{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:KCA3BRMD7EEJCDU5BB4RZ7BRR7","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":"116cb37aed64bd4248cf049d18aefa74431937456f1e19881d00c9da79393ef9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-01T02:48:12Z","title_canon_sha256":"4517372cab340ed49d74e9c4617b402b677c68c5eb3ad7fe25929d0acf38a3a4"},"schema_version":"1.0","source":{"id":"1711.00175","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.00175","created_at":"2026-05-18T00:27:56Z"},{"alias_kind":"arxiv_version","alias_value":"1711.00175v2","created_at":"2026-05-18T00:27:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.00175","created_at":"2026-05-18T00:27:56Z"},{"alias_kind":"pith_short_12","alias_value":"KCA3BRMD7EEJ","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"KCA3BRMD7EEJCDU5","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"KCA3BRMD","created_at":"2026-05-18T12:31:24Z"}],"graph_snapshots":[{"event_id":"sha256:52d1253fde9aa642671c18d853caa06faeb5a4afcb6c790286a7eb8ca55574a3","target":"graph","created_at":"2026-05-18T00:27:56Z","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":"In this paper, we develop a new method to produce explicit formulas for the number $\\tau(n)$ of spanning trees in the undirected circulant graphs $C_{n}(s_1,s_2,\\ldots,s_k)$ and $C_{2n}(s_1,s_2,\\ldots,s_k,n).$ Also, we prove that in both cases the number of spanning trees can be represented in the form $\\tau(n)=p \\,n \\,a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the parity of $n.$ Finally, we find an asymptotic formula for $\\tau(n)$ through the Mahler measure of the associated Laurent polynomial $L(z)=2k-\\sum\\limits_{i=1}^k(z^{s_i}+z^{-s_i})","authors_text":"Alexander Mednykh, Ilya Mednykh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-01T02:48:12Z","title":"The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.00175","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:d36b9142f476c930b159e4dd471ab7f272597d9d34dabc538185f12b70b38b55","target":"record","created_at":"2026-05-18T00:27:56Z","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":"116cb37aed64bd4248cf049d18aefa74431937456f1e19881d00c9da79393ef9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-01T02:48:12Z","title_canon_sha256":"4517372cab340ed49d74e9c4617b402b677c68c5eb3ad7fe25929d0acf38a3a4"},"schema_version":"1.0","source":{"id":"1711.00175","kind":"arxiv","version":2}},"canonical_sha256":"5081b0c583f908910e9d08791cfc318fdf50e19a68d3fe71fab9e43c915ed96a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5081b0c583f908910e9d08791cfc318fdf50e19a68d3fe71fab9e43c915ed96a","first_computed_at":"2026-05-18T00:27:56.821922Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:56.821922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qUTCYVF9HvY6j++Nc1jFEDXIkK8isLKuP/6TC3b50o0nm3P5Y9bfG9b0KewLlOMULVfqso9cc6C2OTsgchEwCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:56.822591Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.00175","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d36b9142f476c930b159e4dd471ab7f272597d9d34dabc538185f12b70b38b55","sha256:52d1253fde9aa642671c18d853caa06faeb5a4afcb6c790286a7eb8ca55574a3"],"state_sha256":"efe43596c149ceef8d6ab8181f0c48f5ef55203821750e4343183b9c9b02edd8"}