{"paper":{"title":"A sharp fixed-size spectral bound for $kK_3$-free graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Joyentanuj Das, Yamini V","submitted_at":"2026-08-06T10:48:15Z","abstract_excerpt":"For a fixed integer $k\\ge2$, we establish a sharp adjacency-spectral upper bound for sufficiently large $m$-edge $kK_3$-free graphs. We prove \\[ \\lambda(G)\\le (k-1)+\\sqrt{m-k(k-1)}. \\] Moreover, equality holds precisely when $(2k-1)\\mid m$ and, up to isolated vertices, $G$ is the join of $K_{2k-1}$ with an independent set of $m/(2k-1)-(k-1)$ vertices. The case $k=2$ was previously known; our argument establishes every fixed $k\\ge3$. The proof requires information beyond first-order spectral stability. We derive an exact nonnegative defect identity at a maximum Perron vertex, use it to bound th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05869","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/2608.05869/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"}