{"paper":{"title":"Positive discrepancy, MaxCut, and eigenvalues of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benny Sudakov, Eero R\\\"aty, Istv\\'an Tomon","submitted_at":"2023-11-03T17:54:27Z","abstract_excerpt":"The positive discrepancy of a graph $G$ of edge density $p=e(G)/\\binom{v(G)}{2}$ is defined as\n  $$\\mbox{disc}^{+}(G)=\\max_{U\\subset V(G)}e(G[U])-p\\binom{|U|}{2}.$$\n  In 1993, Alon proved (using the equivalent terminology of minimum bisections) that if $G$ is $d$-regular on $n$ vertices, and $d=O(n^{1/9})$, then $\\mbox{disc}^{+}(G)=\\Omega(d^{1/2}n)$. We greatly extend this by showing that if $G$ has average degree $d$, then $\\mbox{disc}^{+}(G)=\\Omega(d^{\\frac{1}{2}}n)$ if $d\\in [0,n^{\\frac{2}{3}}]$, $\\Omega(n^2/d)$ if $d\\in [n^{\\frac{2}{3}},n^{\\frac{4}{5}}]$, and $\\Omega(d^{\\frac{1}{4}}n/\\log "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.02070","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2311.02070/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"}