{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:A3O7PYA5JFLQI3UYH32QL7WGPI","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":"d342e97c5909c59d63152f449520dccc2f7097eb6684a15d6908d8d483735444","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-02-27T03:20:46Z","title_canon_sha256":"6284458de42878b6b9efc82a940c3849576d628075e55f342767a6062d0c964b"},"schema_version":"1.0","source":{"id":"2202.13279","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2202.13279","created_at":"2026-07-05T04:00:28Z"},{"alias_kind":"arxiv_version","alias_value":"2202.13279v1","created_at":"2026-07-05T04:00:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.13279","created_at":"2026-07-05T04:00:28Z"},{"alias_kind":"pith_short_12","alias_value":"A3O7PYA5JFLQ","created_at":"2026-07-05T04:00:28Z"},{"alias_kind":"pith_short_16","alias_value":"A3O7PYA5JFLQI3UY","created_at":"2026-07-05T04:00:28Z"},{"alias_kind":"pith_short_8","alias_value":"A3O7PYA5","created_at":"2026-07-05T04:00:28Z"}],"graph_snapshots":[{"event_id":"sha256:fe8d9a38cd42b94f90c389d4c9f93d3be9b4f3b848cefd757648bd80271e3eec","target":"graph","created_at":"2026-07-05T04:00:28Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2202.13279/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $W(D_n)$ denote the walk matrix of the Dynkin graph $D_n$, a tree obtained from the path of order $n-1$ by adding a pendant edge at the second vertex. We prove that $\\text{rank}\\,W(D_n)=n-2$ if $4\\mid n$ and $\\text{rank}\\,W(D_n)=n-1$ otherwise. Furthermore, we prove that the Smith normal form of $W(D_n)$ is $$\\text{diag}[\\underbrace{1,1,\\ldots,1}_{\\lceil\\frac{n}{2}\\rceil},\\underbrace{2,2,\\ldots,2}_{\\lfloor\\frac{n}{2}\\rfloor-1},0]$$ when $4\\nmid n$. This confirms a recent conjecture in [W.Wang, F.Liu, W.Wang, Generalized spectral characterizations of almost controllable graphs, European J. ","authors_text":"Chuanming Wang, Songlin Guo, Wei Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-02-27T03:20:46Z","title":"On the walk matrix of the Dynkin graph $D_n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.13279","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":"sha256:767b9abef56dfe58796e7791fd2c430e901cc5df3b50e07a6d28d987c10c3a9a","target":"record","created_at":"2026-07-05T04:00:28Z","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":"d342e97c5909c59d63152f449520dccc2f7097eb6684a15d6908d8d483735444","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-02-27T03:20:46Z","title_canon_sha256":"6284458de42878b6b9efc82a940c3849576d628075e55f342767a6062d0c964b"},"schema_version":"1.0","source":{"id":"2202.13279","kind":"arxiv","version":1}},"canonical_sha256":"06ddf7e01d4957046e983ef505fec67a013944f556d299e6d4f183861fbf3770","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06ddf7e01d4957046e983ef505fec67a013944f556d299e6d4f183861fbf3770","first_computed_at":"2026-07-05T04:00:28.103436Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:00:28.103436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9jsvN0uOJfnDtfjvnuTdAoFFE+uQcCYLdq2tVfZIOgHaP3FOflxeSdBiqSRK7Glt/wGo2TCYhwW8umHgPpQNDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T04:00:28.103895Z","signed_message":"canonical_sha256_bytes"},"source_id":"2202.13279","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:767b9abef56dfe58796e7791fd2c430e901cc5df3b50e07a6d28d987c10c3a9a","sha256:fe8d9a38cd42b94f90c389d4c9f93d3be9b4f3b848cefd757648bd80271e3eec"],"state_sha256":"da93ecd7fa0c7b065e43c63963185658becdca87f3502c6df224e92fe27a036c"}