### Abstract

The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u, v) and (u′, v′) iff both uu′ ∈ E(G) and vv′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim α(G^{n})/|V(G ^{n})|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 (1996), 290-300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products.

Original language | English (US) |
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Pages (from-to) | 73-87 |

Number of pages | 15 |

Journal | Journal of Graph Theory |

Volume | 54 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2007 |

### Fingerprint

### Keywords

- Independence ratio
- Tensor graph powers
- Vertex transitive graphs

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Graph Theory*,

*54*(1), 73-87. https://doi.org/10.1002/jgt.20194

**Independent sets in tensor graph powers.** / Alon, Noga; Lubetzky, Eyal.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 54, no. 1, pp. 73-87. https://doi.org/10.1002/jgt.20194

}

TY - JOUR

T1 - Independent sets in tensor graph powers

AU - Alon, Noga

AU - Lubetzky, Eyal

PY - 2007/1

Y1 - 2007/1

N2 - The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u, v) and (u′, v′) iff both uu′ ∈ E(G) and vv′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim α(Gn)/|V(G n)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 (1996), 290-300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products.

AB - The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u, v) and (u′, v′) iff both uu′ ∈ E(G) and vv′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim α(Gn)/|V(G n)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 (1996), 290-300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products.

KW - Independence ratio

KW - Tensor graph powers

KW - Vertex transitive graphs

UR - http://www.scopus.com/inward/record.url?scp=33846533064&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846533064&partnerID=8YFLogxK

U2 - 10.1002/jgt.20194

DO - 10.1002/jgt.20194

M3 - Article

VL - 54

SP - 73

EP - 87

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -