Sta...

A Cheeger-Type Inequality on Simplicial Complexes

In this paper, we consider a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterer and Kahle. A Cheeger-type inequality is proved, which is similar to a result on graphs due to Fan Chung. This inequality is then used to study the relationship between coboundary expanders on simplicial complexes and their corresponding eigenvalues, complementing and extend...

New methods for constructing shellable simplicial complexes

A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}...

Minors of simplicial complexes

We extend the notion of a minor from matroids to simplicial complexes. We show that the class of matroids, as well as the class of independence complexes, is characterized by a single forbidden minor. Inspired by a recent result of Aharoni and Berger, we investigate possible ways to extend the matroid intersection theorem to simplicial complexes.

On Isomorphism of Simplicial Complexes and Their Related Algebras