{"paper":{"title":"A spectral threshold for triangle counting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mingqing Zhai, Yuhan Zhang","submitted_at":"2026-06-06T13:25:58Z","abstract_excerpt":"The 1970 spectral extension of Mantel's theorem, proved by Nosal, states that every graph with $m$ edges and spectral radius $\\rho_1>\\sqrt{m}$ contains at least one triangle. Its quantitative refinement by Ning and Zhai later established that any graph $G$ with $m$ edges and spectral radius $\\rho_1\\geq\\sqrt{m}$ contains at least $\\lfloor\\frac{\\sqrt{m}-1}{2}\\rfloor$ triangles, unless $G$ is a complete bipartite graph.\n  In this paper, we further investigate the minimum number of triangles guaranteed under the strengthened spectral condition $\\rho_1\\geq\\sqrt{m}+c$, where $c$ is a positive consta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08163","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/2606.08163/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"}