{"paper":{"title":"A Linear Lower Bound for the Square Energy of Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bojan Mohar, Hitesh Kumar, Saieed Akbari, Shivaramakrishna Pragada","submitted_at":"2024-09-26T18:58:48Z","abstract_excerpt":"Let $G$ be a graph of order $n$ with eigenvalues $\\lambda_1 \\geq \\cdots \\geq\\lambda_n$. Let \\[s^+(G)=\\sum_{\\lambda_i>0} \\lambda_i^2, \\qquad s^-(G)=\\sum_{\\lambda_i<0} \\lambda_i^2.\\] The smaller value, $s(G)=\\min\\{s^+(G), s^-(G)\\}$ is called the \\emph{square energy} of $G$. In 2016, Elphick, Farber, Goldberg and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\\geq n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1, \\ldots, H_k$ be disjoint vertex-induced subgraphs of $G$. In this note, we prove that \\[s^+(G)\\geq\\sum_{i=1}^{k} s^+(H_i) \\quad \\text{ and } "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.18220","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/2409.18220/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"}