### Abstract

It is shown that there exist a sequence of 3-regular graphs {G<inf>n</inf>}<sup>∞</sup><inf>n</inf>=1 and a Hadamard space X such that {G<inf>n</inf>}<sup>∞</sup><inf>n</inf>=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<inf>n</inf>}<sup>∞</sup><inf>n</inf>=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublineartime 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.

Original language | English (US) |
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Pages (from-to) | 1471-1548 |

Number of pages | 78 |

Journal | Duke Mathematical Journal |

Volume | 164 |

Issue number | 8 |

DOIs | |

State | Published - 2015 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*164*(8), 1471-1548. https://doi.org/10.1215/00127094-3119525

**Expanders with respect to Hadamard spaces and random graphs.** / Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 164, no. 8, pp. 1471-1548. https://doi.org/10.1215/00127094-3119525

}

TY - JOUR

T1 - Expanders with respect to Hadamard spaces and random graphs

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2015

Y1 - 2015

N2 - It is shown that there exist a sequence of 3-regular graphs {Gn}∞n=1 and a Hadamard space X such that {Gn}∞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 {Gn}∞n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublineartime 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.

AB - It is shown that there exist a sequence of 3-regular graphs {Gn}∞n=1 and a Hadamard space X such that {Gn}∞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 {Gn}∞n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublineartime 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.

UR - http://www.scopus.com/inward/record.url?scp=84930859553&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930859553&partnerID=8YFLogxK

U2 - 10.1215/00127094-3119525

DO - 10.1215/00127094-3119525

M3 - Article

VL - 164

SP - 1471

EP - 1548

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 8

ER -