{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:XAXEVE46272KY3AZ6LVIPM4SWP","short_pith_number":"pith:XAXEVE46","canonical_record":{"source":{"id":"0704.1782","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-04-13T15:34:43Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"220610006bad3ebc0bff7123d6ff87c78019e5f07551ec5d614b414073662253","abstract_canon_sha256":"b2a17afb983a37d08c61f5a010ef3fa2c35a8e54c5deec4864da62af476ba85c"},"schema_version":"1.0"},"canonical_sha256":"b82e4a939ed7f4ac6c19f2ea87b392b3f37c32514db413f31e88bcac9eb76555","source":{"kind":"arxiv","id":"0704.1782","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0704.1782","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"arxiv_version","alias_value":"0704.1782v1","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.1782","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"pith_short_12","alias_value":"XAXEVE46272K","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"pith_short_16","alias_value":"XAXEVE46272KY3AZ","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"pith_short_8","alias_value":"XAXEVE46","created_at":"2026-06-03T22:06:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:XAXEVE46272KY3AZ6LVIPM4SWP","target":"record","payload":{"canonical_record":{"source":{"id":"0704.1782","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-04-13T15:34:43Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"220610006bad3ebc0bff7123d6ff87c78019e5f07551ec5d614b414073662253","abstract_canon_sha256":"b2a17afb983a37d08c61f5a010ef3fa2c35a8e54c5deec4864da62af476ba85c"},"schema_version":"1.0"},"canonical_sha256":"b82e4a939ed7f4ac6c19f2ea87b392b3f37c32514db413f31e88bcac9eb76555","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T22:06:00.447176Z","signature_b64":"70xDMAly1uz+/DnF8euyWxIUFG7a21OQ7gLgnoVGYbQa0KTPywZC2VgKL1FMqbxvNLx+Gsi/lsBTJcXRf+waDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b82e4a939ed7f4ac6c19f2ea87b392b3f37c32514db413f31e88bcac9eb76555","last_reissued_at":"2026-06-03T22:06:00.446693Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T22:06:00.446693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0704.1782","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":"2026-06-03T22:06:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eRL3C/bQp68qezFZcDGgt/NoCRx6S7PgpbEs38ZDBXQe9lnxZNueRxZryFU1BK0gsClfgHbc9JkFvzxmJW94CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T01:23:15.610235Z"},"content_sha256":"a7294e74ba8a1b6cc04c94923040cd561ac72cccfa15fe7ed703889f16c85ec4","schema_version":"1.0","event_id":"sha256:a7294e74ba8a1b6cc04c94923040cd561ac72cccfa15fe7ed703889f16c85ec4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:XAXEVE46272KY3AZ6LVIPM4SWP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotics of the Euler number of bipartite graphs","license":"","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.CO","authors_text":"Richard Ehrenborg, Yossi Farjoun","submitted_at":"2007-04-13T15:34:43Z","abstract_excerpt":"We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph $G$ with the path $P_m$ in terms of self adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product $P_2 \\Box P_m$, we numerically solve the eigenv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.1782","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0704.1782/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-03T22:06:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"arFB1lPZTUyLHd65XpMZfiiQ6lkPXWQaVXhXbFDzLrOLIX0AIIt8bUHdooGp5Mb0hvyx3fUgL+tDCJcWImtzBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T01:23:15.610635Z"},"content_sha256":"01abc8983d05d88d141d94d92c6f1518be1ce447086efb5099b70b5a732c2147","schema_version":"1.0","event_id":"sha256:01abc8983d05d88d141d94d92c6f1518be1ce447086efb5099b70b5a732c2147"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XAXEVE46272KY3AZ6LVIPM4SWP/bundle.json","state_url":"https://pith.science/pith/XAXEVE46272KY3AZ6LVIPM4SWP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XAXEVE46272KY3AZ6LVIPM4SWP/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":"2026-06-24T01:23:15Z","links":{"resolver":"https://pith.science/pith/XAXEVE46272KY3AZ6LVIPM4SWP","bundle":"https://pith.science/pith/XAXEVE46272KY3AZ6LVIPM4SWP/bundle.json","state":"https://pith.science/pith/XAXEVE46272KY3AZ6LVIPM4SWP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XAXEVE46272KY3AZ6LVIPM4SWP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:XAXEVE46272KY3AZ6LVIPM4SWP","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":"b2a17afb983a37d08c61f5a010ef3fa2c35a8e54c5deec4864da62af476ba85c","cross_cats_sorted":["cs.NA","math.NA"],"license":"","primary_cat":"math.CO","submitted_at":"2007-04-13T15:34:43Z","title_canon_sha256":"220610006bad3ebc0bff7123d6ff87c78019e5f07551ec5d614b414073662253"},"schema_version":"1.0","source":{"id":"0704.1782","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0704.1782","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"arxiv_version","alias_value":"0704.1782v1","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0704.1782","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"pith_short_12","alias_value":"XAXEVE46272K","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"pith_short_16","alias_value":"XAXEVE46272KY3AZ","created_at":"2026-06-03T22:06:00Z"},{"alias_kind":"pith_short_8","alias_value":"XAXEVE46","created_at":"2026-06-03T22:06:00Z"}],"graph_snapshots":[{"event_id":"sha256:01abc8983d05d88d141d94d92c6f1518be1ce447086efb5099b70b5a732c2147","target":"graph","created_at":"2026-06-03T22:06:00Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0704.1782/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph $G$ with the path $P_m$ in terms of self adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product $P_2 \\Box P_m$, we numerically solve the eigenv","authors_text":"Richard Ehrenborg, Yossi Farjoun","cross_cats":["cs.NA","math.NA"],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2007-04-13T15:34:43Z","title":"Asymptotics of the Euler number of bipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.1782","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:a7294e74ba8a1b6cc04c94923040cd561ac72cccfa15fe7ed703889f16c85ec4","target":"record","created_at":"2026-06-03T22:06:00Z","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":"b2a17afb983a37d08c61f5a010ef3fa2c35a8e54c5deec4864da62af476ba85c","cross_cats_sorted":["cs.NA","math.NA"],"license":"","primary_cat":"math.CO","submitted_at":"2007-04-13T15:34:43Z","title_canon_sha256":"220610006bad3ebc0bff7123d6ff87c78019e5f07551ec5d614b414073662253"},"schema_version":"1.0","source":{"id":"0704.1782","kind":"arxiv","version":1}},"canonical_sha256":"b82e4a939ed7f4ac6c19f2ea87b392b3f37c32514db413f31e88bcac9eb76555","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b82e4a939ed7f4ac6c19f2ea87b392b3f37c32514db413f31e88bcac9eb76555","first_computed_at":"2026-06-03T22:06:00.446693Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T22:06:00.446693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"70xDMAly1uz+/DnF8euyWxIUFG7a21OQ7gLgnoVGYbQa0KTPywZC2VgKL1FMqbxvNLx+Gsi/lsBTJcXRf+waDA==","signature_status":"signed_v1","signed_at":"2026-06-03T22:06:00.447176Z","signed_message":"canonical_sha256_bytes"},"source_id":"0704.1782","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a7294e74ba8a1b6cc04c94923040cd561ac72cccfa15fe7ed703889f16c85ec4","sha256:01abc8983d05d88d141d94d92c6f1518be1ce447086efb5099b70b5a732c2147"],"state_sha256":"044e4afad4b449988923c8abb3f20c34595aca8422a52d7d92a69823e59dd045"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yvIv2SCZHlCaiHsY5EHF5oE/GMk4g4kCUDCIsmw+vQte+ql1Qz2DrqWPd89ZpkTUDRmJpoKwownExyQ2EY04Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T01:23:15.612751Z","bundle_sha256":"92512c2ba5c8e8dc6ee5415a86707fda727a520bc6bc36a3ddd6bc84102e5fe6"}}