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In this note, we characterize the equality case in this inequality. Our main result is that for every $n$-vertex graph $G=(V,E)$ and for every $k\\in \\{1,2,\\dots,n-1\\}$, the equality $\\sum_{i=1}^k\\mu_i(G)=|E(G)|+\\binom{k+1}{2}$ holds if and only if $G$ is a threshold graph with clique number $k+1$, where $\\mu_1(G)\\geq \\mu_2(G)\\geq \\c"},"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":"2607.17293","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-19T15:24:59Z","cross_cats_sorted":[],"title_canon_sha256":"5f862d31ba055a478a8ed30338f7f56a824661fa9ccfff3a77553f2513889ccd","abstract_canon_sha256":"18da83e6e346aa4f1f9062eae87bb48174c537e8c14111537892ba8c4d2590d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T01:21:25.770682Z","signature_b64":"klooQOMX9Xf3ouqT1nHimCzwlvu56VEJJZmKjTajFM0v0HdtxSWtthAVM15sxMIh9E97uycPbWiJ4t7bOKGBDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28449253d21bad89063b0ff2a4c192855870aa1b88044856289af6bd37520088","last_reissued_at":"2026-07-21T01:21:25.769869Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T01:21:25.769869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterizing the equality case in Brouwer's inequality for Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xiaodan Chen, Yuhang Cui","submitted_at":"2026-07-19T15:24:59Z","abstract_excerpt":"Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\\binom{k+1}{2}$ for every $k\\in \\{1,2,\\dots,n\\}$, which has been confirmed by Kothari and Tudose (2026) recently. In this note, we characterize the equality case in this inequality. Our main result is that for every $n$-vertex graph $G=(V,E)$ and for every $k\\in \\{1,2,\\dots,n-1\\}$, the equality $\\sum_{i=1}^k\\mu_i(G)=|E(G)|+\\binom{k+1}{2}$ holds if and only if $G$ is a threshold graph with clique number $k+1$, where $\\mu_1(G)\\geq \\mu_2(G)\\geq \\c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17293","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/2607.17293/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":"2607.17293","created_at":"2026-07-21T01:21:25.770271+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.17293v1","created_at":"2026-07-21T01:21:25.770271+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.17293","created_at":"2026-07-21T01:21:25.770271+00:00"},{"alias_kind":"pith_short_12","alias_value":"FBCJEU6SDOWY","created_at":"2026-07-21T01:21:25.770271+00:00"},{"alias_kind":"pith_short_16","alias_value":"FBCJEU6SDOWYSBR3","created_at":"2026-07-21T01:21:25.770271+00:00"},{"alias_kind":"pith_short_8","alias_value":"FBCJEU6S","created_at":"2026-07-21T01:21:25.770271+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV","json":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV.json","graph_json":"https://pith.science/api/pith-number/FBCJEU6SDOWYSBR3B7ZKJQMSQV/graph.json","events_json":"https://pith.science/api/pith-number/FBCJEU6SDOWYSBR3B7ZKJQMSQV/events.json","paper":"https://pith.science/paper/FBCJEU6S"},"agent_actions":{"view_html":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV","download_json":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV.json","view_paper":"https://pith.science/paper/FBCJEU6S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.17293&json=true","fetch_graph":"https://pith.science/api/pith-number/FBCJEU6SDOWYSBR3B7ZKJQMSQV/graph.json","fetch_events":"https://pith.science/api/pith-number/FBCJEU6SDOWYSBR3B7ZKJQMSQV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV/action/storage_attestation","attest_author":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV/action/author_attestation","sign_citation":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV/action/citation_signature","submit_replication":"https://pith.science/pith/FBCJEU6SDOWYSBR3B7ZKJQMSQV/action/replication_record"}},"created_at":"2026-07-21T01:21:25.770271+00:00","updated_at":"2026-07-21T01:21:25.770271+00:00"}