Cheeger type inequalities for high dimensional simplicial complexes

TL;DR

This paper extends Cheeger inequalities, originally from geometry and graph theory, to high-dimensional simplicial complexes, providing a combinatorial perspective on these inequalities.

Contribution

It introduces a novel combinatorial analogue of Cheeger inequality for high-dimensional simplicial complexes, expanding the theoretical framework.

Findings

Established a Cheeger-type inequality for high-dimensional complexes

Connected isoperimetric properties with spectral gaps in simplicial complexes

Provided new tools for analyzing high-dimensional combinatorial structures

Abstract

Cheeger inequality is a classical result emerging from the isoperimetric problem in the field of geometry. In the graph theory, a discrete version of Cheeger inequality was also studied deeply and the notion was further extended for higher dimensional simplicial complexes in various directions. In this paper, we consider an analogue of discrete Cheeger inequality for high dimensional simplicial complexes from a combinatorial viewpoint.