{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:6YBSLZVX773TWTBFLISZ3E6XCU","short_pith_number":"pith:6YBSLZVX","canonical_record":{"source":{"id":"1601.03683","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-01-14T18:02:32Z","cross_cats_sorted":[],"title_canon_sha256":"9c3ded762de93a441a02a87d681207d90928ef87e26567bdc9f1b6ff41b0f8f3","abstract_canon_sha256":"d004129b73c554a93dd52d8147066e31d8ea68815941e7c814bf12d98a63e67d"},"schema_version":"1.0"},"canonical_sha256":"f60325e6b7fff73b4c255a259d93d7152f00ac3c8e8f34135e2c338557a849e6","source":{"kind":"arxiv","id":"1601.03683","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03683","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03683v2","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03683","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"pith_short_12","alias_value":"6YBSLZVX773T","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6YBSLZVX773TWTBF","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6YBSLZVX","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:6YBSLZVX773TWTBFLISZ3E6XCU","target":"record","payload":{"canonical_record":{"source":{"id":"1601.03683","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-01-14T18:02:32Z","cross_cats_sorted":[],"title_canon_sha256":"9c3ded762de93a441a02a87d681207d90928ef87e26567bdc9f1b6ff41b0f8f3","abstract_canon_sha256":"d004129b73c554a93dd52d8147066e31d8ea68815941e7c814bf12d98a63e67d"},"schema_version":"1.0"},"canonical_sha256":"f60325e6b7fff73b4c255a259d93d7152f00ac3c8e8f34135e2c338557a849e6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:55.392630Z","signature_b64":"fRZCM+SZagLvEniRDFaeUG2620GbVnIQ46/0lZNJO9rJVFe21NvO3FaLUwcZaq4KTnoM3fA5cJhXCrfeGGioAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f60325e6b7fff73b4c255a259d93d7152f00ac3c8e8f34135e2c338557a849e6","last_reissued_at":"2026-05-18T00:27:55.392029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:55.392029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.03683","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:27:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VBs1GeXM3HT3eys1n94LKMUHg6K1pyBmGo8+kx2QbUe6qrGsIC6ltvbfgFfcdJg5LFhyL7oISW/iiM2fDUdHAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T15:02:21.064736Z"},"content_sha256":"c90f3784d5a5df8684087d5ddfa24581c3feadd7bbd9bc82ad3ccb89eb27aada","schema_version":"1.0","event_id":"sha256:c90f3784d5a5df8684087d5ddfa24581c3feadd7bbd9bc82ad3ccb89eb27aada"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:6YBSLZVX773TWTBFLISZ3E6XCU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The complement of proper power graphs of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrei Gagarin, R. Rajkumar, T. Anitha","submitted_at":"2016-01-14T18:02:32Z","abstract_excerpt":"For a finite group $G$, the proper power graph $\\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \\neq v$ and $u^m=v$ or $v^m=u$ for some positive integer $m$. In this paper, we consider the complement of $\\mathscr{P}^*(G)$, denoted by ${\\overline{\\mathscr{P}^*(G)}}$. We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a cycle, a star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective. Among the other results, we al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03683","kind":"arxiv","version":2},"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":"2026-05-18T00:27:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zLfvFVWnUo8XqYEaQ/68f8zEeEXGoP5Arzm3ZzzdbzMF3FIPPaU1+zGHwgkGfFh+p8JE4mDhWReAyT0oRrOqCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T15:02:21.065098Z"},"content_sha256":"422026e40ec2e355a82f4f9f40d952556e5d3545249b74896c231d7848ba0b54","schema_version":"1.0","event_id":"sha256:422026e40ec2e355a82f4f9f40d952556e5d3545249b74896c231d7848ba0b54"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6YBSLZVX773TWTBFLISZ3E6XCU/bundle.json","state_url":"https://pith.science/pith/6YBSLZVX773TWTBFLISZ3E6XCU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6YBSLZVX773TWTBFLISZ3E6XCU/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":"2026-06-22T15:02:21Z","links":{"resolver":"https://pith.science/pith/6YBSLZVX773TWTBFLISZ3E6XCU","bundle":"https://pith.science/pith/6YBSLZVX773TWTBFLISZ3E6XCU/bundle.json","state":"https://pith.science/pith/6YBSLZVX773TWTBFLISZ3E6XCU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6YBSLZVX773TWTBFLISZ3E6XCU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:6YBSLZVX773TWTBFLISZ3E6XCU","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":"d004129b73c554a93dd52d8147066e31d8ea68815941e7c814bf12d98a63e67d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-01-14T18:02:32Z","title_canon_sha256":"9c3ded762de93a441a02a87d681207d90928ef87e26567bdc9f1b6ff41b0f8f3"},"schema_version":"1.0","source":{"id":"1601.03683","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03683","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03683v2","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03683","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"pith_short_12","alias_value":"6YBSLZVX773T","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6YBSLZVX773TWTBF","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6YBSLZVX","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:422026e40ec2e355a82f4f9f40d952556e5d3545249b74896c231d7848ba0b54","target":"graph","created_at":"2026-05-18T00:27:55Z","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":"For a finite group $G$, the proper power graph $\\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \\neq v$ and $u^m=v$ or $v^m=u$ for some positive integer $m$. In this paper, we consider the complement of $\\mathscr{P}^*(G)$, denoted by ${\\overline{\\mathscr{P}^*(G)}}$. We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a cycle, a star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective. Among the other results, we al","authors_text":"Andrei Gagarin, R. Rajkumar, T. Anitha","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-01-14T18:02:32Z","title":"The complement of proper power graphs of finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03683","kind":"arxiv","version":2},"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:c90f3784d5a5df8684087d5ddfa24581c3feadd7bbd9bc82ad3ccb89eb27aada","target":"record","created_at":"2026-05-18T00:27:55Z","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":"d004129b73c554a93dd52d8147066e31d8ea68815941e7c814bf12d98a63e67d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-01-14T18:02:32Z","title_canon_sha256":"9c3ded762de93a441a02a87d681207d90928ef87e26567bdc9f1b6ff41b0f8f3"},"schema_version":"1.0","source":{"id":"1601.03683","kind":"arxiv","version":2}},"canonical_sha256":"f60325e6b7fff73b4c255a259d93d7152f00ac3c8e8f34135e2c338557a849e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f60325e6b7fff73b4c255a259d93d7152f00ac3c8e8f34135e2c338557a849e6","first_computed_at":"2026-05-18T00:27:55.392029Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:55.392029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fRZCM+SZagLvEniRDFaeUG2620GbVnIQ46/0lZNJO9rJVFe21NvO3FaLUwcZaq4KTnoM3fA5cJhXCrfeGGioAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:55.392630Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.03683","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c90f3784d5a5df8684087d5ddfa24581c3feadd7bbd9bc82ad3ccb89eb27aada","sha256:422026e40ec2e355a82f4f9f40d952556e5d3545249b74896c231d7848ba0b54"],"state_sha256":"4833cd49661581c27cbee072be6bffb4160b9d954a2a2acda789ab7272f99877"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GFOJSYq2WY6qmrbmGrtCBox5vJhv2J1x0E69yUh+/00x9h4CQu7iKvK5S+VDtW79yKnNF6rCjQME1UhyzrZtAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T15:02:21.067090Z","bundle_sha256":"16a0a5209c9a0c1dbed94af06a9f43be8e0d4ef7ebbb4c2b0cec453c5cffd7ec"}}