{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:EBUUDUH4KSCHHBGNK7MU4WO5IF","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":"fb1e1a686f77468bf3a0826642abe4aba0af2e6d4a6872512642580fe3553310","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-12T14:51:49Z","title_canon_sha256":"14f558af6d600ebcc7e0e80c8e35bef42f9777c4c1aff0aab864e8b49d1cd0d2"},"schema_version":"1.0","source":{"id":"1705.04610","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.04610","created_at":"2026-05-18T00:44:37Z"},{"alias_kind":"arxiv_version","alias_value":"1705.04610v1","created_at":"2026-05-18T00:44:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04610","created_at":"2026-05-18T00:44:37Z"},{"alias_kind":"pith_short_12","alias_value":"EBUUDUH4KSCH","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"EBUUDUH4KSCHHBGN","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"EBUUDUH4","created_at":"2026-05-18T12:31:12Z"}],"graph_snapshots":[{"event_id":"sha256:ab650dcd0f831af23a526a76a3286a09cc7e1aeb0f26f27589a0a3dff407b225","target":"graph","created_at":"2026-05-18T00:44:37Z","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":"Let $\\mathbb{Z}_{p^s}$ be the residue class ring of integers modulo $p^s$, where $p$ is a prime number and $s$ is a positive integer. Using matrix representation and the inner rank of a matrix, we study the intersection, join, dimension formula and dual subspaces on vector subspaces of $\\mathbb{Z}^n_{p^s}$. Based on these results, we investigate the Grassmann graph $G_{p^s}(n,m)$ over $\\mathbb{Z}_{p^s}$. $G_{p^s}(n,m)$ is a connected vertex-transitive graph, and we determine its valency, clique number and maximum cliques. Finally, we characterize the automorphisms of $G_{p^s}(n,m)$.","authors_text":"Benjian Lv, Kaishun Wang, Li-Ping Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-12T14:51:49Z","title":"Vector spaces and Grassmann graphs over residue class rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04610","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:b2cd2db531161bddd6c22d71d6ccf9e17243424cb9d359f78ae344986ebaef29","target":"record","created_at":"2026-05-18T00:44:37Z","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":"fb1e1a686f77468bf3a0826642abe4aba0af2e6d4a6872512642580fe3553310","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-12T14:51:49Z","title_canon_sha256":"14f558af6d600ebcc7e0e80c8e35bef42f9777c4c1aff0aab864e8b49d1cd0d2"},"schema_version":"1.0","source":{"id":"1705.04610","kind":"arxiv","version":1}},"canonical_sha256":"206941d0fc54847384cd57d94e59dd4141705834695caaf01d43b970cc6ed222","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"206941d0fc54847384cd57d94e59dd4141705834695caaf01d43b970cc6ed222","first_computed_at":"2026-05-18T00:44:37.665394Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:37.665394Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EpzNLKnL2GbgdMsFy0CbU7GSrYqDwZz43Zm7KqXRiZAuUEJp4eEVKhM5aRGVcATlXI8RlEfndd8bk/yLGRoRDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:37.666018Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.04610","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2cd2db531161bddd6c22d71d6ccf9e17243424cb9d359f78ae344986ebaef29","sha256:ab650dcd0f831af23a526a76a3286a09cc7e1aeb0f26f27589a0a3dff407b225"],"state_sha256":"4f8fb64549c8a265971a8582580ab9dd0aa5b0458ceb8386b1807043126748bb"}