{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:6KN5KHG2FZNGKKMTGRMDJ2LSR2","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":"a7b4a62615916750f9c9c135cf2f4523fb2876345c6c69056e62d114828900b5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-11-18T16:34:51Z","title_canon_sha256":"f8ade57848e386b771939e4c2e7496b365a0ba583ca7b635a615bd33a47feccc"},"schema_version":"1.0","source":{"id":"2011.09385","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.09385","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"arxiv_version","alias_value":"2011.09385v1","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.09385","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_12","alias_value":"6KN5KHG2FZNG","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_16","alias_value":"6KN5KHG2FZNGKKMT","created_at":"2026-07-05T01:52:43Z"},{"alias_kind":"pith_short_8","alias_value":"6KN5KHG2","created_at":"2026-07-05T01:52:43Z"}],"graph_snapshots":[{"event_id":"sha256:6b681a980a865b4274d2c6ce417af88bc9de3c04a04c1b8374080157061bb310","target":"graph","created_at":"2026-07-05T01:52:43Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2011.09385/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix and use this to upper-bound the spectral radius of the non-backtracking matrix and to give a lower bound on the spectrum. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number","authors_text":"Cory Glover, Mark Kempton","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-11-18T16:34:51Z","title":"Spectral properties of the non-backtracking matrix of a graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.09385","kind":"arxiv","version":1},"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":"sha256:7764a0ecf80ed5811c809fde4bc48ca4c2e3943e0844ea46e7b6d051645dc1b5","target":"record","created_at":"2026-07-05T01:52:43Z","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":"a7b4a62615916750f9c9c135cf2f4523fb2876345c6c69056e62d114828900b5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-11-18T16:34:51Z","title_canon_sha256":"f8ade57848e386b771939e4c2e7496b365a0ba583ca7b635a615bd33a47feccc"},"schema_version":"1.0","source":{"id":"2011.09385","kind":"arxiv","version":1}},"canonical_sha256":"f29bd51cda2e5a652993345834e9728eb2d42978a165e0c6d3931123fff31332","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f29bd51cda2e5a652993345834e9728eb2d42978a165e0c6d3931123fff31332","first_computed_at":"2026-07-05T01:52:43.534237Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:52:43.534237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"51Fsjsnyg9x7H+7rBo/wNaJ09NwvHbxvX1xdLziQBubRDou8aZ0e9XDOkkvxB/aaeQIwjR6zD71w592GLOgIDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:52:43.534606Z","signed_message":"canonical_sha256_bytes"},"source_id":"2011.09385","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7764a0ecf80ed5811c809fde4bc48ca4c2e3943e0844ea46e7b6d051645dc1b5","sha256:6b681a980a865b4274d2c6ce417af88bc9de3c04a04c1b8374080157061bb310"],"state_sha256":"dd4fa0e892f077d8cfc1d3cae6f750cc1c9028e42c7c02ea487906a017b7be6a"}