{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:3NSKYB3H5B76MGLXUBJXVJD632","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":"2d69ea84d2ad6180bbd72edf47f7a105bd158f91d5c68c71065089887b37667f","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-18T11:12:43Z","title_canon_sha256":"727b74f54a98464779bdd4c68c28a4c50711fa3c659ca7f0507a6f4f1e66e994"},"schema_version":"1.0","source":{"id":"1210.5091","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5091","created_at":"2026-05-18T02:57:00Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5091v2","created_at":"2026-05-18T02:57:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5091","created_at":"2026-05-18T02:57:00Z"},{"alias_kind":"pith_short_12","alias_value":"3NSKYB3H5B76","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"3NSKYB3H5B76MGLX","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"3NSKYB3H","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:c54fbf5af38f7f75b2756aa5842982136145711cdd1276ed764041c12cd9937c","target":"graph","created_at":"2026-05-18T02:57:00Z","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":"Here we have investigated a few properties of the eigenvalues of normalized (geometric) graph Laplacian in different graphs. Preservation of eigenvalue 1 from a particular subgraph to the entire graph, the spectrum of the graph constructed with triangles share a common vertex have been addressed. Further using the number and degrees of common neighbors between vertices some new upper bounds for the largest eigenvalue have been introduced.","authors_text":"Anirban Banerjee","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-18T11:12:43Z","title":"A few properties of the eigenvalues of normalized graph Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5091","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":"sha256:5ae43f293444213b3a55aa926f2f1cdb1ef379ffd37e5d9dd0f4bee22a141896","target":"record","created_at":"2026-05-18T02:57:00Z","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":"2d69ea84d2ad6180bbd72edf47f7a105bd158f91d5c68c71065089887b37667f","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-18T11:12:43Z","title_canon_sha256":"727b74f54a98464779bdd4c68c28a4c50711fa3c659ca7f0507a6f4f1e66e994"},"schema_version":"1.0","source":{"id":"1210.5091","kind":"arxiv","version":2}},"canonical_sha256":"db64ac0767e87fe61977a0537aa47edea7693e36558fadeac5e55e1bc5c7e882","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db64ac0767e87fe61977a0537aa47edea7693e36558fadeac5e55e1bc5c7e882","first_computed_at":"2026-05-18T02:57:00.128253Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:00.128253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"APNQ+v7Y3cFTFL8Bs5HVE2ngqiO5w+1fKsnssIPeZ9+PaLTFr2Q84UvX4UT7R/4NxuGFCNynnPbCUwBCk8B5BA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:00.128745Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.5091","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ae43f293444213b3a55aa926f2f1cdb1ef379ffd37e5d9dd0f4bee22a141896","sha256:c54fbf5af38f7f75b2756aa5842982136145711cdd1276ed764041c12cd9937c"],"state_sha256":"27e9f79d19c839d8df6613aa57d130289337e6121e194c9de1ad5e07ec7142dc"}