{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:2HI6ZJEXQ7RTLAY4F2UWUPUKY4","short_pith_number":"pith:2HI6ZJEX","canonical_record":{"source":{"id":"1801.00111","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-30T10:38:42Z","cross_cats_sorted":[],"title_canon_sha256":"3109a070fd737188e941124463dd7bbb5842fc0371ce2c5e563780af2a063916","abstract_canon_sha256":"e0036db4ef7196dd338be4cdf6503f0b051ed1c78d1531771247ec44ec64ccee"},"schema_version":"1.0"},"canonical_sha256":"d1d1eca49787e335831c2ea96a3e8ac7306c947f3114d530fb36cf6589928a47","source":{"kind":"arxiv","id":"1801.00111","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00111","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00111v1","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00111","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"2HI6ZJEXQ7RT","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"2HI6ZJEXQ7RTLAY4","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"2HI6ZJEX","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:2HI6ZJEXQ7RTLAY4F2UWUPUKY4","target":"record","payload":{"canonical_record":{"source":{"id":"1801.00111","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-30T10:38:42Z","cross_cats_sorted":[],"title_canon_sha256":"3109a070fd737188e941124463dd7bbb5842fc0371ce2c5e563780af2a063916","abstract_canon_sha256":"e0036db4ef7196dd338be4cdf6503f0b051ed1c78d1531771247ec44ec64ccee"},"schema_version":"1.0"},"canonical_sha256":"d1d1eca49787e335831c2ea96a3e8ac7306c947f3114d530fb36cf6589928a47","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:59.391485Z","signature_b64":"vsz9JpDgjWoj5wo9QLa/A23v+Q+INySWqeghTMvPZIX7snGDOaAE9d6IM1tSfeklv+Ab+jku5/uyosu/VauDAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1d1eca49787e335831c2ea96a3e8ac7306c947f3114d530fb36cf6589928a47","last_reissued_at":"2026-05-18T00:26:59.390753Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:59.390753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.00111","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mS1qSmR7sFDmWxbILTpvoXA06rxoZAcAfudGfR6KQEwUyR4xwgNLNWGOgr7KaXADOCsvZt0NqzC9QtL/7F9kAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:01:21.935550Z"},"content_sha256":"21aee3ad05c6e3d792d89533e362544cfc5b4ea57cca333bb1de214e3b783c97","schema_version":"1.0","event_id":"sha256:21aee3ad05c6e3d792d89533e362544cfc5b4ea57cca333bb1de214e3b783c97"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:2HI6ZJEXQ7RTLAY4F2UWUPUKY4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inverting non-invertible trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jozef \\v{S}ir\\'a\\v{n}, So\\v{n}a Pavl\\'ikov\\'a","submitted_at":"2017-12-30T10:38:42Z","abstract_excerpt":"If a graph has a non-singular adjacency matrix, then one may use the inverse matrix to define a (labeled) graph that may be considered to be the inverse graph to the original one. It has been known that an adjacency matrix of a tree is non-singular if and only if the tree has a unique perfect matching; in this case the determinant of the matrix turns out to be $\\pm 1$ and the inverse of the tree was shown to be `switching-equivalent' to a simple graph [C. Godsil, Inverses of Trees, Combinatorica 5 (1985), 33--39]. Using generalized inverses of symmetric matrices (that coincide with Moore-Penro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00111","kind":"arxiv","version":1},"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":"2026-05-18T00:26:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jc/yc12H8+YUzWbw3YyzCk2IJNQKzr4c4ryHZoW22ZXcR0nSjSyuCUpwWbmmG4iilz6LEb5G4+Nhhg2l47h4Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:01:21.935898Z"},"content_sha256":"727caa0bab2b20f1ac5b65920f4fa04e0c659f7ce277c7c544cb256ac3fa9be9","schema_version":"1.0","event_id":"sha256:727caa0bab2b20f1ac5b65920f4fa04e0c659f7ce277c7c544cb256ac3fa9be9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2HI6ZJEXQ7RTLAY4F2UWUPUKY4/bundle.json","state_url":"https://pith.science/pith/2HI6ZJEXQ7RTLAY4F2UWUPUKY4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2HI6ZJEXQ7RTLAY4F2UWUPUKY4/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":"2026-05-30T09:01:21Z","links":{"resolver":"https://pith.science/pith/2HI6ZJEXQ7RTLAY4F2UWUPUKY4","bundle":"https://pith.science/pith/2HI6ZJEXQ7RTLAY4F2UWUPUKY4/bundle.json","state":"https://pith.science/pith/2HI6ZJEXQ7RTLAY4F2UWUPUKY4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2HI6ZJEXQ7RTLAY4F2UWUPUKY4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:2HI6ZJEXQ7RTLAY4F2UWUPUKY4","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":"e0036db4ef7196dd338be4cdf6503f0b051ed1c78d1531771247ec44ec64ccee","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-30T10:38:42Z","title_canon_sha256":"3109a070fd737188e941124463dd7bbb5842fc0371ce2c5e563780af2a063916"},"schema_version":"1.0","source":{"id":"1801.00111","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00111","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00111v1","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00111","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"2HI6ZJEXQ7RT","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"2HI6ZJEXQ7RTLAY4","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"2HI6ZJEX","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:727caa0bab2b20f1ac5b65920f4fa04e0c659f7ce277c7c544cb256ac3fa9be9","target":"graph","created_at":"2026-05-18T00:26:59Z","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":"If a graph has a non-singular adjacency matrix, then one may use the inverse matrix to define a (labeled) graph that may be considered to be the inverse graph to the original one. It has been known that an adjacency matrix of a tree is non-singular if and only if the tree has a unique perfect matching; in this case the determinant of the matrix turns out to be $\\pm 1$ and the inverse of the tree was shown to be `switching-equivalent' to a simple graph [C. Godsil, Inverses of Trees, Combinatorica 5 (1985), 33--39]. Using generalized inverses of symmetric matrices (that coincide with Moore-Penro","authors_text":"Jozef \\v{S}ir\\'a\\v{n}, So\\v{n}a Pavl\\'ikov\\'a","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-30T10:38:42Z","title":"Inverting non-invertible trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00111","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":"sha256:21aee3ad05c6e3d792d89533e362544cfc5b4ea57cca333bb1de214e3b783c97","target":"record","created_at":"2026-05-18T00:26:59Z","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":"e0036db4ef7196dd338be4cdf6503f0b051ed1c78d1531771247ec44ec64ccee","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-30T10:38:42Z","title_canon_sha256":"3109a070fd737188e941124463dd7bbb5842fc0371ce2c5e563780af2a063916"},"schema_version":"1.0","source":{"id":"1801.00111","kind":"arxiv","version":1}},"canonical_sha256":"d1d1eca49787e335831c2ea96a3e8ac7306c947f3114d530fb36cf6589928a47","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d1d1eca49787e335831c2ea96a3e8ac7306c947f3114d530fb36cf6589928a47","first_computed_at":"2026-05-18T00:26:59.390753Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:59.390753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vsz9JpDgjWoj5wo9QLa/A23v+Q+INySWqeghTMvPZIX7snGDOaAE9d6IM1tSfeklv+Ab+jku5/uyosu/VauDAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:59.391485Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.00111","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21aee3ad05c6e3d792d89533e362544cfc5b4ea57cca333bb1de214e3b783c97","sha256:727caa0bab2b20f1ac5b65920f4fa04e0c659f7ce277c7c544cb256ac3fa9be9"],"state_sha256":"8818480b6714754c9961139d7d2daaf52a21a2a2acd3a16930a9c4dc6e2db75a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pGZj91OrgYQUbYaclpkb1TdCrXFEIHdbFeY/bcoh3ifxuytEducioG7K2u89s90qC+3UsMFitxi8OTZsfZSeBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T09:01:21.937897Z","bundle_sha256":"ea292ed702fa9fa0aaa75e22a533838f7908bc34db64e6c9055cd3ebd78c3dc7"}}