{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XVHICZGHHUFQRASUHLVHSNV2ET","short_pith_number":"pith:XVHICZGH","canonical_record":{"source":{"id":"1611.07822","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-23T14:43:50Z","cross_cats_sorted":[],"title_canon_sha256":"69e7661b5256f2464d455949d38fb6be926db8756b5aae516dd974d01ae5673e","abstract_canon_sha256":"5bf128765a0f36210d62166a628270c97a069cb333a2ff134711eae8eb5f2f4a"},"schema_version":"1.0"},"canonical_sha256":"bd4e8164c73d0b0882543aea7936ba24d0b06051ff178c73a9de7261904bff7c","source":{"kind":"arxiv","id":"1611.07822","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07822","created_at":"2026-05-18T00:53:24Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07822v2","created_at":"2026-05-18T00:53:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07822","created_at":"2026-05-18T00:53:24Z"},{"alias_kind":"pith_short_12","alias_value":"XVHICZGHHUFQ","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XVHICZGHHUFQRASU","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XVHICZGH","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XVHICZGHHUFQRASUHLVHSNV2ET","target":"record","payload":{"canonical_record":{"source":{"id":"1611.07822","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-23T14:43:50Z","cross_cats_sorted":[],"title_canon_sha256":"69e7661b5256f2464d455949d38fb6be926db8756b5aae516dd974d01ae5673e","abstract_canon_sha256":"5bf128765a0f36210d62166a628270c97a069cb333a2ff134711eae8eb5f2f4a"},"schema_version":"1.0"},"canonical_sha256":"bd4e8164c73d0b0882543aea7936ba24d0b06051ff178c73a9de7261904bff7c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:24.002609Z","signature_b64":"UhuuJY+A5c9yT3KmTyh837vkg7DSLjNS+y+RcZ0UD3uo108ULywq4cfAWTHtFsOQxhWhN2V320kuqPlxc+2zDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd4e8164c73d0b0882543aea7936ba24d0b06051ff178c73a9de7261904bff7c","last_reissued_at":"2026-05-18T00:53:24.002109Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:24.002109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.07822","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":"2026-05-18T00:53:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6zyLiI777IL2ov2SVQ4dMlCeF8EBn/mEI0/NbKwLZ3eUFHq+DURTai6XRamv9ZDstUga2zjS00/grYHgcdGqCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:41:16.593362Z"},"content_sha256":"d99f2031d2aa1321d94c9ad370bcfa7768b904d94fca975787ea8303f7d0e950","schema_version":"1.0","event_id":"sha256:d99f2031d2aa1321d94c9ad370bcfa7768b904d94fca975787ea8303f7d0e950"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XVHICZGHHUFQRASUHLVHSNV2ET","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The power index of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kaishun Wang, Min Feng, Xuanlong Ma","submitted_at":"2016-11-23T14:43:50Z","abstract_excerpt":"The {\\em power index} $\\Theta(\\Gamma)$ of a graph $\\Gamma$ is the least order of a group $G$ such that $\\Gamma$ can embed into the power graph of $G$. Furthermore, this group $G$ is {\\em $\\Gamma$-optimal} if $G$ has order $\\Theta(\\Gamma)$. We say that $\\Gamma$ is {\\em power-critical} if its order equals to $\\Theta(\\Gamma)$. This paper focuses on the power indices of complete graphs, complete bipartite graphs and $1$-factors. We classify all power-critical graphs $\\Gamma'$ in these three families, and give a necessary and sufficient condition for $\\Gamma'$-optimal groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07822","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":"2026-05-18T00:53:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W2D2li9NY3Ve3bGHI+CpLrkNpoie617Rfkzoabz/DFxksEtREWU6ckujaY4u2z/SxWh+mZhKhkw8yPIEDfXgAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:41:16.593709Z"},"content_sha256":"7f13c2b943704a382dc910512a59751a1207d260c7a10c81c9eff22ecb9c39b8","schema_version":"1.0","event_id":"sha256:7f13c2b943704a382dc910512a59751a1207d260c7a10c81c9eff22ecb9c39b8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XVHICZGHHUFQRASUHLVHSNV2ET/bundle.json","state_url":"https://pith.science/pith/XVHICZGHHUFQRASUHLVHSNV2ET/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XVHICZGHHUFQRASUHLVHSNV2ET/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":"2026-06-04T19:41:16Z","links":{"resolver":"https://pith.science/pith/XVHICZGHHUFQRASUHLVHSNV2ET","bundle":"https://pith.science/pith/XVHICZGHHUFQRASUHLVHSNV2ET/bundle.json","state":"https://pith.science/pith/XVHICZGHHUFQRASUHLVHSNV2ET/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XVHICZGHHUFQRASUHLVHSNV2ET/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XVHICZGHHUFQRASUHLVHSNV2ET","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":"5bf128765a0f36210d62166a628270c97a069cb333a2ff134711eae8eb5f2f4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-23T14:43:50Z","title_canon_sha256":"69e7661b5256f2464d455949d38fb6be926db8756b5aae516dd974d01ae5673e"},"schema_version":"1.0","source":{"id":"1611.07822","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07822","created_at":"2026-05-18T00:53:24Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07822v2","created_at":"2026-05-18T00:53:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07822","created_at":"2026-05-18T00:53:24Z"},{"alias_kind":"pith_short_12","alias_value":"XVHICZGHHUFQ","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XVHICZGHHUFQRASU","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XVHICZGH","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:7f13c2b943704a382dc910512a59751a1207d260c7a10c81c9eff22ecb9c39b8","target":"graph","created_at":"2026-05-18T00:53:24Z","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":"The {\\em power index} $\\Theta(\\Gamma)$ of a graph $\\Gamma$ is the least order of a group $G$ such that $\\Gamma$ can embed into the power graph of $G$. Furthermore, this group $G$ is {\\em $\\Gamma$-optimal} if $G$ has order $\\Theta(\\Gamma)$. We say that $\\Gamma$ is {\\em power-critical} if its order equals to $\\Theta(\\Gamma)$. This paper focuses on the power indices of complete graphs, complete bipartite graphs and $1$-factors. We classify all power-critical graphs $\\Gamma'$ in these three families, and give a necessary and sufficient condition for $\\Gamma'$-optimal groups.","authors_text":"Kaishun Wang, Min Feng, Xuanlong Ma","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-23T14:43:50Z","title":"The power index of a graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07822","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:d99f2031d2aa1321d94c9ad370bcfa7768b904d94fca975787ea8303f7d0e950","target":"record","created_at":"2026-05-18T00:53:24Z","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":"5bf128765a0f36210d62166a628270c97a069cb333a2ff134711eae8eb5f2f4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-23T14:43:50Z","title_canon_sha256":"69e7661b5256f2464d455949d38fb6be926db8756b5aae516dd974d01ae5673e"},"schema_version":"1.0","source":{"id":"1611.07822","kind":"arxiv","version":2}},"canonical_sha256":"bd4e8164c73d0b0882543aea7936ba24d0b06051ff178c73a9de7261904bff7c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd4e8164c73d0b0882543aea7936ba24d0b06051ff178c73a9de7261904bff7c","first_computed_at":"2026-05-18T00:53:24.002109Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:24.002109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UhuuJY+A5c9yT3KmTyh837vkg7DSLjNS+y+RcZ0UD3uo108ULywq4cfAWTHtFsOQxhWhN2V320kuqPlxc+2zDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:24.002609Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.07822","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d99f2031d2aa1321d94c9ad370bcfa7768b904d94fca975787ea8303f7d0e950","sha256:7f13c2b943704a382dc910512a59751a1207d260c7a10c81c9eff22ecb9c39b8"],"state_sha256":"3c40f96c44541b87635aa3c3ded3a2502a60fb7d5436108ba2de2c8fc40d40a3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PnAs0lgKDEuJuZR6FuyOENVrzHXl7MUIcZpkmaOJYhhGcilb6k3B4zs1IcRf047BzQJaZVZKl3EKS6GlfBKdCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T19:41:16.595666Z","bundle_sha256":"7f9059bca3605c449e7c8429d2ce327c5442ebd53088517bd7cc3e8738d56f18"}}