{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ZEYVQJJSNR6AFPFANUY7HU4FLS","short_pith_number":"pith:ZEYVQJJS","canonical_record":{"source":{"id":"1309.5629","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-22T18:48:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4b32a88997f2b0d390ad9ea0c32af8cb320e803fd7d4fa4d106ad6cd8369a0ab","abstract_canon_sha256":"8b5b81d6d93a6469c0e105d592a65368eb7a190aa28e1f21e26a07a19941d9b4"},"schema_version":"1.0"},"canonical_sha256":"c9315825326c7c02bca06d31f3d3855c95ab6fecbe3a7e78dee25489d648f837","source":{"kind":"arxiv","id":"1309.5629","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5629","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5629v1","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5629","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"pith_short_12","alias_value":"ZEYVQJJSNR6A","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZEYVQJJSNR6AFPFA","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZEYVQJJS","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ZEYVQJJSNR6AFPFANUY7HU4FLS","target":"record","payload":{"canonical_record":{"source":{"id":"1309.5629","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-22T18:48:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4b32a88997f2b0d390ad9ea0c32af8cb320e803fd7d4fa4d106ad6cd8369a0ab","abstract_canon_sha256":"8b5b81d6d93a6469c0e105d592a65368eb7a190aa28e1f21e26a07a19941d9b4"},"schema_version":"1.0"},"canonical_sha256":"c9315825326c7c02bca06d31f3d3855c95ab6fecbe3a7e78dee25489d648f837","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:35.725054Z","signature_b64":"+5v+9afzDiqFJNGO1FyyzWGky5Le9f/Vj4pxCBrf+ghMo8sv/5TDmEfkAUhQ3wVmGm8m5e1C6Kq0NEPbEAuvAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9315825326c7c02bca06d31f3d3855c95ab6fecbe3a7e78dee25489d648f837","last_reissued_at":"2026-05-18T03:12:35.724265Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:35.724265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.5629","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xxxp5OMw+KYVt6PXKQjHcMnYeto2nVcLh58P0HYOoOzYFxNiq7yfhPcfLevKupUm77J9jF5MNtNao4y6UuxGDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:51:14.450504Z"},"content_sha256":"31958d7230067a56c91f646211fabbb044b8c50997f92908d2c6be7da082af66","schema_version":"1.0","event_id":"sha256:31958d7230067a56c91f646211fabbb044b8c50997f92908d2c6be7da082af66"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ZEYVQJJSNR6AFPFANUY7HU4FLS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Groups having complete bipartite divisor graphs for their conjugacy class sizes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Pablo Spiga, Roghayeh Hafezieh","submitted_at":"2013-09-22T18:48:07Z","abstract_excerpt":"Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G-Z (where Z denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p,n} being an edge if gcd(p,n)\\neq 1.\n  In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K_{2,5}, giving a solution to a question of Taeri."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5629","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jBnx7ueVxYjsJj1PIyq5nNaluiBOtjuU9kVluezFgMEYw2O8C1iTdLwMjANzXWhoo8RtQgJTU7PQ7tKyKkZnCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:51:14.451234Z"},"content_sha256":"57cfd8d7610f9cebbfcd418e4f39258601784cb505df8c90825aef80313ec11d","schema_version":"1.0","event_id":"sha256:57cfd8d7610f9cebbfcd418e4f39258601784cb505df8c90825aef80313ec11d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZEYVQJJSNR6AFPFANUY7HU4FLS/bundle.json","state_url":"https://pith.science/pith/ZEYVQJJSNR6AFPFANUY7HU4FLS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZEYVQJJSNR6AFPFANUY7HU4FLS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T07:51:14Z","links":{"resolver":"https://pith.science/pith/ZEYVQJJSNR6AFPFANUY7HU4FLS","bundle":"https://pith.science/pith/ZEYVQJJSNR6AFPFANUY7HU4FLS/bundle.json","state":"https://pith.science/pith/ZEYVQJJSNR6AFPFANUY7HU4FLS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZEYVQJJSNR6AFPFANUY7HU4FLS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ZEYVQJJSNR6AFPFANUY7HU4FLS","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":"8b5b81d6d93a6469c0e105d592a65368eb7a190aa28e1f21e26a07a19941d9b4","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-22T18:48:07Z","title_canon_sha256":"4b32a88997f2b0d390ad9ea0c32af8cb320e803fd7d4fa4d106ad6cd8369a0ab"},"schema_version":"1.0","source":{"id":"1309.5629","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5629","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5629v1","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5629","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"pith_short_12","alias_value":"ZEYVQJJSNR6A","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZEYVQJJSNR6AFPFA","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZEYVQJJS","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:57cfd8d7610f9cebbfcd418e4f39258601784cb505df8c90825aef80313ec11d","target":"graph","created_at":"2026-05-18T03:12:35Z","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, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G-Z (where Z denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p,n} being an edge if gcd(p,n)\\neq 1.\n  In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K_{2,5}, giving a solution to a question of Taeri.","authors_text":"Pablo Spiga, Roghayeh Hafezieh","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-22T18:48:07Z","title":"Groups having complete bipartite divisor graphs for their conjugacy class sizes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5629","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":"sha256:31958d7230067a56c91f646211fabbb044b8c50997f92908d2c6be7da082af66","target":"record","created_at":"2026-05-18T03:12:35Z","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":"8b5b81d6d93a6469c0e105d592a65368eb7a190aa28e1f21e26a07a19941d9b4","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-22T18:48:07Z","title_canon_sha256":"4b32a88997f2b0d390ad9ea0c32af8cb320e803fd7d4fa4d106ad6cd8369a0ab"},"schema_version":"1.0","source":{"id":"1309.5629","kind":"arxiv","version":1}},"canonical_sha256":"c9315825326c7c02bca06d31f3d3855c95ab6fecbe3a7e78dee25489d648f837","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9315825326c7c02bca06d31f3d3855c95ab6fecbe3a7e78dee25489d648f837","first_computed_at":"2026-05-18T03:12:35.724265Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:35.724265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+5v+9afzDiqFJNGO1FyyzWGky5Le9f/Vj4pxCBrf+ghMo8sv/5TDmEfkAUhQ3wVmGm8m5e1C6Kq0NEPbEAuvAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:35.725054Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.5629","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31958d7230067a56c91f646211fabbb044b8c50997f92908d2c6be7da082af66","sha256:57cfd8d7610f9cebbfcd418e4f39258601784cb505df8c90825aef80313ec11d"],"state_sha256":"51c987d1e0705d538f2eaf5c5055f905e67416d41b23204071a2e4ef0ec040a3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o2oD6wrI38mJPG0dLll2joRFBmipX/yu9ZNv2cDY4Fl5XTqmjlVCQoksV4RveAGamY03T4lJGfqbGQMUCXdABg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T07:51:14.459736Z","bundle_sha256":"60b08769e8f494d7bd460c805577c5178bc35242f01b8150ee6e25d8df185ed9"}}