{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KDLVKLAQU6DKEIGUGEACKEX5YO","short_pith_number":"pith:KDLVKLAQ","canonical_record":{"source":{"id":"1505.08133","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-05-29T18:16:47Z","cross_cats_sorted":[],"title_canon_sha256":"d19b3dc8e7c8f4ef1b63b49ecaaa5e0f7a24ded981d686e7304ac5c9a8574401","abstract_canon_sha256":"db188a9955e06e5f5c64923fd95c945e06d4403c1325e3b0f2fbcf0870f9373d"},"schema_version":"1.0"},"canonical_sha256":"50d7552c10a786a220d431002512fdc39fc6680ec45ccfe610d2fe7e5902e9e0","source":{"kind":"arxiv","id":"1505.08133","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.08133","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"arxiv_version","alias_value":"1505.08133v2","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.08133","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"pith_short_12","alias_value":"KDLVKLAQU6DK","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KDLVKLAQU6DKEIGU","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KDLVKLAQ","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KDLVKLAQU6DKEIGUGEACKEX5YO","target":"record","payload":{"canonical_record":{"source":{"id":"1505.08133","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-05-29T18:16:47Z","cross_cats_sorted":[],"title_canon_sha256":"d19b3dc8e7c8f4ef1b63b49ecaaa5e0f7a24ded981d686e7304ac5c9a8574401","abstract_canon_sha256":"db188a9955e06e5f5c64923fd95c945e06d4403c1325e3b0f2fbcf0870f9373d"},"schema_version":"1.0"},"canonical_sha256":"50d7552c10a786a220d431002512fdc39fc6680ec45ccfe610d2fe7e5902e9e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:55:52.318821Z","signature_b64":"7dfIj0wX15hCjo2Z6Ydg8g3CTHYWqec3wDftDsu5J8D7TTsAQU10oVCvJfl6yOIgbk2S4zndFgcE8p1jMxueCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50d7552c10a786a220d431002512fdc39fc6680ec45ccfe610d2fe7e5902e9e0","last_reissued_at":"2026-05-18T01:55:52.318232Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:55:52.318232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.08133","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:55:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ub81ZCicVbosdF54Q3Gg1BfJIwRraf/l+Tc6XSeGkY1GzQT38H/XTntsGwThOdoRt6Z8p5AgudmiuF5z7ppaDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:46:08.079138Z"},"content_sha256":"4b758ec5994212acf6983f0c3a73e11acf5db044fc2c16941754116763afaddb","schema_version":"1.0","event_id":"sha256:4b758ec5994212acf6983f0c3a73e11acf5db044fc2c16941754116763afaddb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KDLVKLAQU6DKEIGUGEACKEX5YO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectrum of Laplacians for Graphs with Self-Loops","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Behcet Acikmese","submitted_at":"2015-05-29T18:16:47Z","abstract_excerpt":"This note introduces a result on the location of eigenvalues, i.e., the spectrum, of the Laplacian for a family of undirected graphs with self-loops. We extend on the known results for the spectrum of undirected graphs without self-loops or multiple edges. For this purpose, we introduce a new concept of pseudo-connected graphs and apply a lifting of the graph with self-loops to a graph without self-loops, which is then used to quantify the spectrum of the Laplacian for the graph with self-loops."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.08133","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:55:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6lwogRrtzxZXYmArF36U/MFgpeGy7exusk8cdzYPV7eP7fTivWe+mxL4dt9Bi7UX+9aXA8rh7mHvSbTodtyVDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:46:08.079490Z"},"content_sha256":"f6405b4729848674ca4435badb8ddf2152e4b990541faa2fcac330cd7838fb2e","schema_version":"1.0","event_id":"sha256:f6405b4729848674ca4435badb8ddf2152e4b990541faa2fcac330cd7838fb2e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KDLVKLAQU6DKEIGUGEACKEX5YO/bundle.json","state_url":"https://pith.science/pith/KDLVKLAQU6DKEIGUGEACKEX5YO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KDLVKLAQU6DKEIGUGEACKEX5YO/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T07:46:08Z","links":{"resolver":"https://pith.science/pith/KDLVKLAQU6DKEIGUGEACKEX5YO","bundle":"https://pith.science/pith/KDLVKLAQU6DKEIGUGEACKEX5YO/bundle.json","state":"https://pith.science/pith/KDLVKLAQU6DKEIGUGEACKEX5YO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KDLVKLAQU6DKEIGUGEACKEX5YO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KDLVKLAQU6DKEIGUGEACKEX5YO","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":"db188a9955e06e5f5c64923fd95c945e06d4403c1325e3b0f2fbcf0870f9373d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-05-29T18:16:47Z","title_canon_sha256":"d19b3dc8e7c8f4ef1b63b49ecaaa5e0f7a24ded981d686e7304ac5c9a8574401"},"schema_version":"1.0","source":{"id":"1505.08133","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.08133","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"arxiv_version","alias_value":"1505.08133v2","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.08133","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"pith_short_12","alias_value":"KDLVKLAQU6DK","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KDLVKLAQU6DKEIGU","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KDLVKLAQ","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:f6405b4729848674ca4435badb8ddf2152e4b990541faa2fcac330cd7838fb2e","target":"graph","created_at":"2026-05-18T01:55:52Z","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":"This note introduces a result on the location of eigenvalues, i.e., the spectrum, of the Laplacian for a family of undirected graphs with self-loops. We extend on the known results for the spectrum of undirected graphs without self-loops or multiple edges. For this purpose, we introduce a new concept of pseudo-connected graphs and apply a lifting of the graph with self-loops to a graph without self-loops, which is then used to quantify the spectrum of the Laplacian for the graph with self-loops.","authors_text":"Behcet Acikmese","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-05-29T18:16:47Z","title":"Spectrum of Laplacians for Graphs with Self-Loops"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.08133","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":"sha256:4b758ec5994212acf6983f0c3a73e11acf5db044fc2c16941754116763afaddb","target":"record","created_at":"2026-05-18T01:55:52Z","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":"db188a9955e06e5f5c64923fd95c945e06d4403c1325e3b0f2fbcf0870f9373d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-05-29T18:16:47Z","title_canon_sha256":"d19b3dc8e7c8f4ef1b63b49ecaaa5e0f7a24ded981d686e7304ac5c9a8574401"},"schema_version":"1.0","source":{"id":"1505.08133","kind":"arxiv","version":2}},"canonical_sha256":"50d7552c10a786a220d431002512fdc39fc6680ec45ccfe610d2fe7e5902e9e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50d7552c10a786a220d431002512fdc39fc6680ec45ccfe610d2fe7e5902e9e0","first_computed_at":"2026-05-18T01:55:52.318232Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:55:52.318232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7dfIj0wX15hCjo2Z6Ydg8g3CTHYWqec3wDftDsu5J8D7TTsAQU10oVCvJfl6yOIgbk2S4zndFgcE8p1jMxueCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:55:52.318821Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.08133","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b758ec5994212acf6983f0c3a73e11acf5db044fc2c16941754116763afaddb","sha256:f6405b4729848674ca4435badb8ddf2152e4b990541faa2fcac330cd7838fb2e"],"state_sha256":"f4462c0a45b1377d7fb9ef3d473d15c428680cec0a749fec1d2d70d4c31482e5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cZgAo2HlEjX5LBn/D0ld7+pEiy/HX4WiBhmrw9g2waOS49llZ03QMLfzRUFQgHsRHSrXPxnlvdrQ2QpZip33Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T07:46:08.081295Z","bundle_sha256":"a423b585dad3cf487550b45a07cda518b85ac9e35e58edb87da5f50c1ae4262e"}}