{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:35XSTRXZYTQHCJKAOAI3R33M55","short_pith_number":"pith:35XSTRXZ","schema_version":"1.0","canonical_sha256":"df6f29c6f9c4e07125407011b8ef6cef4c3c3eda73e0157542cf3fd951355faa","source":{"kind":"arxiv","id":"2503.16882","version":2},"attestation_state":"computed","paper":{"title":"Vertex Partitioning and $p$-Energy of Graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bojan Mohar, Hitesh Kumar, Saieed Akbari, Shivaramakrishna Pragada","submitted_at":"2025-03-21T06:33:17Z","abstract_excerpt":"For a Hermitian matrix $A$ of order $n$ with eigenvalues $\\lambda_1(A)\\ge \\cdots\\ge \\lambda_n(A)$, define \\[ \\mathcal{E}_p^+(A)=\\sum_{\\lambda_i > 0} \\lambda_i^p(A), \\quad \\mathcal{E}_p^-(A)=\\sum_{\\lambda_i<0} |\\lambda_i(A)|^p,\\] to be the positive and the negative $p$-energy of $A$, respectively. In this note, first we show that if $A=[A_{ij}]_{i,j=1}^k$, where $A_{ii}$ are square matrices, then \\[ \\mathcal{E}_p^+(A)\\geq \\sum_{i=1}^{k} \\mathcal{E}_p^+(A_{ii}), \\quad \\mathcal{E}_p^-(A)\\geq \\sum_{i=1}^{k} \\mathcal{E}_p^-(A_{ii}),\\] for any real number $p\\geq 1$. We then apply the previous inequa"},"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":"2503.16882","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2025-03-21T06:33:17Z","cross_cats_sorted":[],"title_canon_sha256":"83ca7843707765c4b17e19d179b8d43b786ea6a1201e4ddbd5919d8191ebde5a","abstract_canon_sha256":"bd5cef5bdb039a9a00adb4ffa62b01984fe4c9cc5e3266e52254166efc5b9bf1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:25:09.136036Z","signature_b64":"9bNPX+ZH5pYkxIPRrDV5n8tDjYdmzV9ySJDurrrePCmfklOpUperyF2eqM7ao+T2gg5VaC3vZnX2wjEMRcl5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df6f29c6f9c4e07125407011b8ef6cef4c3c3eda73e0157542cf3fd951355faa","last_reissued_at":"2026-07-05T11:25:09.135529Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:25:09.135529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vertex Partitioning and $p$-Energy of Graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bojan Mohar, Hitesh Kumar, Saieed Akbari, Shivaramakrishna Pragada","submitted_at":"2025-03-21T06:33:17Z","abstract_excerpt":"For a Hermitian matrix $A$ of order $n$ with eigenvalues $\\lambda_1(A)\\ge \\cdots\\ge \\lambda_n(A)$, define \\[ \\mathcal{E}_p^+(A)=\\sum_{\\lambda_i > 0} \\lambda_i^p(A), \\quad \\mathcal{E}_p^-(A)=\\sum_{\\lambda_i<0} |\\lambda_i(A)|^p,\\] to be the positive and the negative $p$-energy of $A$, respectively. In this note, first we show that if $A=[A_{ij}]_{i,j=1}^k$, where $A_{ii}$ are square matrices, then \\[ \\mathcal{E}_p^+(A)\\geq \\sum_{i=1}^{k} \\mathcal{E}_p^+(A_{ii}), \\quad \\mathcal{E}_p^-(A)\\geq \\sum_{i=1}^{k} \\mathcal{E}_p^-(A_{ii}),\\] for any real number $p\\geq 1$. We then apply the previous inequa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.16882","kind":"arxiv","version":2},"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/2503.16882/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":"2503.16882","created_at":"2026-07-05T11:25:09.135602+00:00"},{"alias_kind":"arxiv_version","alias_value":"2503.16882v2","created_at":"2026-07-05T11:25:09.135602+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.16882","created_at":"2026-07-05T11:25:09.135602+00:00"},{"alias_kind":"pith_short_12","alias_value":"35XSTRXZYTQH","created_at":"2026-07-05T11:25:09.135602+00:00"},{"alias_kind":"pith_short_16","alias_value":"35XSTRXZYTQHCJKA","created_at":"2026-07-05T11:25:09.135602+00:00"},{"alias_kind":"pith_short_8","alias_value":"35XSTRXZ","created_at":"2026-07-05T11:25:09.135602+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.07264","citing_title":"Refinement of a conjecture on positive square energy of graphs","ref_index":3,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55","json":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55.json","graph_json":"https://pith.science/api/pith-number/35XSTRXZYTQHCJKAOAI3R33M55/graph.json","events_json":"https://pith.science/api/pith-number/35XSTRXZYTQHCJKAOAI3R33M55/events.json","paper":"https://pith.science/paper/35XSTRXZ"},"agent_actions":{"view_html":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55","download_json":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55.json","view_paper":"https://pith.science/paper/35XSTRXZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2503.16882&json=true","fetch_graph":"https://pith.science/api/pith-number/35XSTRXZYTQHCJKAOAI3R33M55/graph.json","fetch_events":"https://pith.science/api/pith-number/35XSTRXZYTQHCJKAOAI3R33M55/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55/action/timestamp_anchor","attest_storage":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55/action/storage_attestation","attest_author":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55/action/author_attestation","sign_citation":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55/action/citation_signature","submit_replication":"https://pith.science/pith/35XSTRXZYTQHCJKAOAI3R33M55/action/replication_record"}},"created_at":"2026-07-05T11:25:09.135602+00:00","updated_at":"2026-07-05T11:25:09.135602+00:00"}