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Define \\[s^+(G)=\\sum_{\\lambda_i >0} \\lambda_i^2(G), \\quad s^-(G)=\\sum_{\\lambda_i<0} \\lambda_i^2(G).\\] It was conjectured by Elphick, Farber, Goldberg and Wocjan that for every connected graph $G$ of order $n$, $s^+(G) \\ge n-1.$ We verify this conjecture for graphs with domination number at most 2. We then strengthen the conjecture as follows: if $G$ is a connected graph of order $n$ and size $m \\geq n+1$, then $s^+(G) \\geq n$. We prove this conjecture for claw-free graphs and graphs with diamet"},"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":"2506.07264","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-06-08T19:58:29Z","cross_cats_sorted":[],"title_canon_sha256":"0c41d42fbde519b07c72642a5b54efb60598374f48745d2507e6f98d0910db20","abstract_canon_sha256":"072372b7216702c005f68a9043f75a8d04b8ef99cd53c74453ca8e8279dfde45"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:18:24.105602Z","signature_b64":"3ULjYwVaC6oTEEmPn9ES4y2H9IKdg31voy9FrRyczxFr7YeH0j9huFPVFLofAGx4jRUtPmCcU1UhcmjIF77UDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e8b5fe79c62856fb5401e3a72d8a4f572bc6a0ea3556a917c8080e6122bcdb0c","last_reissued_at":"2026-07-05T11:18:24.105087Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:18:24.105087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Refinement of a conjecture on positive 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, Shengtong Zhang, Shivaramakrishna Pragada","submitted_at":"2025-06-08T19:58:29Z","abstract_excerpt":"Let $G$ be a simple graph of order $n$ with eigenvalues $\\lambda_1(G)\\geq \\cdots \\geq \\lambda_n(G)$. Define \\[s^+(G)=\\sum_{\\lambda_i >0} \\lambda_i^2(G), \\quad s^-(G)=\\sum_{\\lambda_i<0} \\lambda_i^2(G).\\] It was conjectured by Elphick, Farber, Goldberg and Wocjan that for every connected graph $G$ of order $n$, $s^+(G) \\ge n-1.$ We verify this conjecture for graphs with domination number at most 2. We then strengthen the conjecture as follows: if $G$ is a connected graph of order $n$ and size $m \\geq n+1$, then $s^+(G) \\geq n$. 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