### Abstract

Let G = (V,E) be a simple undirected graph. Define G ^{n} , the n-th power of G, as the graph on the vertex set V ^{n} in which two vertices (u _{1}, . . . , u _{n} ) and (v _{1}, . . . , v _{n} ) are adjacent if and only if u _{i} is adjacent to v _{i} in G for every i. We give a characterization of all independent sets in such graphs whenever G is connected and non-bipartite. Consider the stationary measure of the simple random walk on G ^{n} . We show that every independent set is almost contained with respect to this measure in a junta, a cylinder of constant co-dimension. Moreover we show that the projection of that junta defines a nearly independent set, i.e. it spans few edges (this also guarantees that it is not trivially the entire vertex-set). Our approach is based on an analog of Fourier analysis for product spaces combined with spectral techniques and on a powerful invariance principle presented in [MoOO]. This principle has already been shown in [DiMR] to imply that independent sets in such graph products have an influential coordinate. In this work we prove that in fact there is a set of few coordinates and a junta on them that capture the independent set almost completely.

Original language | English (US) |
---|---|

Pages (from-to) | 77-97 |

Number of pages | 21 |

Journal | Geometric and Functional Analysis |

Volume | 18 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2008 |

### Fingerprint

### Keywords

- Discrete Fourier analysis
- Independent sets
- Intersecting families
- Product graphs

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis

### Cite this

*Geometric and Functional Analysis*,

*18*(1), 77-97. https://doi.org/10.1007/s00039-008-0651-1

**Independent sets in graph powers are almost contained in juntas.** / Dinur, Irit; Friedgut, Ehud; Regev, Oded.

Research output: Contribution to journal › Article

*Geometric and Functional Analysis*, vol. 18, no. 1, pp. 77-97. https://doi.org/10.1007/s00039-008-0651-1

}

TY - JOUR

T1 - Independent sets in graph powers are almost contained in juntas

AU - Dinur, Irit

AU - Friedgut, Ehud

AU - Regev, Oded

PY - 2008/4

Y1 - 2008/4

N2 - Let G = (V,E) be a simple undirected graph. Define G n , the n-th power of G, as the graph on the vertex set V n in which two vertices (u 1, . . . , u n ) and (v 1, . . . , v n ) are adjacent if and only if u i is adjacent to v i in G for every i. We give a characterization of all independent sets in such graphs whenever G is connected and non-bipartite. Consider the stationary measure of the simple random walk on G n . We show that every independent set is almost contained with respect to this measure in a junta, a cylinder of constant co-dimension. Moreover we show that the projection of that junta defines a nearly independent set, i.e. it spans few edges (this also guarantees that it is not trivially the entire vertex-set). Our approach is based on an analog of Fourier analysis for product spaces combined with spectral techniques and on a powerful invariance principle presented in [MoOO]. This principle has already been shown in [DiMR] to imply that independent sets in such graph products have an influential coordinate. In this work we prove that in fact there is a set of few coordinates and a junta on them that capture the independent set almost completely.

AB - Let G = (V,E) be a simple undirected graph. Define G n , the n-th power of G, as the graph on the vertex set V n in which two vertices (u 1, . . . , u n ) and (v 1, . . . , v n ) are adjacent if and only if u i is adjacent to v i in G for every i. We give a characterization of all independent sets in such graphs whenever G is connected and non-bipartite. Consider the stationary measure of the simple random walk on G n . We show that every independent set is almost contained with respect to this measure in a junta, a cylinder of constant co-dimension. Moreover we show that the projection of that junta defines a nearly independent set, i.e. it spans few edges (this also guarantees that it is not trivially the entire vertex-set). Our approach is based on an analog of Fourier analysis for product spaces combined with spectral techniques and on a powerful invariance principle presented in [MoOO]. This principle has already been shown in [DiMR] to imply that independent sets in such graph products have an influential coordinate. In this work we prove that in fact there is a set of few coordinates and a junta on them that capture the independent set almost completely.

KW - Discrete Fourier analysis

KW - Independent sets

KW - Intersecting families

KW - Product graphs

UR - http://www.scopus.com/inward/record.url?scp=42749093780&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=42749093780&partnerID=8YFLogxK

U2 - 10.1007/s00039-008-0651-1

DO - 10.1007/s00039-008-0651-1

M3 - Article

AN - SCOPUS:42749093780

VL - 18

SP - 77

EP - 97

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 1

ER -