## Abstract

Let A be the adjacency matrix of a d-regular graph of order n and girth g and d = λ_{1} ≥ ⋯ ≥ λ_{n} its eigenvalues. Then ∑^{n}_{j=2} λ^{i}_{j} = nt_{i} - d^{i}, for i = 0, 1, ⋯, g - 1, where t_{i} is the number of closed walks of length i on the d-regular infinite tree. Here we consider distributions on the real line, whose ith moment is also nt_{i} - d^{i} for all i = 0, 1, ⋯, g - 1. We investigate distributional analogues of several extremal graph problems involving the parameters n, d, g, and Λ = max λ_{2}, λ_{n}. Surprisingly, perhaps, many similarities hold between the graphical and the distributional situations. Specifically, we show in the case of distributions that the least possible n, given d, g is exactly the (trivial graph-theoretic) Moore bound. We also ask how small Λ can be, given d, g, and n, and improve the best known bound for graphs whose girth exceeds their diameter.

Original language | English |
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Article number | 92088 |

Pages (from-to) | 340-363 |

Number of pages | 24 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 84 |

Issue number | 2 |

DOIs | |

State | Published - 2002 |

### Bibliographical note

Funding Information:1Supported in part by grants from the US–Israel Binational Science Fund and from the Israeli Academy of Science.