## 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) |
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Pages (from-to) | 77-97 |

Number of pages | 21 |

Journal | Geometric and Functional Analysis |

Volume | 18 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2008 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Geometry and Topology