{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:32A5STYGBMJWI7W7NENYJH4OQ3","short_pith_number":"pith:32A5STYG","schema_version":"1.0","canonical_sha256":"de81d94f060b13647edf691b849f8e86febc1ff463ba6346573ce866e8558854","source":{"kind":"arxiv","id":"2605.24668","version":1},"attestation_state":"computed","paper":{"title":"A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Jing Zeng","submitted_at":"2026-05-23T17:10:37Z","abstract_excerpt":"Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues, respectively. Akbari, Kumar, Mohar, Pragada, and Zhang conjectured that, when $k$ is odd, the value of $k$ modulo $4$ determines which of $s^+(G)$ and $s^-(G)$ is greater than $n$. More precisely, if $k\\equiv 3\\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\\equiv 1\\pmod 4$, then $s^+(G)<n<s^-(G)$. We confirm this conjecture."},"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":"2605.24668","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-23T17:10:37Z","cross_cats_sorted":[],"title_canon_sha256":"bb070e5d41d039e8e8c37487435c7c6200f2ed71393a89f14b61b4c708db04fa","abstract_canon_sha256":"2bae8710eca0fbcac5b9f4a6346d2c0ffa9eb59e88581e1d616991297a8c0f50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:03:52.249173Z","signature_b64":"MjQWE0ROOzaaTzYH5k+1D4LrRTUi4KVBPPGlZukWmXmQYujJ4U/hKC5MEgA0ocIY5WV2VJDpEB1ulv/6AW/GAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de81d94f060b13647edf691b849f8e86febc1ff463ba6346573ce866e8558854","last_reissued_at":"2026-05-26T01:03:52.248151Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:03:52.248151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Jing Zeng","submitted_at":"2026-05-23T17:10:37Z","abstract_excerpt":"Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues, respectively. Akbari, Kumar, Mohar, Pragada, and Zhang conjectured that, when $k$ is odd, the value of $k$ modulo $4$ determines which of $s^+(G)$ and $s^-(G)$ is greater than $n$. More precisely, if $k\\equiv 3\\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\\equiv 1\\pmod 4$, then $s^+(G)<n<s^-(G)$. We confirm this conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24668","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/2605.24668/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":"2605.24668","created_at":"2026-05-26T01:03:52.248345+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.24668v1","created_at":"2026-05-26T01:03:52.248345+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.24668","created_at":"2026-05-26T01:03:52.248345+00:00"},{"alias_kind":"pith_short_12","alias_value":"32A5STYGBMJW","created_at":"2026-05-26T01:03:52.248345+00:00"},{"alias_kind":"pith_short_16","alias_value":"32A5STYGBMJWI7W7","created_at":"2026-05-26T01:03:52.248345+00:00"},{"alias_kind":"pith_short_8","alias_value":"32A5STYG","created_at":"2026-05-26T01:03:52.248345+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3","json":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3.json","graph_json":"https://pith.science/api/pith-number/32A5STYGBMJWI7W7NENYJH4OQ3/graph.json","events_json":"https://pith.science/api/pith-number/32A5STYGBMJWI7W7NENYJH4OQ3/events.json","paper":"https://pith.science/paper/32A5STYG"},"agent_actions":{"view_html":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3","download_json":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3.json","view_paper":"https://pith.science/paper/32A5STYG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.24668&json=true","fetch_graph":"https://pith.science/api/pith-number/32A5STYGBMJWI7W7NENYJH4OQ3/graph.json","fetch_events":"https://pith.science/api/pith-number/32A5STYGBMJWI7W7NENYJH4OQ3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3/action/storage_attestation","attest_author":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3/action/author_attestation","sign_citation":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3/action/citation_signature","submit_replication":"https://pith.science/pith/32A5STYGBMJWI7W7NENYJH4OQ3/action/replication_record"}},"created_at":"2026-05-26T01:03:52.248345+00:00","updated_at":"2026-05-26T01:03:52.248345+00:00"}