### Abstract

It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number D(G) of nested quantifiers in such a formula can serve as a measure for the "first order complexity" of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is θ(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log* n, the inverse of the TOWER-function. The general picture, however, is still a mystery.

Original language | English (US) |
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Pages (from-to) | 119-145 |

Number of pages | 27 |

Journal | Random Structures and Algorithms |

Volume | 26 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 2005 |

### ASJC Scopus subject areas

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

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## Cite this

*Random Structures and Algorithms*,

*26*(1-2), 119-145. https://doi.org/10.1002/rsa.20049