{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:K3TMDKQLLEAZ4LPTOBN7UU3DEZ","short_pith_number":"pith:K3TMDKQL","canonical_record":{"source":{"id":"1305.2729","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-13T10:19:01Z","cross_cats_sorted":[],"title_canon_sha256":"b3e463aed9ce4b0c1f891baa38d0b2e4ec3c31754c74eb4cd6b8d5fcc47dda5b","abstract_canon_sha256":"2fcb3d607c85fa32aab72ad6e8e8a9a74205cf812d8aa2c7f4e603ea3f493f5e"},"schema_version":"1.0"},"canonical_sha256":"56e6c1aa0b59019e2df3705bfa53632670c53ce4fcb9e89ab8b32e8397940fa5","source":{"kind":"arxiv","id":"1305.2729","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2729","created_at":"2026-05-18T03:25:57Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2729v1","created_at":"2026-05-18T03:25:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2729","created_at":"2026-05-18T03:25:57Z"},{"alias_kind":"pith_short_12","alias_value":"K3TMDKQLLEAZ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"K3TMDKQLLEAZ4LPT","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"K3TMDKQL","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:K3TMDKQLLEAZ4LPTOBN7UU3DEZ","target":"record","payload":{"canonical_record":{"source":{"id":"1305.2729","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-13T10:19:01Z","cross_cats_sorted":[],"title_canon_sha256":"b3e463aed9ce4b0c1f891baa38d0b2e4ec3c31754c74eb4cd6b8d5fcc47dda5b","abstract_canon_sha256":"2fcb3d607c85fa32aab72ad6e8e8a9a74205cf812d8aa2c7f4e603ea3f493f5e"},"schema_version":"1.0"},"canonical_sha256":"56e6c1aa0b59019e2df3705bfa53632670c53ce4fcb9e89ab8b32e8397940fa5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:57.022935Z","signature_b64":"9RU3mnJ2nLCPbbB9+3J0+Z4BTd+QadrcvsTQdNwDOLkGii6+4k5YGvcdDYhyd4I9ivREy5cAOt130icdtn5jCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56e6c1aa0b59019e2df3705bfa53632670c53ce4fcb9e89ab8b32e8397940fa5","last_reissued_at":"2026-05-18T03:25:57.022461Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:57.022461Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.2729","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":"2026-05-18T03:25:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CVj0piGTquXM88rgqam/TR1e63pGEGaS34GMCRLMGw6l2e/dqPC+wjYssjgQHfMNpfWmt/daYFxNn9ZpBrodAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T08:22:06.761050Z"},"content_sha256":"801b152b281a962df0d5ffda71ee4dbcfbb80b50f58ba159fd9b0ac6a4d7a4f3","schema_version":"1.0","event_id":"sha256:801b152b281a962df0d5ffda71ee4dbcfbb80b50f58ba159fd9b0ac6a4d7a4f3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:K3TMDKQLLEAZ4LPTOBN7UU3DEZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Connectivity and other invariants of generalized products of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"F. A. Muntaner-Batle, S. C. L\\'opez","submitted_at":"2013-05-13T10:19:01Z","abstract_excerpt":"Figueroa-Centeno et al. introduced the following product of digraphs: let $D$ be a digraph and let $\\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\\in \\Gamma$. Consider any function $h:E(D)\\longrightarrow\\Gamma $. Then the product $D\\otimes_{h} \\Gamma$ is the digraph with vertex set $V(D)\\times V$ and $((a,x),(b,y))\\in E(D\\otimes_h\\Gamma)$ if and only if $(a,b)\\in E(D)$ and $(x,y)\\in E(h (a,b))$.\n  In this paper, we introduce the undirected version of the $\\otimes_h$-product, which is a generalization of the classical direct product of graphs and, motivated by it, we also recov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2729","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":"2026-05-18T03:25:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7kaSGBqQfzvmlu1rnnGKoioU91LRe6FVVFjXYtboweGxqL3+1xDRlidmzuS8/gFDbYtOtwqr8AFvUbM6cu+UAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T08:22:06.761414Z"},"content_sha256":"8a75662a07b4fb956de48002f8993a2b771ff4c333d4f0e2319c319b33b8606a","schema_version":"1.0","event_id":"sha256:8a75662a07b4fb956de48002f8993a2b771ff4c333d4f0e2319c319b33b8606a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K3TMDKQLLEAZ4LPTOBN7UU3DEZ/bundle.json","state_url":"https://pith.science/pith/K3TMDKQLLEAZ4LPTOBN7UU3DEZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K3TMDKQLLEAZ4LPTOBN7UU3DEZ/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":"2026-06-26T08:22:06Z","links":{"resolver":"https://pith.science/pith/K3TMDKQLLEAZ4LPTOBN7UU3DEZ","bundle":"https://pith.science/pith/K3TMDKQLLEAZ4LPTOBN7UU3DEZ/bundle.json","state":"https://pith.science/pith/K3TMDKQLLEAZ4LPTOBN7UU3DEZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K3TMDKQLLEAZ4LPTOBN7UU3DEZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:K3TMDKQLLEAZ4LPTOBN7UU3DEZ","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":"2fcb3d607c85fa32aab72ad6e8e8a9a74205cf812d8aa2c7f4e603ea3f493f5e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-13T10:19:01Z","title_canon_sha256":"b3e463aed9ce4b0c1f891baa38d0b2e4ec3c31754c74eb4cd6b8d5fcc47dda5b"},"schema_version":"1.0","source":{"id":"1305.2729","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2729","created_at":"2026-05-18T03:25:57Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2729v1","created_at":"2026-05-18T03:25:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2729","created_at":"2026-05-18T03:25:57Z"},{"alias_kind":"pith_short_12","alias_value":"K3TMDKQLLEAZ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"K3TMDKQLLEAZ4LPT","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"K3TMDKQL","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:8a75662a07b4fb956de48002f8993a2b771ff4c333d4f0e2319c319b33b8606a","target":"graph","created_at":"2026-05-18T03:25:57Z","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":"Figueroa-Centeno et al. introduced the following product of digraphs: let $D$ be a digraph and let $\\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\\in \\Gamma$. Consider any function $h:E(D)\\longrightarrow\\Gamma $. Then the product $D\\otimes_{h} \\Gamma$ is the digraph with vertex set $V(D)\\times V$ and $((a,x),(b,y))\\in E(D\\otimes_h\\Gamma)$ if and only if $(a,b)\\in E(D)$ and $(x,y)\\in E(h (a,b))$.\n  In this paper, we introduce the undirected version of the $\\otimes_h$-product, which is a generalization of the classical direct product of graphs and, motivated by it, we also recov","authors_text":"F. A. Muntaner-Batle, S. C. L\\'opez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-13T10:19:01Z","title":"Connectivity and other invariants of generalized products of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2729","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":"sha256:801b152b281a962df0d5ffda71ee4dbcfbb80b50f58ba159fd9b0ac6a4d7a4f3","target":"record","created_at":"2026-05-18T03:25:57Z","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":"2fcb3d607c85fa32aab72ad6e8e8a9a74205cf812d8aa2c7f4e603ea3f493f5e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-13T10:19:01Z","title_canon_sha256":"b3e463aed9ce4b0c1f891baa38d0b2e4ec3c31754c74eb4cd6b8d5fcc47dda5b"},"schema_version":"1.0","source":{"id":"1305.2729","kind":"arxiv","version":1}},"canonical_sha256":"56e6c1aa0b59019e2df3705bfa53632670c53ce4fcb9e89ab8b32e8397940fa5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"56e6c1aa0b59019e2df3705bfa53632670c53ce4fcb9e89ab8b32e8397940fa5","first_computed_at":"2026-05-18T03:25:57.022461Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:57.022461Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9RU3mnJ2nLCPbbB9+3J0+Z4BTd+QadrcvsTQdNwDOLkGii6+4k5YGvcdDYhyd4I9ivREy5cAOt130icdtn5jCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:57.022935Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.2729","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:801b152b281a962df0d5ffda71ee4dbcfbb80b50f58ba159fd9b0ac6a4d7a4f3","sha256:8a75662a07b4fb956de48002f8993a2b771ff4c333d4f0e2319c319b33b8606a"],"state_sha256":"e27639d931c1eca7fd798e578c1b1b5b9c304bcf2166d4fd9f66d520fa202d76"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1iNYR2iCoXTbLgz103n/uA+LsuNNAEhxuK7tTOwccAomHvC0v62DsSETmSzW2Gwvs0NRc6f7DCFjGXGSvFSSCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T08:22:06.763303Z","bundle_sha256":"886c7d5331ba56e8b47e96c721562260601a2f41f1319c5595c83d6b124fadb3"}}