{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:GHBBLF2XXKY5ADGSPYMZN2IYHA","short_pith_number":"pith:GHBBLF2X","schema_version":"1.0","canonical_sha256":"31c2159757bab1d00cd27e1996e9183810ed556d131180afc1bf8c5b66513d4f","source":{"kind":"arxiv","id":"1105.2675","version":2},"attestation_state":"computed","paper":{"title":"Dual complementary polynomials of graphs and combinatorial interpretation on the values of the Tutte polynomial at positive integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Beifang Chen","submitted_at":"2011-05-13T09:58:30Z","abstract_excerpt":"We introduce a modular (integral) complementary polynomial $\\kappa(G;x,y)$ ($\\kappa_{\\mathbbm z}(G;x,y)$) of two variables of a graph $G$ by counting the number of modular (integral) complementary tension-flows (CTF) of $G$ with an orientation $\\epsilon$. We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron $\\Delta_\\textsc{ctf}(G,\\epsilon)$, the complementary open 0-1 polytope $\\Delta^+_\\textsc{ctf}(G,\\epsilon)$, and the complementary open lattice polytopes $\\Delta^\\rho_\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1105.2675","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-13T09:58:30Z","cross_cats_sorted":[],"title_canon_sha256":"9baeadcda88a5c7a0ff46d55be653024e74655c104f2868f2f1c4256d88e3ab7","abstract_canon_sha256":"f2bb36a812043e6114cd97e3dbf55fd0b0c0bf630a60076f1eaf969f68ce64d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:32.050558Z","signature_b64":"O8mbXBJ6vTKvLtCQl3ktfJLIa1ugAs+YlB7nNSu9qFbZwyddD27XsgNDdkaTo/u3ZiST2k4Y6J0zJtLrlQGaBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31c2159757bab1d00cd27e1996e9183810ed556d131180afc1bf8c5b66513d4f","last_reissued_at":"2026-05-18T03:21:32.050105Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:32.050105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dual complementary polynomials of graphs and combinatorial interpretation on the values of the Tutte polynomial at positive integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Beifang Chen","submitted_at":"2011-05-13T09:58:30Z","abstract_excerpt":"We introduce a modular (integral) complementary polynomial $\\kappa(G;x,y)$ ($\\kappa_{\\mathbbm z}(G;x,y)$) of two variables of a graph $G$ by counting the number of modular (integral) complementary tension-flows (CTF) of $G$ with an orientation $\\epsilon$. We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron $\\Delta_\\textsc{ctf}(G,\\epsilon)$, the complementary open 0-1 polytope $\\Delta^+_\\textsc{ctf}(G,\\epsilon)$, and the complementary open lattice polytopes $\\Delta^\\rho_\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2675","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1105.2675","created_at":"2026-05-18T03:21:32.050177+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.2675v2","created_at":"2026-05-18T03:21:32.050177+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.2675","created_at":"2026-05-18T03:21:32.050177+00:00"},{"alias_kind":"pith_short_12","alias_value":"GHBBLF2XXKY5","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"GHBBLF2XXKY5ADGS","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"GHBBLF2X","created_at":"2026-05-18T12:26:28.662955+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA","json":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA.json","graph_json":"https://pith.science/api/pith-number/GHBBLF2XXKY5ADGSPYMZN2IYHA/graph.json","events_json":"https://pith.science/api/pith-number/GHBBLF2XXKY5ADGSPYMZN2IYHA/events.json","paper":"https://pith.science/paper/GHBBLF2X"},"agent_actions":{"view_html":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA","download_json":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA.json","view_paper":"https://pith.science/paper/GHBBLF2X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.2675&json=true","fetch_graph":"https://pith.science/api/pith-number/GHBBLF2XXKY5ADGSPYMZN2IYHA/graph.json","fetch_events":"https://pith.science/api/pith-number/GHBBLF2XXKY5ADGSPYMZN2IYHA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA/action/storage_attestation","attest_author":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA/action/author_attestation","sign_citation":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA/action/citation_signature","submit_replication":"https://pith.science/pith/GHBBLF2XXKY5ADGSPYMZN2IYHA/action/replication_record"}},"created_at":"2026-05-18T03:21:32.050177+00:00","updated_at":"2026-05-18T03:21:32.050177+00:00"}