{"paper":{"title":"On a theorem of Nosal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"V. Nikiforov","submitted_at":"2021-04-25T14:22:17Z","abstract_excerpt":"Let $G$ be a graph with $m$ edges and spectral radius $\\lambda_{1}$. Let $bk\\left( G\\right) $ stand for the maximal number of triangles with a common edge in $G$.\n  In 1970 Nosal proved that if $\\lambda_{1}^{2}>m,$ then $G$ contains a triangle. In this paper we show that the same premise implies that \\[ bk\\left( G\\right) >\\frac{1}{12}\\sqrt[4]{m}. \\] This result settles a conjecture of Zhai, Lin, and Shu.\n  Write $\\lambda_{2}$ for the second largest eigenvalue of $G$. Recently, Lin, Ning, and Wu showed that if $G$ is a triangle-free graph of order at least three, then \\[ \\lambda_{1}^{2}+\\lambda"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.12171","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/2104.12171/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"}