{"paper":{"title":"Partition density, star arboricity, and sums of Laplacian eigenvalues of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Lew","submitted_at":"2024-10-06T17:35:30Z","abstract_excerpt":"Let $G=(V,E)$ be a graph on $n$ vertices, and let $\\lambda_1(L(G))\\ge \\cdots\\ge \\lambda_{n-1}(L(G))\\ge \\lambda_n(L(G))=0$ be the eigenvalues of its Laplacian matrix $L(G)$. Brouwer conjectured that for every $1\\le k\\le n$, $\\sum_{i=1}^k \\lambda_i(L(G)) \\le |E|+\\binom{k+1}{2}$. Here, we prove the following weak version of Brouwer's conjecture: For every $1\\leq k \\leq n$, \\[\n  \\sum_{i=1}^k \\lambda_i(L(G)) \\leq\n  |E|+k^2+15k\\log{k}+65k. \\] For a graph $G=(V,E)$, we define its partition density $\\tilde{\\rho}(G)$ as the maximum, over all subgraphs $H$ of $G$, of the ratio between the number of edge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.04563","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/2410.04563/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"}