## Abstract

We study the diameter of LPS Ramanujan graphs X_{p,q}. We show that the diameter of the bipartite Ramanujan graphs is greater than (4/3)log_{p}(n)+O(1), where n is the number of vertices of X_{p,q}. We also construct an infinite family of (p+1)-regular LPS Ramanujan graphs X_{p,m} such that the diameter of these graphs is greater than or equal to ⌊(4/3)log_{p}(n)⌋. On the other hand, for any k-regular Ramanujan graph we show that only a tiny fraction of all pairs of vertices have distance greater than (1+ϵ) log_{k–1}(n). We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically (4/3)log_{k–1}(n) and log_{k–1}(n), respectively.

Original language | English (US) |
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Pages (from-to) | 427-446 |

Number of pages | 20 |

Journal | Combinatorica |

Volume | 39 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2019 |

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics