Measured expanders

TL;DR

This paper investigates measured graphs with vertex measures, establishing Cheeger inequalities in specific cases, and explores their relation to classical and generalized expanders, motivated by applications in Roe algebras.

Contribution

It proves Cheeger inequalities for measured graphs under certain conditions and relates measured expanders to Tessera's generalized expanders, expanding the theory of graph expansion.

Findings

Cheeger inequality holds for measures from random walks.

Cheeger inequality holds when measure ratios are bounded.

Measured expanders are related to Tessera's generalized expanders.

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 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. The current paper is motivated primarily by our previous work on the rigidity problem for Roe algebras.