{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KFR3WLOWAUNC2FZMMYO53DILS7","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":"f15fdb7eca235aca335a12e1bfcbfa784025729f2bf57e3813d94f782f7f05c6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-19T02:28:42Z","title_canon_sha256":"f2d45959f2700e33d0ea8fa61cc9ce538586d135afc2c3fb64dc083c6a7bd474"},"schema_version":"1.0","source":{"id":"1309.4837","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4837","created_at":"2026-05-18T02:55:49Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4837v2","created_at":"2026-05-18T02:55:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4837","created_at":"2026-05-18T02:55:49Z"},{"alias_kind":"pith_short_12","alias_value":"KFR3WLOWAUNC","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KFR3WLOWAUNC2FZM","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KFR3WLOW","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:d6f5c395da18bffad515dec0a93fd8169499f7cc8255b89951b2c64b5bd9b47d","target":"graph","created_at":"2026-05-18T02:55:49Z","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":"Let $G$ be a $k$-degenerate graph of order $n.$ It is well-known that $G\\ $has no more edges than $S_{n,k},$ the join of a complete graph of order $k$ and an independent set of order $n-k.$ In this note it is shown that $S_{n,k}$ is extremal for some spectral parameters of $G$ as well. More precisely, letting $\\mu\\left( H\\right) $ and $q\\left( H\\right) $ denote the largest eigenvalues of the adjacency matrix and the signless Laplacian of a graph $H,$ the inequalities \\[ \\mu\\left( G\\right) <\\mu\\left( S_{n,k}\\right) \\text{ and }q\\left( G\\right) <q\\left( S_{n,k}\\right) \\] hold, unless $G=S_{n,k}$","authors_text":"V. Nikiforov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-19T02:28:42Z","title":"Maxima of the Q-index: degenerate graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4837","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:ca1b8dc7ac5b9f51ff7ad490b55c65078e0e72ce5bef939e5f5f795bbc96307c","target":"record","created_at":"2026-05-18T02:55:49Z","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":"f15fdb7eca235aca335a12e1bfcbfa784025729f2bf57e3813d94f782f7f05c6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-19T02:28:42Z","title_canon_sha256":"f2d45959f2700e33d0ea8fa61cc9ce538586d135afc2c3fb64dc083c6a7bd474"},"schema_version":"1.0","source":{"id":"1309.4837","kind":"arxiv","version":2}},"canonical_sha256":"5163bb2dd6051a2d172c661ddd8d0b97c9b3e4ae0734e45e2d748f19472f1a5c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5163bb2dd6051a2d172c661ddd8d0b97c9b3e4ae0734e45e2d748f19472f1a5c","first_computed_at":"2026-05-18T02:55:49.804337Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:49.804337Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zw3be1Tq7f+4nW+PiufTGO0lK9+UzmsUcLub4T+Cd/9C+h9+SJK6M4DFowaqV8vdtnS+tq35T4HdtXelo7RcBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:49.804831Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.4837","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ca1b8dc7ac5b9f51ff7ad490b55c65078e0e72ce5bef939e5f5f795bbc96307c","sha256:d6f5c395da18bffad515dec0a93fd8169499f7cc8255b89951b2c64b5bd9b47d"],"state_sha256":"9c6818bc72be21b4ef387d12e6d0f5c25c093634cef1b53db2420c1fd63d487f"}