{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:CVFRDBR2ML6XRE7FJOJLYLYVRH","short_pith_number":"pith:CVFRDBR2","canonical_record":{"source":{"id":"2507.05700","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2025-07-08T06:21:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"338c5fea30b8ae64f341c7c535e05aaf7a4245959434017e2a75c76587b66526","abstract_canon_sha256":"1e91a94f5438bf9329b133e37dd3a3045ad9890a7340d4fcf3b9896e85fb9f95"},"schema_version":"1.0"},"canonical_sha256":"154b11863a62fd7893e54b92bc2f1589c70184b67e16295c5b3c683eb12361e1","source":{"kind":"arxiv","id":"2507.05700","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.05700","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"arxiv_version","alias_value":"2507.05700v2","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.05700","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"pith_short_12","alias_value":"CVFRDBR2ML6X","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"pith_short_16","alias_value":"CVFRDBR2ML6XRE7F","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"pith_short_8","alias_value":"CVFRDBR2","created_at":"2026-07-05T11:39:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:CVFRDBR2ML6XRE7FJOJLYLYVRH","target":"record","payload":{"canonical_record":{"source":{"id":"2507.05700","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2025-07-08T06:21:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"338c5fea30b8ae64f341c7c535e05aaf7a4245959434017e2a75c76587b66526","abstract_canon_sha256":"1e91a94f5438bf9329b133e37dd3a3045ad9890a7340d4fcf3b9896e85fb9f95"},"schema_version":"1.0"},"canonical_sha256":"154b11863a62fd7893e54b92bc2f1589c70184b67e16295c5b3c683eb12361e1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:39:13.153863Z","signature_b64":"eRvIORjQeGXP4mXN+l2cj2IgHan3P5tK+BAtdAlC8g9QEpLQ2tSMrLTtx8kdMDboAobB/dbDq+S5N+d8t8cYAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"154b11863a62fd7893e54b92bc2f1589c70184b67e16295c5b3c683eb12361e1","last_reissued_at":"2026-07-05T11:39:13.153273Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:39:13.153273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.05700","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":"2026-07-05T11:39:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"feErGvoHea78KXvvTSGBLnVgnmFlSNGT84JMHjRUya+Uv9OFifqzgJBB/APcn90uZAfmoB3gQYhFwXkULODGAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T20:53:52.661477Z"},"content_sha256":"92a102e18a73edbe0fb9df53d66b463ce439c32db2b4f728a6870d510095b606","schema_version":"1.0","event_id":"sha256:92a102e18a73edbe0fb9df53d66b463ce439c32db2b4f728a6870d510095b606"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:CVFRDBR2ML6XRE7FJOJLYLYVRH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Comparing the $\\mathrm{v}$-number and $h$-polynomials of edge ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Adam Van Tuyl, Kamalesh Saha","submitted_at":"2025-07-08T06:21:21Z","abstract_excerpt":"In this paper, we compare the $\\mathrm{v}$-numbers and the degree of the $h$-polynomials associated with edge ideals of connected graphs. We prove that the $\\mathrm{v}$-number can be arbitrarily larger or smaller than the degree of the $h$-polynomial for the edge ideal of a connected graph. We also establish that for any pair of positive integers $(v,d)$ with $v \\leq d$, there exists a connected graph $H(v,d)$ with the $\\mathrm{v}$-number equal to $v$ and the degree of $h$-polynomial equal to $d$. Additionally, we show that the sum of the $\\mathrm{v}$-number and the degree of the $h$-polynomia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.05700","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.05700/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":"2026-07-05T11:39:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+HJ/gd4/ZyXC3vzAAXIoUzN3bqigexe6QL9uf+U2UBD2wgp9q4BfQ8KKgthiGERi1MIcu1fggdT0h6t7F1aIBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T20:53:52.662014Z"},"content_sha256":"63a75fb2204e798f614aee3dd66be686f08d4cc25b4e11ae3200408c0c89f18b","schema_version":"1.0","event_id":"sha256:63a75fb2204e798f614aee3dd66be686f08d4cc25b4e11ae3200408c0c89f18b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CVFRDBR2ML6XRE7FJOJLYLYVRH/bundle.json","state_url":"https://pith.science/pith/CVFRDBR2ML6XRE7FJOJLYLYVRH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CVFRDBR2ML6XRE7FJOJLYLYVRH/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":"2026-08-08T20:53:52Z","links":{"resolver":"https://pith.science/pith/CVFRDBR2ML6XRE7FJOJLYLYVRH","bundle":"https://pith.science/pith/CVFRDBR2ML6XRE7FJOJLYLYVRH/bundle.json","state":"https://pith.science/pith/CVFRDBR2ML6XRE7FJOJLYLYVRH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CVFRDBR2ML6XRE7FJOJLYLYVRH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:CVFRDBR2ML6XRE7FJOJLYLYVRH","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":"1e91a94f5438bf9329b133e37dd3a3045ad9890a7340d4fcf3b9896e85fb9f95","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2025-07-08T06:21:21Z","title_canon_sha256":"338c5fea30b8ae64f341c7c535e05aaf7a4245959434017e2a75c76587b66526"},"schema_version":"1.0","source":{"id":"2507.05700","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.05700","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"arxiv_version","alias_value":"2507.05700v2","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.05700","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"pith_short_12","alias_value":"CVFRDBR2ML6X","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"pith_short_16","alias_value":"CVFRDBR2ML6XRE7F","created_at":"2026-07-05T11:39:13Z"},{"alias_kind":"pith_short_8","alias_value":"CVFRDBR2","created_at":"2026-07-05T11:39:13Z"}],"graph_snapshots":[{"event_id":"sha256:63a75fb2204e798f614aee3dd66be686f08d4cc25b4e11ae3200408c0c89f18b","target":"graph","created_at":"2026-07-05T11:39:13Z","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/2507.05700/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we compare the $\\mathrm{v}$-numbers and the degree of the $h$-polynomials associated with edge ideals of connected graphs. We prove that the $\\mathrm{v}$-number can be arbitrarily larger or smaller than the degree of the $h$-polynomial for the edge ideal of a connected graph. We also establish that for any pair of positive integers $(v,d)$ with $v \\leq d$, there exists a connected graph $H(v,d)$ with the $\\mathrm{v}$-number equal to $v$ and the degree of $h$-polynomial equal to $d$. Additionally, we show that the sum of the $\\mathrm{v}$-number and the degree of the $h$-polynomia","authors_text":"Adam Van Tuyl, Kamalesh Saha","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2025-07-08T06:21:21Z","title":"Comparing the $\\mathrm{v}$-number and $h$-polynomials of edge ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.05700","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":"sha256:92a102e18a73edbe0fb9df53d66b463ce439c32db2b4f728a6870d510095b606","target":"record","created_at":"2026-07-05T11:39:13Z","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":"1e91a94f5438bf9329b133e37dd3a3045ad9890a7340d4fcf3b9896e85fb9f95","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2025-07-08T06:21:21Z","title_canon_sha256":"338c5fea30b8ae64f341c7c535e05aaf7a4245959434017e2a75c76587b66526"},"schema_version":"1.0","source":{"id":"2507.05700","kind":"arxiv","version":2}},"canonical_sha256":"154b11863a62fd7893e54b92bc2f1589c70184b67e16295c5b3c683eb12361e1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"154b11863a62fd7893e54b92bc2f1589c70184b67e16295c5b3c683eb12361e1","first_computed_at":"2026-07-05T11:39:13.153273Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:39:13.153273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eRvIORjQeGXP4mXN+l2cj2IgHan3P5tK+BAtdAlC8g9QEpLQ2tSMrLTtx8kdMDboAobB/dbDq+S5N+d8t8cYAg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:39:13.153863Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.05700","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:92a102e18a73edbe0fb9df53d66b463ce439c32db2b4f728a6870d510095b606","sha256:63a75fb2204e798f614aee3dd66be686f08d4cc25b4e11ae3200408c0c89f18b"],"state_sha256":"a2a33cf1473692d833865ee565fd98c6a0f523152c70080577f18b3fd39f7fa1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mc6PEcV++nKXcCBF+4J5HdIIpk/WdMdHD4SEgXzHbVlX5Yw74nXnkSrpt15m7VrxLxscksQKxSDvF8WMEgIXAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T20:53:52.667166Z","bundle_sha256":"ef697dff62a5f92495e25cb8b266e1743d890c5481127a1c462d01c778931def"}}