{"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"}