## Abstract

It is shown that the edges of any n-point vertex expander can be replaced by new edges so that the resulting graph is an edge expander, and such that any two vertices that are joined by a new edge are at distance O (√log n) in the original graph. This result is optimal, and is shown to have various geometric consequences. In particular, it is used to obtain an alternative perspective on the recent algorithm of Arora et al. [Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, 2004, pp. 222-231.] for approximating the edge expansion of a graph, and to give a nearly optimal lower bound on the ratio between the observable diameter and the diameter of doubling metric measure spaces which are quasisymmetrically embeddable in Hilbert space.

Original language | American English |
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Pages (from-to) | 273-303 |

Number of pages | 31 |

Journal | Journal of Functional Analysis |

Volume | 227 |

Issue number | 2 |

DOIs | |

State | Published - 15 Oct 2005 |

Externally published | Yes |

### Bibliographical note

Funding Information:∗ Corresponding author. Fax: +1 425 936 7329. E-mail addresses: anaor@microsoft.com (A. Naor), rabani@cs.technion.ac.il (Y. Rabani), sinclair@cs.berkeley.edu (A. Sinclair). 1This work was done while the author was on sabbatical leave at Cornell University. Part of this work was done while the author was visiting Microsoft Research, Redmond. 2Supported in part by NSF ITR grant CCR-0121555. Part of this work was done while the author was on sabbatical leave at Microsoft Research, Redmond.

## Keywords

- Edge expansion
- Observable diameter
- Quasisymmetric embeddings
- Vertex expansion