### 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 | English (US) |
---|---|

Pages (from-to) | 273-303 |

Number of pages | 31 |

Journal | Journal of Functional Analysis |

Volume | 227 |

Issue number | 2 |

DOIs | |

State | Published - Oct 15 2005 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*227*(2), 273-303. https://doi.org/10.1016/j.jfa.2005.04.003

**Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs.** / Naor, Assaf; Rabani, Yuval; Sinclair, Alistair.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 227, no. 2, pp. 273-303. https://doi.org/10.1016/j.jfa.2005.04.003

}

TY - JOUR

T1 - Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs

AU - Naor, Assaf

AU - Rabani, Yuval

AU - Sinclair, Alistair

PY - 2005/10/15

Y1 - 2005/10/15

N2 - 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.

AB - 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.

KW - Edge expansion

KW - Observable diameter

KW - Quasisymmetric embeddings

KW - Vertex expansion

UR - http://www.scopus.com/inward/record.url?scp=25444434660&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=25444434660&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2005.04.003

DO - 10.1016/j.jfa.2005.04.003

M3 - Article

VL - 227

SP - 273

EP - 303

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -