### Abstract

The independence numbers of powers of graphs have been long studied, under several definitions of graph products, and in particular, under the strong graph product. We show that the series of independence numbers in strong powers of a fixed graph can exhibit a complex structure, implying that the Shannon capacity of a graph cannot be approximated (up to a subpolynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the series. This is true even if this prefix shows a significant increase of the independence number at a given power, after which it stabilizes for a while.

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

Pages (from-to) | 2172-2176 |

Number of pages | 5 |

Journal | IEEE Transactions on Information Theory |

Volume | 52 |

Issue number | 5 |

DOIs | |

State | Published - May 2006 |

### Keywords

- Graph powers
- Shannon capacity

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Information Systems

### Cite this

**The Shannon capacity of a graph and the independence numbers of its powers.** / Alon, Noga; Lubetzky, Eyal.

Research output: Contribution to journal › Article

*IEEE Transactions on Information Theory*, vol. 52, no. 5, pp. 2172-2176. https://doi.org/10.1109/TIT.2006.872856

}

TY - JOUR

T1 - The Shannon capacity of a graph and the independence numbers of its powers

AU - Alon, Noga

AU - Lubetzky, Eyal

PY - 2006/5

Y1 - 2006/5

N2 - The independence numbers of powers of graphs have been long studied, under several definitions of graph products, and in particular, under the strong graph product. We show that the series of independence numbers in strong powers of a fixed graph can exhibit a complex structure, implying that the Shannon capacity of a graph cannot be approximated (up to a subpolynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the series. This is true even if this prefix shows a significant increase of the independence number at a given power, after which it stabilizes for a while.

AB - The independence numbers of powers of graphs have been long studied, under several definitions of graph products, and in particular, under the strong graph product. We show that the series of independence numbers in strong powers of a fixed graph can exhibit a complex structure, implying that the Shannon capacity of a graph cannot be approximated (up to a subpolynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the series. This is true even if this prefix shows a significant increase of the independence number at a given power, after which it stabilizes for a while.

KW - Graph powers

KW - Shannon capacity

UR - http://www.scopus.com/inward/record.url?scp=33646061628&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646061628&partnerID=8YFLogxK

U2 - 10.1109/TIT.2006.872856

DO - 10.1109/TIT.2006.872856

M3 - Article

AN - SCOPUS:33646061628

VL - 52

SP - 2172

EP - 2176

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 5

ER -