{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:6PTZHLX6QCTMQ5BJFW3ELI4RUK","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":"4c5a33043e0aab70e338591f993ebd37bc9730f2d8f54c0ab4b2366082ab4d99","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.AC","submitted_at":"2013-05-27T12:45:22Z","title_canon_sha256":"f8af8d5ba98580bd47aee061140ed9971d00c78fee750baa74e9cb0d15e26087"},"schema_version":"1.0","source":{"id":"1305.6199","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.6199","created_at":"2026-05-18T03:17:24Z"},{"alias_kind":"arxiv_version","alias_value":"1305.6199v2","created_at":"2026-05-18T03:17:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6199","created_at":"2026-05-18T03:17:24Z"},{"alias_kind":"pith_short_12","alias_value":"6PTZHLX6QCTM","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6PTZHLX6QCTMQ5BJ","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6PTZHLX6","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:535cc987a663a25754ab700c383efab4492243a614cae2643319c4eb215ad185","target":"graph","created_at":"2026-05-18T03:17:24Z","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 $R$ be a commutative ring and $M$ be an $R$-module, and let $Z(M)$ be the set of all zero-divisors on $M$. In 2008, D.F. Anderson and A. Badawi introduced the regular graph of $R$. In this paper, we generalize the regular graph of $R$ to the \\textit{$M$-regular graph} of $R$, denoted by $M$-$Reg(\\Gamma(R))$. It is the undirected graph with all $M$-regular elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\\in Z(M)$. The basic properties and possible structures of the $M$-$Reg(\\Gamma(R))$ are studied. We determine the girth of the $M$-regular ","authors_text":"F. Heydari, M.J. Nikmehr","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.AC","submitted_at":"2013-05-27T12:45:22Z","title":"The M-Regular Graph of a Commutative Ring"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6199","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":"sha256:1860e1e3f5154e43c7ad20a90b34a033487a08454105c01d7e34dc3a26b7fd24","target":"record","created_at":"2026-05-18T03:17:24Z","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":"4c5a33043e0aab70e338591f993ebd37bc9730f2d8f54c0ab4b2366082ab4d99","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.AC","submitted_at":"2013-05-27T12:45:22Z","title_canon_sha256":"f8af8d5ba98580bd47aee061140ed9971d00c78fee750baa74e9cb0d15e26087"},"schema_version":"1.0","source":{"id":"1305.6199","kind":"arxiv","version":2}},"canonical_sha256":"f3e793aefe80a6c874292db645a391a2ac417e115df7084e38aae99752a2d0dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3e793aefe80a6c874292db645a391a2ac417e115df7084e38aae99752a2d0dd","first_computed_at":"2026-05-18T03:17:24.241894Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:17:24.241894Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7Km4L7WPhfQL7AacdA8KZne6A4gEBN3PSvbuD76q6E7FFb2wDHsVI1ojmU+UHT6XAdJ5aMlr3WpA72ZCFnleDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:17:24.242444Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.6199","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1860e1e3f5154e43c7ad20a90b34a033487a08454105c01d7e34dc3a26b7fd24","sha256:535cc987a663a25754ab700c383efab4492243a614cae2643319c4eb215ad185"],"state_sha256":"8e5997cb98e7c6cdce6d7bfed87781258a79a63820f1bdb6286f1a018ef9bed0"}