{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6PKSFVAYVJWD75OMXOAZRQMMSO","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":"f9fb26013f09e2f0b4204700bfe04ee8ba0756dd73a693bf89f190341366de8e","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-22T08:16:22Z","title_canon_sha256":"54060e37302a351b2991d493ec824fdb72add2980358d4ff523af5451b2aaf69"},"schema_version":"1.0","source":{"id":"1402.5494","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.5494","created_at":"2026-05-18T02:58:20Z"},{"alias_kind":"arxiv_version","alias_value":"1402.5494v1","created_at":"2026-05-18T02:58:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.5494","created_at":"2026-05-18T02:58:20Z"},{"alias_kind":"pith_short_12","alias_value":"6PKSFVAYVJWD","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6PKSFVAYVJWD75OM","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6PKSFVAY","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:8207cd25772d78ddf6d1a7f8a6f3d14d5df40f7144ea10fc02b4d114518db5c5","target":"graph","created_at":"2026-05-18T02:58:20Z","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":"Given a finite group G, we say that a subset C of G is power-closed if, for every x in C and y in <x> with <x>=<y>, we have that y lies in C.\n  In this paper we are interested in finite Cayley digraphs Cay(G,C) over G with connection set C, where C is a union of conjugacy classes of G. We show that each eigenvalue of Cay(G,C) is integral if and only if C is power-closed. This result will follow from a discussion of some more general rationality conditions on the eigenvalues of Cay(G,C).","authors_text":"Chris Godsil, Pablo Spiga","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-22T08:16:22Z","title":"Rationality conditions for the eigenvalues of normal finite Cayley graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5494","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:be15b0438833f0b981fc5d7efd8a32586151d8676727a208cc3f729daaa64499","target":"record","created_at":"2026-05-18T02:58:20Z","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":"f9fb26013f09e2f0b4204700bfe04ee8ba0756dd73a693bf89f190341366de8e","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-22T08:16:22Z","title_canon_sha256":"54060e37302a351b2991d493ec824fdb72add2980358d4ff523af5451b2aaf69"},"schema_version":"1.0","source":{"id":"1402.5494","kind":"arxiv","version":1}},"canonical_sha256":"f3d522d418aa6c3ff5ccbb8198c18c93941de9f0a01a0a724562de4372c03850","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3d522d418aa6c3ff5ccbb8198c18c93941de9f0a01a0a724562de4372c03850","first_computed_at":"2026-05-18T02:58:20.958604Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:20.958604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NrkOhc5Yre3Ay+0Q5YbSnaCuDvRnH1ojYFwWrko9C9G6HbjLGeat8/mRAsx9TKL/2PBq2DiUiKypzdPFzKXGBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:20.959122Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.5494","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be15b0438833f0b981fc5d7efd8a32586151d8676727a208cc3f729daaa64499","sha256:8207cd25772d78ddf6d1a7f8a6f3d14d5df40f7144ea10fc02b4d114518db5c5"],"state_sha256":"f2d833d4a0b21b071ba39c70e3fb78285ce1936aaf848c812f53d79e84817704"}