{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:7TIN4UUHO26X6AMCSPLLPUYE6S","short_pith_number":"pith:7TIN4UUH","canonical_record":{"source":{"id":"1602.01996","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-05T12:00:38Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"6f494c4a2c50f1440d2076fda6adf049d09fde0c95474f5345cfd4c027909b88","abstract_canon_sha256":"67bb865a465109f0609f8c393bd4c2a72bfa1532a9d681dd295837a261a6460f"},"schema_version":"1.0"},"canonical_sha256":"fcd0de528776bd7f018293d6b7d304f4aa3d936c26647ea88d273cafc92c8c04","source":{"kind":"arxiv","id":"1602.01996","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.01996","created_at":"2026-05-18T01:08:12Z"},{"alias_kind":"arxiv_version","alias_value":"1602.01996v1","created_at":"2026-05-18T01:08:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.01996","created_at":"2026-05-18T01:08:12Z"},{"alias_kind":"pith_short_12","alias_value":"7TIN4UUHO26X","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7TIN4UUHO26X6AMC","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7TIN4UUH","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:7TIN4UUHO26X6AMCSPLLPUYE6S","target":"record","payload":{"canonical_record":{"source":{"id":"1602.01996","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-05T12:00:38Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"6f494c4a2c50f1440d2076fda6adf049d09fde0c95474f5345cfd4c027909b88","abstract_canon_sha256":"67bb865a465109f0609f8c393bd4c2a72bfa1532a9d681dd295837a261a6460f"},"schema_version":"1.0"},"canonical_sha256":"fcd0de528776bd7f018293d6b7d304f4aa3d936c26647ea88d273cafc92c8c04","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:12.028424Z","signature_b64":"h8DkFixlyuPDefLEwkD4K8Vptevs9O+KMXS+p7xB+Hw6SHVPxRvngfFIkcZd94jEsyS+rkK2v6z6T2gbLhb3DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcd0de528776bd7f018293d6b7d304f4aa3d936c26647ea88d273cafc92c8c04","last_reissued_at":"2026-05-18T01:08:12.027967Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:12.027967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.01996","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":"2026-05-18T01:08:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cHhTv3UQHGLnRuvQWPUs1jhYmwPk/X/McZMKPeXsfnI9XBRPX/m1OsEoOJEd+VB8fofp2Hja/D77y9z+FsGTBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T05:40:19.300177Z"},"content_sha256":"e8b979c8b3f7e6675559e39aec8206702e55e06fb7cb588f20414e0bbe741943","schema_version":"1.0","event_id":"sha256:e8b979c8b3f7e6675559e39aec8206702e55e06fb7cb588f20414e0bbe741943"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:7TIN4UUHO26X6AMCSPLLPUYE6S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting spanning trees on fractal graphs and their asymptotic complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.CO","authors_text":"Jason A. Anema, Konstantinos Tsougkas","submitted_at":"2016-02-05T12:00:38Z","abstract_excerpt":"Using the method of spectral decimation and a modified version of Kirchhoff's Matrix-Tree Theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in Theorem \\ref{thm:maintheoremfull}. We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpinski Gasket, a non post critically finite analog of the Sierpinski Gasket, the Diamond fractal, and the Hexagasket. For each example, the a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01996","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":"2026-05-18T01:08:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kc4r5xb/nuxWsUaaV+KO9lC5z+gujydgbw5HixcdVlSCM21p41Bkt6rTAvJrRRp0HkbmkRPCc56oaSDkTTPaBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T05:40:19.300537Z"},"content_sha256":"acfa185b531ca63b5a83083d1b1dc32376fbe54234b2d8b69f7ca72eec989711","schema_version":"1.0","event_id":"sha256:acfa185b531ca63b5a83083d1b1dc32376fbe54234b2d8b69f7ca72eec989711"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7TIN4UUHO26X6AMCSPLLPUYE6S/bundle.json","state_url":"https://pith.science/pith/7TIN4UUHO26X6AMCSPLLPUYE6S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7TIN4UUHO26X6AMCSPLLPUYE6S/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":"2026-06-24T05:40:19Z","links":{"resolver":"https://pith.science/pith/7TIN4UUHO26X6AMCSPLLPUYE6S","bundle":"https://pith.science/pith/7TIN4UUHO26X6AMCSPLLPUYE6S/bundle.json","state":"https://pith.science/pith/7TIN4UUHO26X6AMCSPLLPUYE6S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7TIN4UUHO26X6AMCSPLLPUYE6S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7TIN4UUHO26X6AMCSPLLPUYE6S","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":"67bb865a465109f0609f8c393bd4c2a72bfa1532a9d681dd295837a261a6460f","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-05T12:00:38Z","title_canon_sha256":"6f494c4a2c50f1440d2076fda6adf049d09fde0c95474f5345cfd4c027909b88"},"schema_version":"1.0","source":{"id":"1602.01996","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.01996","created_at":"2026-05-18T01:08:12Z"},{"alias_kind":"arxiv_version","alias_value":"1602.01996v1","created_at":"2026-05-18T01:08:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.01996","created_at":"2026-05-18T01:08:12Z"},{"alias_kind":"pith_short_12","alias_value":"7TIN4UUHO26X","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7TIN4UUHO26X6AMC","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7TIN4UUH","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:acfa185b531ca63b5a83083d1b1dc32376fbe54234b2d8b69f7ca72eec989711","target":"graph","created_at":"2026-05-18T01:08:12Z","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":"Using the method of spectral decimation and a modified version of Kirchhoff's Matrix-Tree Theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in Theorem \\ref{thm:maintheoremfull}. We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpinski Gasket, a non post critically finite analog of the Sierpinski Gasket, the Diamond fractal, and the Hexagasket. For each example, the a","authors_text":"Jason A. Anema, Konstantinos Tsougkas","cross_cats":["math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-05T12:00:38Z","title":"Counting spanning trees on fractal graphs and their asymptotic complexity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01996","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:e8b979c8b3f7e6675559e39aec8206702e55e06fb7cb588f20414e0bbe741943","target":"record","created_at":"2026-05-18T01:08:12Z","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":"67bb865a465109f0609f8c393bd4c2a72bfa1532a9d681dd295837a261a6460f","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-05T12:00:38Z","title_canon_sha256":"6f494c4a2c50f1440d2076fda6adf049d09fde0c95474f5345cfd4c027909b88"},"schema_version":"1.0","source":{"id":"1602.01996","kind":"arxiv","version":1}},"canonical_sha256":"fcd0de528776bd7f018293d6b7d304f4aa3d936c26647ea88d273cafc92c8c04","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fcd0de528776bd7f018293d6b7d304f4aa3d936c26647ea88d273cafc92c8c04","first_computed_at":"2026-05-18T01:08:12.027967Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:12.027967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h8DkFixlyuPDefLEwkD4K8Vptevs9O+KMXS+p7xB+Hw6SHVPxRvngfFIkcZd94jEsyS+rkK2v6z6T2gbLhb3DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:12.028424Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.01996","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8b979c8b3f7e6675559e39aec8206702e55e06fb7cb588f20414e0bbe741943","sha256:acfa185b531ca63b5a83083d1b1dc32376fbe54234b2d8b69f7ca72eec989711"],"state_sha256":"ffde37f53a908f079741c8a8ab1ae23e1c30bbeb321d18b54f4826c6b10ce305"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4Nuba2maE2LZ+gCF1Bx1oEulPPDNsEzLKpSdaWh8oRwqXWOCmUSLPZgVIWYbhKDQCpCjh04gLdQ/dIGM+8f4CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T05:40:19.302469Z","bundle_sha256":"071bd73f23738e05c74255d95925df6217a8b3b8ffd12fc3a3ece90d22ca1435"}}