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The eigenvalues of $\\DL(G)$ are ordered as $\\partial^{L}_1(G)\\ge \\partial^{L}_2(G)\\ge \\cdots \\ge \\partial^{L}_n(G)=0$. Building on the chromatic lower bound $\\partial^{L}_1(G)\\ge n+\\ceil{n/\\chi}$ and subsequent developments, we prove a \\emph{color-class majorization principle}: if $(\\ell_1,\\dots,\\ell_\\chi)$ are the color-cla"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2604.10785","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-04-12T19:22:19Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"c54740941a9a3925309c9a7a3db2654a89e748454c4bc60479663edceb0415a5","abstract_canon_sha256":"84ab21c81cdd029d0294f1f0d687836c233ea83ed2f9d6a40fd3cc983d50bb8f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:00:37.841476Z","signature_b64":"zuXCJUDo3oJqZRoXk8ehdwsHqd//g/3JtL+MJTd3cwi5XBB9FbwU42BHCjBh8ITTOiQJ/slX1NvvgPK04MxCBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b86925a0bbed8245f9977504db65d074bfd3a9e71817369dc12c156c52cd715d","last_reissued_at":"2026-05-20T00:00:37.840974Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:00:37.840974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal chromatic bounds for distance Laplacian eigenvalues","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Color class sizes from an optimal coloring bound the first several distance Laplacian eigenvalues of a connected graph.","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Bilal Ahmad Rather","submitted_at":"2026-04-12T19:22:19Z","abstract_excerpt":"For a connected simple graph $G$ on $n$ vertices with chromatic number $\\chi$, the distance Laplacian matrix is $\\DL(G)=\\operatorname{diag}(\\Tr_G(v_1),\\dots,\\Tr_G(v_n))-D(G)$, where $D(G)$ is the distance matrix and $\\Tr_G(v)=\\sum_{u\\in V(G)} d_G(u,v)$ is the transmission. The eigenvalues of $\\DL(G)$ are ordered as $\\partial^{L}_1(G)\\ge \\partial^{L}_2(G)\\ge \\cdots \\ge \\partial^{L}_n(G)=0$. Building on the chromatic lower bound $\\partial^{L}_1(G)\\ge n+\\ceil{n/\\chi}$ and subsequent developments, we prove a \\emph{color-class majorization principle}: if $(\\ell_1,\\dots,\\ell_\\chi)$ are the color-cla"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"If (ℓ1,…,ℓχ) are the color-class sizes in an optimal χ-coloring with ℓ1≥⋯≥ℓχ, then the first ℓ1−1 distance Laplacian eigenvalues satisfy ∂^L_i(G)≥n+ℓ1 for 1≤i≤ℓ1−1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The graph is connected and the given partition is an optimal proper χ-coloring whose class sizes are ordered decreasingly; the proof relies on this ordering to apply the majorization step to the distance matrix.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves that for a χ-chromatic graph the first ℓ1−1 distance Laplacian eigenvalues satisfy ∂^L_i(G) ≥ n + ℓ1 where ℓ1 is the largest color-class size, refining distribution and extremal results.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Color class sizes from an optimal coloring bound the first several distance Laplacian eigenvalues of a connected graph.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"379d9768d3d74ec2716fdb94394d0635068cc08d52933e8009fc92de1eab680b"},"source":{"id":"2604.10785","kind":"arxiv","version":2},"verdict":{"id":"1b1309cc-4de1-40e4-b02a-3b4687b67422","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:01:10.438737Z","strongest_claim":"If (ℓ1,…,ℓχ) are the color-class sizes in an optimal χ-coloring with ℓ1≥⋯≥ℓχ, then the first ℓ1−1 distance Laplacian eigenvalues satisfy ∂^L_i(G)≥n+ℓ1 for 1≤i≤ℓ1−1.","one_line_summary":"Proves that for a χ-chromatic graph the first ℓ1−1 distance Laplacian eigenvalues satisfy ∂^L_i(G) ≥ n + ℓ1 where ℓ1 is the largest color-class size, refining distribution and extremal results.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The graph is connected and the given partition is an optimal proper χ-coloring whose class sizes are ordered decreasingly; the proof relies on this ordering to apply the majorization step to the distance matrix.","pith_extraction_headline":"Color class sizes from an optimal coloring bound the first several distance Laplacian eigenvalues of a connected graph."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.10785/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":15,"sample":[{"doi":"","year":2013,"title":"M. 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