## Abstract

We prove that, for fixed k ≥ 3, the following classes of labeled n-vertex graphs are asymptotically equicardinal: graphs of diameter k, connected graphs of diameter at least k, and (not necessarily connected) graphs with a shortest path of length at least k. An asymptotically exact approximation of the number of such n-vertex graphs is obtained, and an explicit error estimate in the approximation is found. Thus, the estimates are improved for the asymptotic approximation of the number of n-vertex graphs of fixed diameter k earlier obtained by Füredi and Kim. It is shown that almost all graphs of diameter k have a unique pair of diametrical vertices but almost all graphs of diameter 2 have more than one pair of such vertices.

Original language | English |
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Pages (from-to) | 204-214 |

Number of pages | 11 |

Journal | Journal of Applied and Industrial Mathematics |

Volume | 11 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Apr 2017 |

## Keywords

- graph
- graph diameter
- labeled graph
- number of graphs
- ordinary graph
- shortest path