Abstract
It is shown that there exist a sequence of -regular graphs and a Hadamard space such that forms an expander sequence with respect to , yet random regular graphs are not expanders with respect to . This answers a question of the second author and Silberman. The graphs 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.
Citation
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
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