{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:RFVTXR62ZAEKTLCT5G3LUJXKXK","short_pith_number":"pith:RFVTXR62","canonical_record":{"source":{"id":"1804.08808","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-24T01:58:18Z","cross_cats_sorted":[],"title_canon_sha256":"7f4b5b2817f562d5b673386389b1e0162fe2b48401a0747ab885fdda5c380766","abstract_canon_sha256":"f7305d2d64cf8d64b802fde2274613cc72d2ab91ba69332572833304abcfe70f"},"schema_version":"1.0"},"canonical_sha256":"896b3bc7dac808a9ac53e9b6ba26eaba9e648ee5a31792e0e7c1816862d21cb0","source":{"kind":"arxiv","id":"1804.08808","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.08808","created_at":"2026-05-18T00:17:37Z"},{"alias_kind":"arxiv_version","alias_value":"1804.08808v1","created_at":"2026-05-18T00:17:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.08808","created_at":"2026-05-18T00:17:37Z"},{"alias_kind":"pith_short_12","alias_value":"RFVTXR62ZAEK","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RFVTXR62ZAEKTLCT","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RFVTXR62","created_at":"2026-05-18T12:32:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:RFVTXR62ZAEKTLCT5G3LUJXKXK","target":"record","payload":{"canonical_record":{"source":{"id":"1804.08808","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-24T01:58:18Z","cross_cats_sorted":[],"title_canon_sha256":"7f4b5b2817f562d5b673386389b1e0162fe2b48401a0747ab885fdda5c380766","abstract_canon_sha256":"f7305d2d64cf8d64b802fde2274613cc72d2ab91ba69332572833304abcfe70f"},"schema_version":"1.0"},"canonical_sha256":"896b3bc7dac808a9ac53e9b6ba26eaba9e648ee5a31792e0e7c1816862d21cb0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:37.274829Z","signature_b64":"AJzxuLYefoedgeCQ4UJxu+oyLyyDpM9Qn5yUlqh3/qgIiF8nvIfI8D/mREfKbo/GvvOeT6uR761/YUogO1N8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"896b3bc7dac808a9ac53e9b6ba26eaba9e648ee5a31792e0e7c1816862d21cb0","last_reissued_at":"2026-05-18T00:17:37.274121Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:37.274121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.08808","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":"2026-05-18T00:17:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YarnWk2GCvyy/iG5yidfpBJkS4Bky/+HAbub6OXrGS/8osKGabK19tp0cHty/7BzS2Z+yW3EQwmw8m/R17HIDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T19:38:39.974562Z"},"content_sha256":"ee1a1f612981384e84398fa60e670910827aeb3db76428b0b52292f2e016c827","schema_version":"1.0","event_id":"sha256:ee1a1f612981384e84398fa60e670910827aeb3db76428b0b52292f2e016c827"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:RFVTXR62ZAEKTLCT5G3LUJXKXK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The $(p,q)$-spectral radii of $(r,s)$-directed hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lele Liu, Linyuan Lu","submitted_at":"2018-04-24T01:58:18Z","abstract_excerpt":"An $(r,s)$-directed hypergraph is a directed hypergraph with $r$ vertices in tail and $s$ vertices in head of each arc. Let $G$ be an $(r,s)$-directed hypergraph. For any real numbers $p$, $q\\geq 1$, we define the $(p,q)$-spectral radius $\\lambda_{p,q}(G)$ as \\[ \\lambda_{p,q}(G):=\\max_{||{\\bf x}||_p=||{\\bf y}||_q=1} \\sum_{e\\in E(G)}\\Bigg(\\prod_{u\\in T(e)}x_u\\Bigg)\\Bigg(\\prod_{v\\in H(e)}y_v\\Bigg), \\] where ${\\bf x}=(x_1, \\ldots, x_m)^{{\\rm T}}$, ${\\bf y}=(y_1,\\ldots, y_n)^{{\\rm T}}$ are real vectors; and $T(e)$, $H(e)$ are the tail and head of arc $e$, respectively. We study some properties abo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08808","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":"2026-05-18T00:17:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MBo1hLJB5A/enc57wFl1Pu4uMJm1kiRmgE7EOMKJ9FQAX4Fa8FffpMxeRMA2dzCeWVZQYHdmXXq3+3kfaqJWCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T19:38:39.974902Z"},"content_sha256":"cea182199d570cdfee946eefd092023766948711f79e67dd7fe6520ecbb38210","schema_version":"1.0","event_id":"sha256:cea182199d570cdfee946eefd092023766948711f79e67dd7fe6520ecbb38210"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RFVTXR62ZAEKTLCT5G3LUJXKXK/bundle.json","state_url":"https://pith.science/pith/RFVTXR62ZAEKTLCT5G3LUJXKXK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RFVTXR62ZAEKTLCT5G3LUJXKXK/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":"2026-06-01T19:38:39Z","links":{"resolver":"https://pith.science/pith/RFVTXR62ZAEKTLCT5G3LUJXKXK","bundle":"https://pith.science/pith/RFVTXR62ZAEKTLCT5G3LUJXKXK/bundle.json","state":"https://pith.science/pith/RFVTXR62ZAEKTLCT5G3LUJXKXK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RFVTXR62ZAEKTLCT5G3LUJXKXK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:RFVTXR62ZAEKTLCT5G3LUJXKXK","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":"f7305d2d64cf8d64b802fde2274613cc72d2ab91ba69332572833304abcfe70f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-24T01:58:18Z","title_canon_sha256":"7f4b5b2817f562d5b673386389b1e0162fe2b48401a0747ab885fdda5c380766"},"schema_version":"1.0","source":{"id":"1804.08808","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.08808","created_at":"2026-05-18T00:17:37Z"},{"alias_kind":"arxiv_version","alias_value":"1804.08808v1","created_at":"2026-05-18T00:17:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.08808","created_at":"2026-05-18T00:17:37Z"},{"alias_kind":"pith_short_12","alias_value":"RFVTXR62ZAEK","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RFVTXR62ZAEKTLCT","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RFVTXR62","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:cea182199d570cdfee946eefd092023766948711f79e67dd7fe6520ecbb38210","target":"graph","created_at":"2026-05-18T00:17:37Z","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":"An $(r,s)$-directed hypergraph is a directed hypergraph with $r$ vertices in tail and $s$ vertices in head of each arc. Let $G$ be an $(r,s)$-directed hypergraph. For any real numbers $p$, $q\\geq 1$, we define the $(p,q)$-spectral radius $\\lambda_{p,q}(G)$ as \\[ \\lambda_{p,q}(G):=\\max_{||{\\bf x}||_p=||{\\bf y}||_q=1} \\sum_{e\\in E(G)}\\Bigg(\\prod_{u\\in T(e)}x_u\\Bigg)\\Bigg(\\prod_{v\\in H(e)}y_v\\Bigg), \\] where ${\\bf x}=(x_1, \\ldots, x_m)^{{\\rm T}}$, ${\\bf y}=(y_1,\\ldots, y_n)^{{\\rm T}}$ are real vectors; and $T(e)$, $H(e)$ are the tail and head of arc $e$, respectively. We study some properties abo","authors_text":"Lele Liu, Linyuan Lu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-24T01:58:18Z","title":"The $(p,q)$-spectral radii of $(r,s)$-directed hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08808","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:ee1a1f612981384e84398fa60e670910827aeb3db76428b0b52292f2e016c827","target":"record","created_at":"2026-05-18T00:17:37Z","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":"f7305d2d64cf8d64b802fde2274613cc72d2ab91ba69332572833304abcfe70f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-24T01:58:18Z","title_canon_sha256":"7f4b5b2817f562d5b673386389b1e0162fe2b48401a0747ab885fdda5c380766"},"schema_version":"1.0","source":{"id":"1804.08808","kind":"arxiv","version":1}},"canonical_sha256":"896b3bc7dac808a9ac53e9b6ba26eaba9e648ee5a31792e0e7c1816862d21cb0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"896b3bc7dac808a9ac53e9b6ba26eaba9e648ee5a31792e0e7c1816862d21cb0","first_computed_at":"2026-05-18T00:17:37.274121Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:37.274121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AJzxuLYefoedgeCQ4UJxu+oyLyyDpM9Qn5yUlqh3/qgIiF8nvIfI8D/mREfKbo/GvvOeT6uR761/YUogO1N8CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:37.274829Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.08808","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee1a1f612981384e84398fa60e670910827aeb3db76428b0b52292f2e016c827","sha256:cea182199d570cdfee946eefd092023766948711f79e67dd7fe6520ecbb38210"],"state_sha256":"339f7c6f9cd93e70e5fca411a5806835f2f37e69551035ee5be452ee83c3404c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wVlJq/jXTg2FbOXsHxG2yOjNEhGsSRZIsmb6759PhXPHPcfy8hWn/vckJk1P7ixVZ5ucwSMvjqgFYZqmGV06Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T19:38:39.976780Z","bundle_sha256":"4fb18bc347f28b93e82a897154766d4af351535388b3190e414073869124580b"}}