{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:SV62SFOZBQBGX6XV4H35W3JKRM","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":"de23a7850276362de792608416a536179c6be776acf945b8ae7226bafd5fa2a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-16T15:20:10Z","title_canon_sha256":"05472f6aa8a075108c4e460fea6770dce553a811294745037281a57058ec178a"},"schema_version":"1.0","source":{"id":"1611.05311","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.05311","created_at":"2026-05-18T00:47:56Z"},{"alias_kind":"arxiv_version","alias_value":"1611.05311v2","created_at":"2026-05-18T00:47:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05311","created_at":"2026-05-18T00:47:56Z"},{"alias_kind":"pith_short_12","alias_value":"SV62SFOZBQBG","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"SV62SFOZBQBGX6XV","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"SV62SFOZ","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:0aa431c3fb513bfa64524fdadf031f48cfdbfde80bc916f42d070d02f146f729","target":"graph","created_at":"2026-05-18T00:47:56Z","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":"In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly four distinct normalized Laplacian eigenvalues, and find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.","authors_text":"Qiongxiang Huang, Xueyi Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-16T15:20:10Z","title":"On graphs with three or four distinct normalized Laplacian eigenvalues"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05311","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:f94f0b79c1f3bc07fafed3430d3f65cfc7e35067ee65b49238422df1973faab2","target":"record","created_at":"2026-05-18T00:47:56Z","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":"de23a7850276362de792608416a536179c6be776acf945b8ae7226bafd5fa2a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-16T15:20:10Z","title_canon_sha256":"05472f6aa8a075108c4e460fea6770dce553a811294745037281a57058ec178a"},"schema_version":"1.0","source":{"id":"1611.05311","kind":"arxiv","version":2}},"canonical_sha256":"957da915d90c026bfaf5e1f7db6d2a8b015c11b50dcc4cfcb391a0699e95ecc4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"957da915d90c026bfaf5e1f7db6d2a8b015c11b50dcc4cfcb391a0699e95ecc4","first_computed_at":"2026-05-18T00:47:56.655875Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:47:56.655875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R2UIrED0f3XLDjGqqkCuNJlLM5LLFIfCGmNkGsDy9u68zhDucY5gFiYzr1HngZBM/UymWbo4Bv0LPXGO0Vp9CA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:47:56.656306Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.05311","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f94f0b79c1f3bc07fafed3430d3f65cfc7e35067ee65b49238422df1973faab2","sha256:0aa431c3fb513bfa64524fdadf031f48cfdbfde80bc916f42d070d02f146f729"],"state_sha256":"3c767b594ac06ca0cb7d53c7b84bb3ce2432b8d4599fa908926d79c69d079add"}