A continuous analogue of the girth problem

Alon Amit*, Shlomo Hoory, Nathan Linial

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Let A be the adjacency matrix of a d-regular graph of order n and girth g and d = λ1 ≥ ⋯ ≥ λn its eigenvalues. Then ∑nj=2 λij = nti - di, for i = 0, 1, ⋯, g - 1, where ti 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 nti - di 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 λ2, λn. 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 languageAmerican English
Article number92088
Pages (from-to)340-363
Number of pages24
JournalJournal of Combinatorial Theory. Series B
Issue number2
StatePublished - 2002

Bibliographical note

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


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