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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2410.04563","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-10-06T17:35:30Z","cross_cats_sorted":[],"title_canon_sha256":"9647718e57129f58c3253ba7585d068c9de9904bc022a2f44f8cc31c1ea76f3c","abstract_canon_sha256":"82502baae1beaeb2cfda846c307f51f93004fc63e562c66ab50e750b676f9ea5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:16:36.092270Z","signature_b64":"A3Rawdt0muKoNcAzp31bmhNo7yvE/zCGKU5pjcf1EiyvbiBRe+Y/2jfWSYj+8eMjw6+EWVT7qredVNcceIdxBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa812b0611287a8ea7ee95985308ba7800ae29fb5fc2d71f5ee884fa38f8fd64","last_reissued_at":"2026-07-05T09:16:36.091771Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:16:36.091771Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2410.04563","created_at":"2026-07-05T09:16:36.091834+00:00"},{"alias_kind":"arxiv_version","alias_value":"2410.04563v1","created_at":"2026-07-05T09:16:36.091834+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.04563","created_at":"2026-07-05T09:16:36.091834+00:00"},{"alias_kind":"pith_short_12","alias_value":"VKASWBQRFB5I","created_at":"2026-07-05T09:16:36.091834+00:00"},{"alias_kind":"pith_short_16","alias_value":"VKASWBQRFB5I5J7O","created_at":"2026-07-05T09:16:36.091834+00:00"},{"alias_kind":"pith_short_8","alias_value":"VKASWBQR","created_at":"2026-07-05T09:16:36.091834+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2508.07550","citing_title":"Remarks on the Brouwer Conjecture","ref_index":21,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA","json":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA.json","graph_json":"https://pith.science/api/pith-number/VKASWBQRFB5I5J7OSWMFGCF2PA/graph.json","events_json":"https://pith.science/api/pith-number/VKASWBQRFB5I5J7OSWMFGCF2PA/events.json","paper":"https://pith.science/paper/VKASWBQR"},"agent_actions":{"view_html":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA","download_json":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA.json","view_paper":"https://pith.science/paper/VKASWBQR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2410.04563&json=true","fetch_graph":"https://pith.science/api/pith-number/VKASWBQRFB5I5J7OSWMFGCF2PA/graph.json","fetch_events":"https://pith.science/api/pith-number/VKASWBQRFB5I5J7OSWMFGCF2PA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA/action/storage_attestation","attest_author":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA/action/author_attestation","sign_citation":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA/action/citation_signature","submit_replication":"https://pith.science/pith/VKASWBQRFB5I5J7OSWMFGCF2PA/action/replication_record"}},"created_at":"2026-07-05T09:16:36.091834+00:00","updated_at":"2026-07-05T09:16:36.091834+00:00"}