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A bi-Cayley graph $\\bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph $\\bcay(G,T),$ $\\bcay(G,S) \\cong \\bcay(G,T)$ implies that $T = g S^\\alpha$ for some $g \\in G$ and $\\alpha \\in \\aut(G)$. A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in"},"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":"1308.6812","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-30T17:58:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6ebecfc092b59de2274a9dd5b64a90c56261ae8c84ecf41a45464f76e8845ddc","abstract_canon_sha256":"ee8b67828acc86ab3522b43275fa360261db21ed047ba968a3c855ada16f3acd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:14:30.836735Z","signature_b64":"SB3uJ1qS3fTqPMG8+mQYmc1JSTWNvMzHS+/HTPyDj3hmI9zSu3Kz43fPomxuJrp49vB0lllndPRTi3REXi9hAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf00ff5d15da48eb04adfb1bc20ba01b4f331f9c759209ccbb66cafbec94948f","last_reissued_at":"2026-05-18T03:14:30.836101Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:14:30.836101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A classification of nilpotent 3-BCI groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Hiroki Koike, Istv\\'an Kov\\'acs","submitted_at":"2013-08-30T17:58:52Z","abstract_excerpt":"Given a finite group $G$ and a subset $S\\subseteq G,$ the bi-Cayley graph $\\bcay(G,S)$ is the graph whose vertex set is $G \\times \\{0,1\\}$ and edge set is $\\{\\{(x,0),(s x,1)\\} : x \\in G, s\\in S \\}$. A bi-Cayley graph $\\bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph $\\bcay(G,T),$ $\\bcay(G,S) \\cong \\bcay(G,T)$ implies that $T = g S^\\alpha$ for some $g \\in G$ and $\\alpha \\in \\aut(G)$. A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.6812","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":"1308.6812","created_at":"2026-05-18T03:14:30.836199+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.6812v1","created_at":"2026-05-18T03:14:30.836199+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.6812","created_at":"2026-05-18T03:14:30.836199+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z4AP6XIV3JEO","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z4AP6XIV3JEOWBFN","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z4AP6XIV","created_at":"2026-05-18T12:28:09.283467+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/Z4AP6XIV3JEOWBFN7MN4EC5ADN","json":"https://pith.science/pith/Z4AP6XIV3JEOWBFN7MN4EC5ADN.json","graph_json":"https://pith.science/api/pith-number/Z4AP6XIV3JEOWBFN7MN4EC5ADN/graph.json","events_json":"https://pith.science/api/pith-number/Z4AP6XIV3JEOWBFN7MN4EC5ADN/events.json","paper":"https://pith.science/paper/Z4AP6XIV"},"agent_actions":{"view_html":"https://pith.science/pith/Z4AP6XIV3JEOWBFN7MN4EC5ADN","download_json":"https://pith.science/pith/Z4AP6XIV3JEOWBFN7MN4EC5ADN.json","view_paper":"https://pith.science/paper/Z4AP6XIV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.6812&json=true","fetch_graph":"https://pith.science/api/pith-number/Z4AP6XIV3JEOWBFN7MN4EC5ADN/graph.json","fetch_events":"https://pith.science/api/pith-number/Z4AP6XIV3JEOWBFN7MN4EC5ADN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z4AP6XIV3JEOWBFN7MN4EC5ADN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z4AP6XIV3JEOWBFN7MN4EC5ADN/action/storage_attestation","attest_author":"https://pith.science/pith/Z4AP6XIV3JEOWBFN7MN4EC5ADN/action/author_attestation","sign_citation":"https://pith.science/pith/Z4AP6XIV3JEOWBFN7MN4EC5ADN/action/citation_signature","submit_replication":"https://pith.science/pith/Z4AP6XIV3JEOWBFN7MN4EC5ADN/action/replication_record"}},"created_at":"2026-05-18T03:14:30.836199+00:00","updated_at":"2026-05-18T03:14:30.836199+00:00"}