{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:H22QUOWXM3DQTP6HZIUP2ALBSQ","short_pith_number":"pith:H22QUOWX","canonical_record":{"source":{"id":"1609.04692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-15T15:10:50Z","cross_cats_sorted":[],"title_canon_sha256":"a7c436a8e5ba32ce52016615826c97c472cb073925388c46e62a4f7675e799f2","abstract_canon_sha256":"607195efb9d0b255b17747653d5000ccf04a273c90da42c0cc68330dc8cd5f6a"},"schema_version":"1.0"},"canonical_sha256":"3eb50a3ad766c709bfc7ca28fd0161940704cf722051b442a175c7618ada3d99","source":{"kind":"arxiv","id":"1609.04692","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.04692","created_at":"2026-05-18T00:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"1609.04692v1","created_at":"2026-05-18T00:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04692","created_at":"2026-05-18T00:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"H22QUOWXM3DQ","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"H22QUOWXM3DQTP6H","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"H22QUOWX","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:H22QUOWXM3DQTP6HZIUP2ALBSQ","target":"record","payload":{"canonical_record":{"source":{"id":"1609.04692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-15T15:10:50Z","cross_cats_sorted":[],"title_canon_sha256":"a7c436a8e5ba32ce52016615826c97c472cb073925388c46e62a4f7675e799f2","abstract_canon_sha256":"607195efb9d0b255b17747653d5000ccf04a273c90da42c0cc68330dc8cd5f6a"},"schema_version":"1.0"},"canonical_sha256":"3eb50a3ad766c709bfc7ca28fd0161940704cf722051b442a175c7618ada3d99","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:18.938272Z","signature_b64":"xmNVCXgqRF7guuVORDZjdV7lvWdzV2apoWkUH6tYhLKCtBAcMg43gPpiIWc7FPHTndvidmW77nwkXIKMBIZ0BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3eb50a3ad766c709bfc7ca28fd0161940704cf722051b442a175c7618ada3d99","last_reissued_at":"2026-05-18T00:07:18.937652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:18.937652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.04692","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":"2026-05-18T00:07:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jbBploXZ6X5+EJuPt48F0Vqr2ZjzH09MOIIItb8GtitoNbTbDMigIRhSh5vQUSWBelQRIcP4xDgEH1MiHbyTDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:58:03.210669Z"},"content_sha256":"b6c7d22135e9ca5136675945f1120d4c21ba5a5d11f788d635f6cf7742e8d298","schema_version":"1.0","event_id":"sha256:b6c7d22135e9ca5136675945f1120d4c21ba5a5d11f788d635f6cf7742e8d298"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:H22QUOWXM3DQTP6HZIUP2ALBSQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Method for Computing the Edge-Hyper-Wiener Index of Partial Cubes and an Algorithm for Benzenoid Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Niko Tratnik","submitted_at":"2016-09-15T15:10:50Z","abstract_excerpt":"The edge-hyper-Wiener index of a connected graph $G$ is defined as $WW_e(G) = \\frac{1}{2}\\sum_{\\lbrace e,f\\rbrace \\subseteq E(G)}d(e,f) + \\frac{1}{2}\\sum_{\\lbrace e,f\\rbrace \\subseteq E(G)}d(e,f)^2$. We develop a method for computing the edge-hyper-Wiener index of partial cubes, which constitute a large class of graphs with a lot of applications. It is also shown how the method can be applied to trees. Furthermore, an algorithm for computing the edge-hyper-Wiener index of benzenoid systems is obtained. Finally, the algorithm is used to correct already known closed formulas for the edge-Wiener "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04692","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":"2026-05-18T00:07:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kCvygh4T2CrFIHzc7LDnB3vwEpnqbakm48TiIPXo5HUKQ13yoRX+4hq6qjFVay54kLHiqAoSC+gmwi5rN9fSAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:58:03.211009Z"},"content_sha256":"43a49266177007d5d7b06b69d7ad3c0ce459006708c8678a68d716bf75db91e1","schema_version":"1.0","event_id":"sha256:43a49266177007d5d7b06b69d7ad3c0ce459006708c8678a68d716bf75db91e1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H22QUOWXM3DQTP6HZIUP2ALBSQ/bundle.json","state_url":"https://pith.science/pith/H22QUOWXM3DQTP6HZIUP2ALBSQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H22QUOWXM3DQTP6HZIUP2ALBSQ/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":"2026-06-08T14:58:03Z","links":{"resolver":"https://pith.science/pith/H22QUOWXM3DQTP6HZIUP2ALBSQ","bundle":"https://pith.science/pith/H22QUOWXM3DQTP6HZIUP2ALBSQ/bundle.json","state":"https://pith.science/pith/H22QUOWXM3DQTP6HZIUP2ALBSQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H22QUOWXM3DQTP6HZIUP2ALBSQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:H22QUOWXM3DQTP6HZIUP2ALBSQ","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":"607195efb9d0b255b17747653d5000ccf04a273c90da42c0cc68330dc8cd5f6a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-15T15:10:50Z","title_canon_sha256":"a7c436a8e5ba32ce52016615826c97c472cb073925388c46e62a4f7675e799f2"},"schema_version":"1.0","source":{"id":"1609.04692","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.04692","created_at":"2026-05-18T00:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"1609.04692v1","created_at":"2026-05-18T00:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04692","created_at":"2026-05-18T00:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"H22QUOWXM3DQ","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"H22QUOWXM3DQTP6H","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"H22QUOWX","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:43a49266177007d5d7b06b69d7ad3c0ce459006708c8678a68d716bf75db91e1","target":"graph","created_at":"2026-05-18T00:07:18Z","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 edge-hyper-Wiener index of a connected graph $G$ is defined as $WW_e(G) = \\frac{1}{2}\\sum_{\\lbrace e,f\\rbrace \\subseteq E(G)}d(e,f) + \\frac{1}{2}\\sum_{\\lbrace e,f\\rbrace \\subseteq E(G)}d(e,f)^2$. We develop a method for computing the edge-hyper-Wiener index of partial cubes, which constitute a large class of graphs with a lot of applications. It is also shown how the method can be applied to trees. Furthermore, an algorithm for computing the edge-hyper-Wiener index of benzenoid systems is obtained. Finally, the algorithm is used to correct already known closed formulas for the edge-Wiener ","authors_text":"Niko Tratnik","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-15T15:10:50Z","title":"A Method for Computing the Edge-Hyper-Wiener Index of Partial Cubes and an Algorithm for Benzenoid Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04692","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:b6c7d22135e9ca5136675945f1120d4c21ba5a5d11f788d635f6cf7742e8d298","target":"record","created_at":"2026-05-18T00:07:18Z","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":"607195efb9d0b255b17747653d5000ccf04a273c90da42c0cc68330dc8cd5f6a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-15T15:10:50Z","title_canon_sha256":"a7c436a8e5ba32ce52016615826c97c472cb073925388c46e62a4f7675e799f2"},"schema_version":"1.0","source":{"id":"1609.04692","kind":"arxiv","version":1}},"canonical_sha256":"3eb50a3ad766c709bfc7ca28fd0161940704cf722051b442a175c7618ada3d99","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3eb50a3ad766c709bfc7ca28fd0161940704cf722051b442a175c7618ada3d99","first_computed_at":"2026-05-18T00:07:18.937652Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:18.937652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xmNVCXgqRF7guuVORDZjdV7lvWdzV2apoWkUH6tYhLKCtBAcMg43gPpiIWc7FPHTndvidmW77nwkXIKMBIZ0BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:18.938272Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.04692","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b6c7d22135e9ca5136675945f1120d4c21ba5a5d11f788d635f6cf7742e8d298","sha256:43a49266177007d5d7b06b69d7ad3c0ce459006708c8678a68d716bf75db91e1"],"state_sha256":"e1a2d4802c0212504e9de77c84d5c1fa21ee554fadc43bcb481cbbb861eec9d6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"28+XDEU9z8cmNo8XVu7451yXDjB0eZDFIzTDrc1Z50gOQMFObN6gtADcYGSVB5pLEOW+xv8wNFEeRAZsltyZAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T14:58:03.213174Z","bundle_sha256":"0db2772ced14147cf85f017fe398a5b368f15a296df0de2eee16e01e28eba09e"}}