{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:YA7SPCQZFZKLWKEIQRUTJL3YL6","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":"3cde6595d00e551b1dcd6c7afce1a1f299e13d298bcb42f5f414f375d0c701f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-08-13T19:03:51Z","title_canon_sha256":"675e216bf4a1e4e99c647008d468e398fc0123308b5b0b4ec97bdd3b7a30aecf"},"schema_version":"1.0","source":{"id":"0908.1956","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.1956","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"arxiv_version","alias_value":"0908.1956v3","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.1956","created_at":"2026-05-18T04:11:49Z"},{"alias_kind":"pith_short_12","alias_value":"YA7SPCQZFZKL","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"YA7SPCQZFZKLWKEI","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"YA7SPCQZ","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:826000dc06f6ece2d33817bbb059cea8b1dd195d33e8d9a321f0960a488869a7","target":"graph","created_at":"2026-05-18T04:11:49Z","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":"We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifte","authors_text":"Art M. Duval, Caroline J. Klivans, Jeremy L. Martin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-08-13T19:03:51Z","title":"Cellular spanning trees and Laplacians of cubical complexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.1956","kind":"arxiv","version":3},"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":"sha256:32ad0de6e6c9393d6213b74520506dfe0f6ba23e82cffa9df5ec07ba34c2bd70","target":"record","created_at":"2026-05-18T04:11:49Z","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":"3cde6595d00e551b1dcd6c7afce1a1f299e13d298bcb42f5f414f375d0c701f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-08-13T19:03:51Z","title_canon_sha256":"675e216bf4a1e4e99c647008d468e398fc0123308b5b0b4ec97bdd3b7a30aecf"},"schema_version":"1.0","source":{"id":"0908.1956","kind":"arxiv","version":3}},"canonical_sha256":"c03f278a192e54bb2888846934af785f8d6af4381f60fb758b1e85c9a16cbefc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c03f278a192e54bb2888846934af785f8d6af4381f60fb758b1e85c9a16cbefc","first_computed_at":"2026-05-18T04:11:49.319221Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:49.319221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ki5rEBO3q0MgoRKxUnvbUxpu4aYufRHYOcLiNxvjBvOCDlSbiZrEgtkzWOJ3fa+Ytud2J1ZEzEceuPWJ1OQJBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:49.319752Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.1956","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:32ad0de6e6c9393d6213b74520506dfe0f6ba23e82cffa9df5ec07ba34c2bd70","sha256:826000dc06f6ece2d33817bbb059cea8b1dd195d33e8d9a321f0960a488869a7"],"state_sha256":"3a9a2c34c67196d24bcd8b800d239d0e3c0f7715e9121a383646468950802e8e"}