{"paper":{"title":"An Alon-Boppana--type bound for very dense graphs, with applications to max-cut","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.CO","authors_text":"Shengtong Zhang","submitted_at":"2025-07-14T08:15:11Z","abstract_excerpt":"For any $\\epsilon > 0$, we show that if $G$ is a regular graph on $n \\gg_\\epsilon 1$ vertices that is $\\epsilon$-far (differs by at least $\\epsilon n^2$ edges) from any Tur\\'{a}n graph, then its second eigenvalue $\\lambda_2$ satisfies $$\\lambda_2 \\geq n^{1/4 - \\epsilon}.$$ The exponent $1/4$ is optimal. Our result generalizes an analogous bound, independently obtained by Balla, R\\\"{a}ty -- Sudakov-Tomon, and Ihringer, which only applies to graphs with density at most $\\frac{1}{2}$. Up to a lower-order factor, this confirms a conjecture of R\\\"{a}ty, Sudakov and Tomon. Our spectral approach has "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.10037","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.10037/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}