Measured expanders
Measured expanders
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 Poincaré inequalities. We prove that the so-called Cheeger inequality holds in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Moreover, we also prove that our measured (asymptotic) expanders are generalised expanders introduced by Tessera. Finally, we present some examples to demonstrate relations and differences between classical expander graphs and the measured ones. This paper is motivated primarily by our previous work on the rigidity problem for Roe algebras.
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
Li, Kang
62945651-4b08-4fa3-a1fa-0eadaddbf6c5
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
Li, Kang
62945651-4b08-4fa3-a1fa-0eadaddbf6c5
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 Poincaré inequalities. We prove that the so-called Cheeger inequality holds in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Moreover, we also prove that our measured (asymptotic) expanders are generalised expanders introduced by Tessera. Finally, we present some examples to demonstrate relations and differences between classical expander graphs and the measured ones. This paper is motivated primarily by our previous work on the rigidity problem for Roe algebras.
Text
2104.06052 (1)
- Accepted Manuscript
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e-pub ahead of print date: 12 March 2022
Identifiers
Local EPrints ID: 457653
URI: http://eprints.soton.ac.uk/id/eprint/457653
ISSN: 1793-5253
PURE UUID: c2d8f425-f387-415e-afb7-c71d4badfaa3
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Date deposited: 14 Jun 2022 16:58
Last modified: 17 Mar 2024 07:20
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Author:
Jiawen Zhang
Author:
Kang Li
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