A Cheeger-Type Inequality on Simplicial Complexes

Isoperimetric inequalities in simplicial complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In...

Higher Dimensional Discrete Cheeger Inequalities

4 For graphs there exists a strong connection between spectral and combinatorial 5 expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the 6 lower bound of which states that λ(G) ≤ h(G), where λ(G) is the second smallest 7 eigenvalue of the Laplacian of a graph G and h(G) is the Cheeger constant measuring 8 the edge expansion of G. We are interested in generalizat...

Amenable Covers, Volume and L-betti Numbers of Aspherical Manifolds

We provide a proof for an inequality between volume and LBetti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau’s theory of L-Betti numbers of R-simplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for ...

Towards Spectral Sparsification of Simplicial Complexes Based on Generalized Effective Resistance

As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher-order interactions between three or more objects in complex systems. There has been a recent surge in activity for the development of data analysis methods applicable to simplicial complexes, including techniques based on computational topology, higher-order random proces...

On Isomorphism of Simplicial Complexes and Their Related Algebras