1 June 2015 Expanders with respect to Hadamard spaces and random graphs
Manor Mendel, Assaf Naor
Duke Math. J. 164(8): 1471-1548 (1 June 2015). DOI: 10.1215/00127094-3119525


It is shown that there exist a sequence of 3 -regular graphs { G n } n = 1 and a Hadamard space X such that { G n } n = 1 forms an expander sequence with respect to X , yet random regular graphs are not expanders with respect to X . This answers a question of the second author and Silberman. The graphs { G n } n = 1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear-time constant-factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.


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Manor Mendel. Assaf Naor. "Expanders with respect to Hadamard spaces and random graphs." Duke Math. J. 164 (8) 1471 - 1548, 1 June 2015. https://doi.org/10.1215/00127094-3119525


Received: 4 July 2013; Revised: 17 July 2014; Published: 1 June 2015
First available in Project Euclid: 28 May 2015

zbMATH: 1316.05109
MathSciNet: MR3352040
Digital Object Identifier: 10.1215/00127094-3119525

Primary: 51F99
Secondary: 05C12 , 05C50 , 46B85

Keywords: bi-Lipschitz embeddings , CAT(0) spaces , Euclidean cones , Expanding graphs , Random graphs

Rights: Copyright © 2015 Duke University Press


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Vol.164 • No. 8 • 1 June 2015
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