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In this work, we provide several new tools for studying square energy encompassing semi-definite optimization, graph operations, and surplus. Using our tools, we prove the following results on the extremal values of $s^{\\pm}(G)$ with a given number of vertices and edges.\n  1. We have $\\min(s^+(G), s^-(G)) \\geq n - \\gamma \\geq \\frac{n}{2}$, where"},"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":"2409.15504","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-09-23T19:48:53Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"b06123691d66950310c17c8db4efd2885635a3d995c85948e91f1cf6e3383bb7","abstract_canon_sha256":"ea6921b72b7927c104ae2b0a4094d1270a4798f90aa9805091c15a3f8a771a0f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:11:48.563109Z","signature_b64":"XzwNkag3yBgLXc0WiMAJVYgBF5DI7PazyUp0HzzqTuLydKS5CQ4dGASQ8GjJNMJ1JBRgNR9/pQcArv0U0+HeAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ed47355d1127271b8cc3eac6167a3888cd7259ce9e20bc13ab667e48b171c1d","last_reissued_at":"2026-07-05T09:11:48.562680Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:11:48.562680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal values for the square energies of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.CO","authors_text":"Shengtong Zhang","submitted_at":"2024-09-23T19:48:53Z","abstract_excerpt":"Let $G$ be a graph with $n$ non-isolated vertices and $m$ edges. 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We have $\\min(s^+(G), s^-(G)) \\geq n - \\gamma \\geq \\frac{n}{2}$, where"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.15504","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/2409.15504/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":"2409.15504","created_at":"2026-07-05T09:11:48.562742+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.15504v2","created_at":"2026-07-05T09:11:48.562742+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.15504","created_at":"2026-07-05T09:11:48.562742+00:00"},{"alias_kind":"pith_short_12","alias_value":"D3KHGVORCJZH","created_at":"2026-07-05T09:11:48.562742+00:00"},{"alias_kind":"pith_short_16","alias_value":"D3KHGVORCJZHDOGM","created_at":"2026-07-05T09:11:48.562742+00:00"},{"alias_kind":"pith_short_8","alias_value":"D3KHGVOR","created_at":"2026-07-05T09:11:48.562742+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":6,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.22730","citing_title":"Path-Minimality of $p$-Energy for Connected Graphs","ref_index":35,"is_internal_anchor":false},{"citing_arxiv_id":"2606.30996","citing_title":"Path-Minimality for Positive $p$-Energies, Laplacian-Type Spectra, and Line Graphs","ref_index":20,"is_internal_anchor":false},{"citing_arxiv_id":"2605.22730","citing_title":"Path-Minimality of $p$-Energy for Connected Graphs","ref_index":35,"is_internal_anchor":false},{"citing_arxiv_id":"2605.24668","citing_title":"A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs","ref_index":19,"is_internal_anchor":false},{"citing_arxiv_id":"2605.22730","citing_title":"Path-Minimality of $p$-Energy for Connected Graphs","ref_index":35,"is_internal_anchor":false},{"citing_arxiv_id":"2604.15656","citing_title":"Positive and negative 3-energies of graphs","ref_index":12,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC","json":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC.json","graph_json":"https://pith.science/api/pith-number/D3KHGVORCJZHDOGMH2WGCZ5DRC/graph.json","events_json":"https://pith.science/api/pith-number/D3KHGVORCJZHDOGMH2WGCZ5DRC/events.json","paper":"https://pith.science/paper/D3KHGVOR"},"agent_actions":{"view_html":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC","download_json":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC.json","view_paper":"https://pith.science/paper/D3KHGVOR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.15504&json=true","fetch_graph":"https://pith.science/api/pith-number/D3KHGVORCJZHDOGMH2WGCZ5DRC/graph.json","fetch_events":"https://pith.science/api/pith-number/D3KHGVORCJZHDOGMH2WGCZ5DRC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC/action/storage_attestation","attest_author":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC/action/author_attestation","sign_citation":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC/action/citation_signature","submit_replication":"https://pith.science/pith/D3KHGVORCJZHDOGMH2WGCZ5DRC/action/replication_record"}},"created_at":"2026-07-05T09:11:48.562742+00:00","updated_at":"2026-07-05T09:11:48.562742+00:00"}