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# Mathematics > Metric Geometry

# Title: Measured expanders

(Submitted on 13 Apr 2021 (v1), last revised 24 Sep 2021 (this version, v2))

Abstract: By measured graphs we mean graphs endowed with a measure on the set of vertices. In this context, we explore the relations between the appropriate Cheeger constant and Poincar\'{e} inequalities. We prove that the expected inequalities hold in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Finally, we prove that our measured (asymptotic) expanders are Tessera's generalised expanders. This concept is motivated by analytic work in [9] and [10].

## Submission history

From: JiaWen Zhang [view email]**[v1]**Tue, 13 Apr 2021 09:29:59 GMT (24kb)

**[v2]**Fri, 24 Sep 2021 03:12:46 GMT (22kb)

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