{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:NL4TXACOTPQBNV75QM7MNYHER5","short_pith_number":"pith:NL4TXACO","schema_version":"1.0","canonical_sha256":"6af93b804e9be016d7fd833ec6e0e48f635169d9bd4c05e913b0fef21fa1b69a","source":{"kind":"arxiv","id":"2409.18220","version":1},"attestation_state":"computed","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 } "},"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":"2409.18220","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-09-26T18:58:48Z","cross_cats_sorted":[],"title_canon_sha256":"2cfc78746d99ea2753a45edbcf0f678c1c40825d822b24817aa653ba4b52e061","abstract_canon_sha256":"69c1feb2c54360773c43da37a42f6b89ef1fac3566406b78b9ba3a576d71c6c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:12:32.860380Z","signature_b64":"BNfgW8sGqsFEGykCfAWy36BsdkL+sWGmg9dgWDnW21XS03cVZS0TrQx3pT02YMMkrPvcHFE+Z3TwCMUozR3VDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6af93b804e9be016d7fd833ec6e0e48f635169d9bd4c05e913b0fef21fa1b69a","last_reissued_at":"2026-07-05T09:12:32.859834Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:12:32.859834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2409.18220","created_at":"2026-07-05T09:12:32.859906+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.18220v1","created_at":"2026-07-05T09:12:32.859906+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.18220","created_at":"2026-07-05T09:12:32.859906+00:00"},{"alias_kind":"pith_short_12","alias_value":"NL4TXACOTPQB","created_at":"2026-07-05T09:12:32.859906+00:00"},{"alias_kind":"pith_short_16","alias_value":"NL4TXACOTPQBNV75","created_at":"2026-07-05T09:12:32.859906+00:00"},{"alias_kind":"pith_short_8","alias_value":"NL4TXACO","created_at":"2026-07-05T09:12:32.859906+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":2,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5","json":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5.json","graph_json":"https://pith.science/api/pith-number/NL4TXACOTPQBNV75QM7MNYHER5/graph.json","events_json":"https://pith.science/api/pith-number/NL4TXACOTPQBNV75QM7MNYHER5/events.json","paper":"https://pith.science/paper/NL4TXACO"},"agent_actions":{"view_html":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5","download_json":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5.json","view_paper":"https://pith.science/paper/NL4TXACO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.18220&json=true","fetch_graph":"https://pith.science/api/pith-number/NL4TXACOTPQBNV75QM7MNYHER5/graph.json","fetch_events":"https://pith.science/api/pith-number/NL4TXACOTPQBNV75QM7MNYHER5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5/action/storage_attestation","attest_author":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5/action/author_attestation","sign_citation":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5/action/citation_signature","submit_replication":"https://pith.science/pith/NL4TXACOTPQBNV75QM7MNYHER5/action/replication_record"}},"created_at":"2026-07-05T09:12:32.859906+00:00","updated_at":"2026-07-05T09:12:32.859906+00:00"}