{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:HLG4IG3NQGKTKQR5GVDCZK4SUZ","short_pith_number":"pith:HLG4IG3N","schema_version":"1.0","canonical_sha256":"3acdc41b6d819535423d35462cab92a65431bbeff2529a71f14eff289977b792","source":{"kind":"arxiv","id":"1705.02161","version":1},"attestation_state":"computed","paper":{"title":"Relative non-commuting graph of a finite ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Dhiren Kumar Basnet, Jutirekha Dutta","submitted_at":"2017-05-05T10:31:33Z","abstract_excerpt":"Let $S$ be a subring of a finite ring $R$ and $C_R(S) = \\{r \\in R : rs = sr \\;\\forall\\; s \\in S\\}$. The relative non-commuting graph of the subring $S$ in $R$, denoted by $\\Gamma_{S, R}$, is a simple undirected graph whose vertex set is $R \\setminus C_R(S)$ and two distinct vertices $a, b$ are adjacent if and only if $a$ or $b \\in S$ and $ab \\neq ba$. In this paper, we discuss some properties of $\\Gamma_{S, R}$, determine diameter, girth, some dominating sets and chromatic index for $\\Gamma_{S, R}$. Also, we derive some connections between $\\Gamma_{S, R}$ and the relative commuting probability"},"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":"1705.02161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-05-05T10:31:33Z","cross_cats_sorted":[],"title_canon_sha256":"28a7e3e03fa7e55f3f007c689853a8dd5e020e19cc084fe859ecd720a02fa4e5","abstract_canon_sha256":"1dfd5aaaffaecbc808db1e19552a3bcb5d8e7ef1103c9cf65a758b8c18820b87"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:00.510258Z","signature_b64":"Mc9Km4Z4qh2iomQmmSpYu4ilDOYc2ym58L8Oxv39ywIFCv2eCeAn0h3B5NFC1f+VhrX/LwdaQY0/cmf0ylIRDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3acdc41b6d819535423d35462cab92a65431bbeff2529a71f14eff289977b792","last_reissued_at":"2026-05-18T00:45:00.509841Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:00.509841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Relative non-commuting graph of a finite ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Dhiren Kumar Basnet, Jutirekha Dutta","submitted_at":"2017-05-05T10:31:33Z","abstract_excerpt":"Let $S$ be a subring of a finite ring $R$ and $C_R(S) = \\{r \\in R : rs = sr \\;\\forall\\; s \\in S\\}$. The relative non-commuting graph of the subring $S$ in $R$, denoted by $\\Gamma_{S, R}$, is a simple undirected graph whose vertex set is $R \\setminus C_R(S)$ and two distinct vertices $a, b$ are adjacent if and only if $a$ or $b \\in S$ and $ab \\neq ba$. In this paper, we discuss some properties of $\\Gamma_{S, R}$, determine diameter, girth, some dominating sets and chromatic index for $\\Gamma_{S, R}$. Also, we derive some connections between $\\Gamma_{S, R}$ and the relative commuting probability"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02161","kind":"arxiv","version":1},"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":"1705.02161","created_at":"2026-05-18T00:45:00.509913+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.02161v1","created_at":"2026-05-18T00:45:00.509913+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.02161","created_at":"2026-05-18T00:45:00.509913+00:00"},{"alias_kind":"pith_short_12","alias_value":"HLG4IG3NQGKT","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_16","alias_value":"HLG4IG3NQGKTKQR5","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_8","alias_value":"HLG4IG3N","created_at":"2026-05-18T12:31:18.294218+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/HLG4IG3NQGKTKQR5GVDCZK4SUZ","json":"https://pith.science/pith/HLG4IG3NQGKTKQR5GVDCZK4SUZ.json","graph_json":"https://pith.science/api/pith-number/HLG4IG3NQGKTKQR5GVDCZK4SUZ/graph.json","events_json":"https://pith.science/api/pith-number/HLG4IG3NQGKTKQR5GVDCZK4SUZ/events.json","paper":"https://pith.science/paper/HLG4IG3N"},"agent_actions":{"view_html":"https://pith.science/pith/HLG4IG3NQGKTKQR5GVDCZK4SUZ","download_json":"https://pith.science/pith/HLG4IG3NQGKTKQR5GVDCZK4SUZ.json","view_paper":"https://pith.science/paper/HLG4IG3N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.02161&json=true","fetch_graph":"https://pith.science/api/pith-number/HLG4IG3NQGKTKQR5GVDCZK4SUZ/graph.json","fetch_events":"https://pith.science/api/pith-number/HLG4IG3NQGKTKQR5GVDCZK4SUZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HLG4IG3NQGKTKQR5GVDCZK4SUZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HLG4IG3NQGKTKQR5GVDCZK4SUZ/action/storage_attestation","attest_author":"https://pith.science/pith/HLG4IG3NQGKTKQR5GVDCZK4SUZ/action/author_attestation","sign_citation":"https://pith.science/pith/HLG4IG3NQGKTKQR5GVDCZK4SUZ/action/citation_signature","submit_replication":"https://pith.science/pith/HLG4IG3NQGKTKQR5GVDCZK4SUZ/action/replication_record"}},"created_at":"2026-05-18T00:45:00.509913+00:00","updated_at":"2026-05-18T00:45:00.509913+00:00"}