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Under the hypothesis that $n$ is prime we : 1) compute the regularity index of $R/I(G)$; 2) compute the Castelnuovo-Mumford regularity when $R/I(G)$ is Cohen-Macaulay; 3) prove that the circulant graphs with $S=\\{1,\\ldots,s\\}$ are sequentially $S_2$ . We end characterizing the Cohen-Macaulay circulant graphs of Krull dimension $2$ and computing their Cohen-Macaulay type and Castelnuovo-Mumford regularity."},"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":"1701.01357","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-01-05T15:40:42Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3aa4893ca264b9613b7bf716e9a730e26e4c3ab6f99806f2973821597ba43ba3","abstract_canon_sha256":"7b26332d5e284d5458048a5a7a68fcce5401352388120383abb27cf29861e3db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:01.185204Z","signature_b64":"6kKGSaSCK6mPLuqBvqvGoO1LtfqhGjlv2Sqv4pbaC/HXQEP5JzDZX9viXHddGOflVuztcPK5UpwfhaL7fvx0AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6b8e049fa30160256724184a5f3d004f894d6bcd63e152798e5a1da6d4ff5cf","last_reissued_at":"2026-05-18T00:43:01.184550Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:01.184550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some algebraic invariants of edge ideal of circulant graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Giancarlo Rinaldo","submitted_at":"2017-01-05T15:40:42Z","abstract_excerpt":"Let $G$ be the circulant graph $C_n(S)$ with $S\\subseteq\\{ 1,\\ldots,\\left \\lfloor\\frac{n}{2}\\right \\rfloor\\}$ and let $I(G)$ be its edge ideal in the ring $K[x_0,\\ldots,x_{n-1}]$. Under the hypothesis that $n$ is prime we : 1) compute the regularity index of $R/I(G)$; 2) compute the Castelnuovo-Mumford regularity when $R/I(G)$ is Cohen-Macaulay; 3) prove that the circulant graphs with $S=\\{1,\\ldots,s\\}$ are sequentially $S_2$ . We end characterizing the Cohen-Macaulay circulant graphs of Krull dimension $2$ and computing their Cohen-Macaulay type and Castelnuovo-Mumford regularity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.01357","kind":"arxiv","version":4},"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":"1701.01357","created_at":"2026-05-18T00:43:01.184644+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.01357v4","created_at":"2026-05-18T00:43:01.184644+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.01357","created_at":"2026-05-18T00:43:01.184644+00:00"},{"alias_kind":"pith_short_12","alias_value":"624OASP2GALA","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"624OASP2GALAEVTS","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"624OASP2","created_at":"2026-05-18T12:31:03.183658+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/624OASP2GALAEVTSIGCKL46QAT","json":"https://pith.science/pith/624OASP2GALAEVTSIGCKL46QAT.json","graph_json":"https://pith.science/api/pith-number/624OASP2GALAEVTSIGCKL46QAT/graph.json","events_json":"https://pith.science/api/pith-number/624OASP2GALAEVTSIGCKL46QAT/events.json","paper":"https://pith.science/paper/624OASP2"},"agent_actions":{"view_html":"https://pith.science/pith/624OASP2GALAEVTSIGCKL46QAT","download_json":"https://pith.science/pith/624OASP2GALAEVTSIGCKL46QAT.json","view_paper":"https://pith.science/paper/624OASP2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.01357&json=true","fetch_graph":"https://pith.science/api/pith-number/624OASP2GALAEVTSIGCKL46QAT/graph.json","fetch_events":"https://pith.science/api/pith-number/624OASP2GALAEVTSIGCKL46QAT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/624OASP2GALAEVTSIGCKL46QAT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/624OASP2GALAEVTSIGCKL46QAT/action/storage_attestation","attest_author":"https://pith.science/pith/624OASP2GALAEVTSIGCKL46QAT/action/author_attestation","sign_citation":"https://pith.science/pith/624OASP2GALAEVTSIGCKL46QAT/action/citation_signature","submit_replication":"https://pith.science/pith/624OASP2GALAEVTSIGCKL46QAT/action/replication_record"}},"created_at":"2026-05-18T00:43:01.184644+00:00","updated_at":"2026-05-18T00:43:01.184644+00:00"}