### 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 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Keywords

- Independence ratio
- Tensor graph powers
- Vertex transitive graphs

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## Cite this

*Journal of Graph Theory*,

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