{"paper":{"title":"A Matching-Number Refinement of Brouwer's Laplacian Eigenvalue Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chunyan Qin, Jing Huang","submitted_at":"2026-07-08T08:00:33Z","abstract_excerpt":"Let $G=(V,E)$ be a finite simple graph with Laplacian eigenvalues\n  $\\lambda_1(L(G))\\ge\\cdots\\ge\\lambda_{|V|}(L(G))$, and define\n  \\[\n  \\eps_k(G)=\n  \\sum_{j=1}^{\\min\\{k,|V|\\}}\\lambda_j(L(G))-|E|.\n  \\]\n  Let $\\nu(G)$ be the matching number of $G$, and let $n(G)$ be the number of\n  non-isolated vertices of $G$. Lew proved that\n  \\(\\eps_k(G)\\le k\\nu(G)+\\lfloor k/2\\rfloor\\), and conjectured that the\n  additive term can be removed in the non-endpoint range. We prove this\n  conjecture:\n  \\[\n  \\eps_k(G)\\le k\\nu(G)\n  \\qquad\n  (1\\le k\\le n(G)-2).\n  \\]\n  We also characterize all equality cases. Up to is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07118","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/2607.07118/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"}