Random walks on simplicial complexes and harmonics†

On Isomorphism of Simplicial Complexes and Their Related Algebras

Walking on the Edge and Cosystolic Expansion

Random walks on regular bounded degree expander graphs have numerous applications. A key property of these walks is that they converge rapidly to the uniform distribution on the vertices. The recent study of expansion of high dimensional simplicial complexes, which are the high dimensional analogues of graphs, calls for the natural generalization of random walks to higher dimensions. In particu...

High Dimensional Combinatorial Random Walks and Colorful Expansion

Random walks on expander graphs have been extensively studied. We define a high order combinatorial random walk on high dimensional simplicial complexes. This walk moves at random between neighboring i-dimensional faces (e.g. edges) of the complex, where two i-dimensional faces are considered neighbors if they share a common (i + 1)-dimensional face (e.g. a triangle). We prove that if the links...

Cohen-Macaulay-ness in codimension for simplicial complexes and expansion functor

In this paper we show that expansion of a Buchsbaum simplicial complex is $CM_t$, for an optimal integer $tgeq 1$. Also, by imposing extra assumptions on a $CM_t$ simplicial complex, we provethat it can be obtained from a Buchsbaum complex.

Vertex Decomposable Simplicial Complexes Associated to Path Graphs

Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure ...