{"paper":{"title":"The spectral gap of dense random regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Konstantin Tikhomirov, Pierre Youssef","submitted_at":"2016-10-06T07:54:39Z","abstract_excerpt":"For any $\\alpha\\in (0,1)$ and any $n^{\\alpha}\\leq d\\leq n/2$, we show that $\\lambda(G)\\leq C_\\alpha \\sqrt{d}$ with probability at least $1-\\frac{1}{n}$, where $G$ is the uniform random $d$-regular graph on $n$ vertices, $\\lambda(G)$ denotes its second largest eigenvalue (in absolute value) and $C_\\alpha$ is a constant depending only on $\\alpha$. Combined with earlier results in this direction covering the case of sparse random graphs, this completely settles the problem of estimating the magnitude of $\\lambda(G)$, up to a multiplicative constant, for all values of $n$ and $d$, confirming a con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.01765","kind":"arxiv","version":2},"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"}