### Abstract

We show that if a sequence of dense graphs G_{n} has the property that for every fixed graph F, the density of copies of F in G_{n} tends to a limit, then there is a natural "limit object," namely a symmetric measurable function W : [0, 1]^{2} → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. We also characterize graph parameters that are obtained as limits of subgraph densities by the "reflection positivity" property. Along the way we introduce a rather general model of random graphs, which seems to be interesting on its own right.

Original language | English |
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Pages (from-to) | 933-957 |

Number of pages | 25 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 96 |

Issue number | 6 |

DOIs | |

Publication status | Published - Nov 1 2006 |

### Keywords

- Convergent graph sequence
- Graph homomorphism
- Limit
- Quasirandom graph

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

Lovász, L., & Szegedy, B. (2006). Limits of dense graph sequences.

*Journal of Combinatorial Theory. Series B*,*96*(6), 933-957. https://doi.org/10.1016/j.jctb.2006.05.002