A Cheeger-Type Inequality on Simplicial Complexes
John Steenbergen, Caroline Klivans, Sayan Mukherjee
TL;DR
This paper establishes a Cheeger-type inequality for simplicial complexes, linking coboundary expansion to eigenvalues, and reveals limitations of coboundary expanders in satisfying classical inequalities.
Contribution
It introduces a Cheeger-type inequality for simplicial complexes and explores its implications for coboundary expanders and their spectral properties.
Findings
Proved a Cheeger-type inequality for simplicial complexes.
Showed coboundary expanders do not satisfy Buser or Cheeger inequalities.
Extended understanding of spectral properties of coboundary expanders.
Abstract
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 extending results found by Gundert and Wagner. In particular, we find these coboundary expanders do not satisfy natural Buser or Cheeger inequalities.
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