{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:BO7CLJS5LR6IMF7UTA3UXIBUYB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d380b49252a2d73bf06bd4668fd0cfb73a40a33f847a81c3ec48a5dd244f2dd5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-10T23:49:56Z","title_canon_sha256":"1b17b5e19d498263924e6dc926a4dab46d029436ec109bf1683d6b1b42bb2ca4"},"schema_version":"1.0","source":{"id":"1804.03752","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.03752","created_at":"2026-05-18T00:08:31Z"},{"alias_kind":"arxiv_version","alias_value":"1804.03752v2","created_at":"2026-05-18T00:08:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.03752","created_at":"2026-05-18T00:08:31Z"},{"alias_kind":"pith_short_12","alias_value":"BO7CLJS5LR6I","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"BO7CLJS5LR6IMF7U","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"BO7CLJS5","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:406b65edf6deba871766f50b5c409cc862390145396cfa240d921a25c7f5e9a8","target":"graph","created_at":"2026-05-18T00:08:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"It is well known that $n/(n - \\mu)$, where $\\mu$ is the spectral radius of a graph with $n$ vertices, is a lower bound for the clique number. We conjecture that $\\mu$ can be replaced in this bound with $\\sqrt{s^+}$, where $s^+$ is the sum of the squares of the positive eigenvalues. We prove this conjecture for various classes of graphs, including triangle-free graphs, and for almost all graphs.","authors_text":"Clive Elphick, Pawel Wocjan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-10T23:49:56Z","title":"Conjectured lower bound for the clique number of a graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03752","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d3fd218bc9af7168dc6e69186eae5555db2426c4b5ace5f0a3b936575dd590de","target":"record","created_at":"2026-05-18T00:08:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d380b49252a2d73bf06bd4668fd0cfb73a40a33f847a81c3ec48a5dd244f2dd5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-10T23:49:56Z","title_canon_sha256":"1b17b5e19d498263924e6dc926a4dab46d029436ec109bf1683d6b1b42bb2ca4"},"schema_version":"1.0","source":{"id":"1804.03752","kind":"arxiv","version":2}},"canonical_sha256":"0bbe25a65d5c7c8617f498374ba034c07c6a1c57be87c3eed4919b6fd6bc4eb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0bbe25a65d5c7c8617f498374ba034c07c6a1c57be87c3eed4919b6fd6bc4eb3","first_computed_at":"2026-05-18T00:08:31.059208Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:31.059208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jRHI/TDL3LDS/RkrfAucvu+HtN8WNeEyybX/E3zIzbRE6lpt67BBmczhtacRE06Ix3xHm2Wfq0vDR0I/8rUSCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:31.059727Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.03752","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d3fd218bc9af7168dc6e69186eae5555db2426c4b5ace5f0a3b936575dd590de","sha256:406b65edf6deba871766f50b5c409cc862390145396cfa240d921a25c7f5e9a8"],"state_sha256":"f540d372de0c44f5cc2f716abc7b49c5f80cadd7cd106f32a88f7cedf9bd0c95"}