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

Abstract

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
Volume84
Issue number2
DOIs
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.

Fingerprint

Dive into the research topics of 'A continuous analogue of the girth problem'. Together they form a unique fingerprint.

Cite this