{"paper":{"title":"Spectral gaps of random graphs and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.PR"],"primary_cat":"math.CO","authors_text":"Christopher Hoffman, Elliot Paquette, Matthew Kahle","submitted_at":"2012-01-02T05:50:04Z","abstract_excerpt":"We study the spectral gap of the Erd\\H{o}s--R\\'enyi random graph through the connectivity threshold. In particular, we show that for any fixed $\\delta > 0$ if $$p \\ge \\frac{(1/2 + \\delta) \\log n}{n},$$ then the normalized graph Laplacian of an Erd\\H{o}s--R\\'enyi graph has all of its nonzero eigenvalues tightly concentrated around $1$. We estimate both the decay rate of the spectral gap to $1$ and the failure probability, up to a constant factor. We also show that the $1/2$ in the above is optimal, and that if $p = \\frac{c \\log n}{n}$ for $c < 1/2,$ then there are eigenvalues of the Laplacian r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0425","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}