### 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 2006 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

*96*(6), 933-957. https://doi.org/10.1016/j.jctb.2006.05.002

**Limits of dense graph sequences.** / Lovász, L.; Szegedy, Balázs.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 96, no. 6, pp. 933-957. https://doi.org/10.1016/j.jctb.2006.05.002

}

TY - JOUR

T1 - Limits of dense graph sequences

AU - Lovász, L.

AU - Szegedy, Balázs

PY - 2006/11

Y1 - 2006/11

N2 - We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn 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.

AB - We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn 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.

KW - Convergent graph sequence

KW - Graph homomorphism

KW - Limit

KW - Quasirandom graph

UR - http://www.scopus.com/inward/record.url?scp=33845276522&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845276522&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2006.05.002

DO - 10.1016/j.jctb.2006.05.002

M3 - Article

AN - SCOPUS:33845276522

VL - 96

SP - 933

EP - 957

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 6

ER -