{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:MLAN2AWIQNORJ2W4JZVGEANVOL","short_pith_number":"pith:MLAN2AWI","canonical_record":{"source":{"id":"1611.07137","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-22T03:25:27Z","cross_cats_sorted":[],"title_canon_sha256":"0e3dfd9a1f9e0c3a5d726523230395f0449caf741ba0ac8e62da68b5d3594d98","abstract_canon_sha256":"fecb4001e14485b023669528f8d3090ebe3cdcf85458daeab62315945cae4504"},"schema_version":"1.0"},"canonical_sha256":"62c0dd02c8835d14eadc4e6a6201b572e59a6f7bebac5ff3baf974677c856fa6","source":{"kind":"arxiv","id":"1611.07137","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07137","created_at":"2026-05-18T00:46:04Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07137v2","created_at":"2026-05-18T00:46:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07137","created_at":"2026-05-18T00:46:04Z"},{"alias_kind":"pith_short_12","alias_value":"MLAN2AWIQNOR","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MLAN2AWIQNORJ2W4","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MLAN2AWI","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:MLAN2AWIQNORJ2W4JZVGEANVOL","target":"record","payload":{"canonical_record":{"source":{"id":"1611.07137","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-22T03:25:27Z","cross_cats_sorted":[],"title_canon_sha256":"0e3dfd9a1f9e0c3a5d726523230395f0449caf741ba0ac8e62da68b5d3594d98","abstract_canon_sha256":"fecb4001e14485b023669528f8d3090ebe3cdcf85458daeab62315945cae4504"},"schema_version":"1.0"},"canonical_sha256":"62c0dd02c8835d14eadc4e6a6201b572e59a6f7bebac5ff3baf974677c856fa6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:04.260573Z","signature_b64":"7qCVB1B5PwO2fCQSffY19rqcnX7W4acU2JeWBrys9i0uukQlK3qdWzX0r18+HAdnsL8MVsT1wJIFpe8gcH2PAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62c0dd02c8835d14eadc4e6a6201b572e59a6f7bebac5ff3baf974677c856fa6","last_reissued_at":"2026-05-18T00:46:04.260110Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:04.260110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.07137","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":"2026-05-18T00:46:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OePtgWZGtQNdhZIJiBkH9s0mY25YYdv1k6hgaHDTv66stfk8iYSoUsBJ7HogaQLc0v7GYzliI59dWI5BzUaRAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:43:41.138561Z"},"content_sha256":"cfe31fc773ac1c4b73b8bd58fa74a44b2d4319d0374869865bde43610de2ff7b","schema_version":"1.0","event_id":"sha256:cfe31fc773ac1c4b73b8bd58fa74a44b2d4319d0374869865bde43610de2ff7b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:MLAN2AWIQNORJ2W4JZVGEANVOL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chunxiang Wang, Jia-Bao Liu, Lin Chen, Shaohui Wang","submitted_at":"2016-11-22T03:25:27Z","abstract_excerpt":"The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of $\\prod_1(G) $ and $\\prod_2(G) $ of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07137","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":"2026-05-18T00:46:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"enwezboRMJToAM8dm8DFrE6419FNgfH5pvruSo+gcyT4uZcGH8SK36iKsqobaiEvov9I2D29z3wb1SiLsvAmDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:43:41.139107Z"},"content_sha256":"e806060f35016a7a385415fbec881d01dde5fa7ed5a893977c715afed9894a66","schema_version":"1.0","event_id":"sha256:e806060f35016a7a385415fbec881d01dde5fa7ed5a893977c715afed9894a66"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MLAN2AWIQNORJ2W4JZVGEANVOL/bundle.json","state_url":"https://pith.science/pith/MLAN2AWIQNORJ2W4JZVGEANVOL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MLAN2AWIQNORJ2W4JZVGEANVOL/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":"2026-05-27T04:43:41Z","links":{"resolver":"https://pith.science/pith/MLAN2AWIQNORJ2W4JZVGEANVOL","bundle":"https://pith.science/pith/MLAN2AWIQNORJ2W4JZVGEANVOL/bundle.json","state":"https://pith.science/pith/MLAN2AWIQNORJ2W4JZVGEANVOL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MLAN2AWIQNORJ2W4JZVGEANVOL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MLAN2AWIQNORJ2W4JZVGEANVOL","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":"fecb4001e14485b023669528f8d3090ebe3cdcf85458daeab62315945cae4504","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-22T03:25:27Z","title_canon_sha256":"0e3dfd9a1f9e0c3a5d726523230395f0449caf741ba0ac8e62da68b5d3594d98"},"schema_version":"1.0","source":{"id":"1611.07137","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07137","created_at":"2026-05-18T00:46:04Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07137v2","created_at":"2026-05-18T00:46:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07137","created_at":"2026-05-18T00:46:04Z"},{"alias_kind":"pith_short_12","alias_value":"MLAN2AWIQNOR","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MLAN2AWIQNORJ2W4","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MLAN2AWI","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:e806060f35016a7a385415fbec881d01dde5fa7ed5a893977c715afed9894a66","target":"graph","created_at":"2026-05-18T00:46:04Z","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 first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of $\\prod_1(G) $ and $\\prod_2(G) $ of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized.","authors_text":"Chunxiang Wang, Jia-Bao Liu, Lin Chen, Shaohui Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-22T03:25:27Z","title":"On extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07137","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:cfe31fc773ac1c4b73b8bd58fa74a44b2d4319d0374869865bde43610de2ff7b","target":"record","created_at":"2026-05-18T00:46:04Z","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":"fecb4001e14485b023669528f8d3090ebe3cdcf85458daeab62315945cae4504","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-22T03:25:27Z","title_canon_sha256":"0e3dfd9a1f9e0c3a5d726523230395f0449caf741ba0ac8e62da68b5d3594d98"},"schema_version":"1.0","source":{"id":"1611.07137","kind":"arxiv","version":2}},"canonical_sha256":"62c0dd02c8835d14eadc4e6a6201b572e59a6f7bebac5ff3baf974677c856fa6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62c0dd02c8835d14eadc4e6a6201b572e59a6f7bebac5ff3baf974677c856fa6","first_computed_at":"2026-05-18T00:46:04.260110Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:04.260110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7qCVB1B5PwO2fCQSffY19rqcnX7W4acU2JeWBrys9i0uukQlK3qdWzX0r18+HAdnsL8MVsT1wJIFpe8gcH2PAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:04.260573Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.07137","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cfe31fc773ac1c4b73b8bd58fa74a44b2d4319d0374869865bde43610de2ff7b","sha256:e806060f35016a7a385415fbec881d01dde5fa7ed5a893977c715afed9894a66"],"state_sha256":"2363e141b5c087d5116c562d4f17ff3a4e83d39f2ad71ccecf8cfeff72a168e6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ltr/gqHWCbJJGu8fSQ6EfWA5a9VahTxngXP62D1nbOBlHln76DDRCjUiKk/DZZP1h2JzPFKm3ebvzt3gtRwJAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T04:43:41.142880Z","bundle_sha256":"0e0fd3af75511722f85ced5016de2c2570f946cb779240b0b91b3abc878bf58c"}}