{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:QIQOAK4ZFSD53JBMUDH4O4WYH6","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":"b05bbb92cc3a775504826f38cbe64fa9044558547f0ac0cec08a4b7bb965193f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-17T08:03:04Z","title_canon_sha256":"5765b6205c264c4d77a700e26d6a3f25b433daa40aec73e4531300d79a259cfe"},"schema_version":"1.0","source":{"id":"1907.07376","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.07376","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"arxiv_version","alias_value":"1907.07376v1","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.07376","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"pith_short_12","alias_value":"QIQOAK4ZFSD5","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"QIQOAK4ZFSD53JBM","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"QIQOAK4Z","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:1d61cf177d7e6fb8cc6fdf3a29c0d15a3596317ce6fc8904857b406fe6e6828a","target":"graph","created_at":"2026-05-17T23:40:22Z","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":"For any connected multigraph $G=(V,E)$ and any $M\\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $\\tau_G(M)$ is obtained, where $\\tau_G(M)$ is the number of spanning trees in $G$ which contain all edges in $M$. Applying this result, we can easily obtain a formula for the number of spanning trees in the line graph or the middle graph of an arbitrary graph. Applying this result, we also show that for any connected graph $G$ with a clique $U$ which is a cut-set of $G$, the number of ","authors_text":"Fengming Dong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-17T08:03:04Z","title":"Formulas counting spanning trees in line graphs and their extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.07376","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":"sha256:25194ec1c60067ba8d106a1090cbb356cff6c85f32cf81be55b163911b5c812e","target":"record","created_at":"2026-05-17T23:40:22Z","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":"b05bbb92cc3a775504826f38cbe64fa9044558547f0ac0cec08a4b7bb965193f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-17T08:03:04Z","title_canon_sha256":"5765b6205c264c4d77a700e26d6a3f25b433daa40aec73e4531300d79a259cfe"},"schema_version":"1.0","source":{"id":"1907.07376","kind":"arxiv","version":1}},"canonical_sha256":"8220e02b992c87dda42ca0cfc772d83fb42b0bae6c12397b70c1d65af2c3ac68","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8220e02b992c87dda42ca0cfc772d83fb42b0bae6c12397b70c1d65af2c3ac68","first_computed_at":"2026-05-17T23:40:22.439244Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:22.439244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YnyOCyBB4LkYVznF9xovfuiIbUtpNQ/rrZE+U12N0IpjSixK22CUzYEpP9xYXTN2dIin3n7dsVRBTJAsGXdrCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:22.439902Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.07376","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:25194ec1c60067ba8d106a1090cbb356cff6c85f32cf81be55b163911b5c812e","sha256:1d61cf177d7e6fb8cc6fdf3a29c0d15a3596317ce6fc8904857b406fe6e6828a"],"state_sha256":"867f9df7cc826d8c20ac1b98feb54964708fd0f4335271c538db9654e90d3915"}