{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:NJKB573PNWVEZMGLX3554FOATF","short_pith_number":"pith:NJKB573P","canonical_record":{"source":{"id":"2506.23112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-06-29T06:31:42Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"9f0246ad2fe55291c79442b17e983be2fa45ab6e75ffdd0eb109d46573b18c83","abstract_canon_sha256":"32b7d0098eaccbab5c649acfc09a4d0d15891ec134c6ea829477784ec20881cf"},"schema_version":"1.0"},"canonical_sha256":"6a541eff6f6daa4cb0cbbefbde15c09945ff63e21460d7e84ae5425438c1add8","source":{"kind":"arxiv","id":"2506.23112","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.23112","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"arxiv_version","alias_value":"2506.23112v1","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.23112","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"pith_short_12","alias_value":"NJKB573PNWVE","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"pith_short_16","alias_value":"NJKB573PNWVEZMGL","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"pith_short_8","alias_value":"NJKB573P","created_at":"2026-07-05T11:29:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:NJKB573PNWVEZMGLX3554FOATF","target":"record","payload":{"canonical_record":{"source":{"id":"2506.23112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-06-29T06:31:42Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"9f0246ad2fe55291c79442b17e983be2fa45ab6e75ffdd0eb109d46573b18c83","abstract_canon_sha256":"32b7d0098eaccbab5c649acfc09a4d0d15891ec134c6ea829477784ec20881cf"},"schema_version":"1.0"},"canonical_sha256":"6a541eff6f6daa4cb0cbbefbde15c09945ff63e21460d7e84ae5425438c1add8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:29:15.783523Z","signature_b64":"AV4QhLsG/eUPC22iUAgyPdn8z76KMsTMMld0omxP05aH0QdAKwhbtHHdrSHS4Pihihp2zuoIH3DyegOGfs1bDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a541eff6f6daa4cb0cbbefbde15c09945ff63e21460d7e84ae5425438c1add8","last_reissued_at":"2026-07-05T11:29:15.783034Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:29:15.783034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.23112","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":"2026-07-05T11:29:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MScXqCBBIf520sk1Z+QrmhSHgEbfRNN7+PxPsbPe8HIuRh9zfj7DoPsbix4/12m8dPY2eBC4KY4UtpvrXCv9AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T16:07:31.472868Z"},"content_sha256":"717865cfcf064fa6589b6475024681df6336d47659833533675fd4a5093b027d","schema_version":"1.0","event_id":"sha256:717865cfcf064fa6589b6475024681df6336d47659833533675fd4a5093b027d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:NJKB573PNWVEZMGLX3554FOATF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inertia indices of signed graphs with given cyclomatic number and given number of pendant vertices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.SP","authors_text":"Fang Duan, Jie Pu","submitted_at":"2025-06-29T06:31:42Z","abstract_excerpt":"Let $\\Gamma=(G, \\sigma)$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $\\sigma: E(G)\\rightarrow \\{+, -\\}$. Denoted by $i_+(\\Gamma)$, $\\theta(\\Gamma)$ and $p(\\Gamma)$ the positive inertia index, the cyclomatic number and the number of pendant vertices of $\\Gamma$, respectively. In this article, we prove that $i_+(\\Gamma)$, $\\theta(\\Gamma)$ and $p(\\Gamma)$ are related by the inequality $i_+(\\Gamma)\\geq \\frac{n-p(\\Gamma)}{2}-\\theta(\\Gamma)$. Furthermore, we completely characterize the signed graph $\\Gamma$ for which $i_+(\\Gamma)=\\frac{n-p(\\Gamma)}{2}-\\theta(\\Gamma)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.23112","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/2506.23112/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":"2026-07-05T11:29:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wbQAQxFaxgyZLeX1YFyQxanWVFCEmL9a403k32LKcLy/GXBxkdLwP3v9aUIqtTeszD3wn/31CekdU+sQyexoCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T16:07:31.473344Z"},"content_sha256":"e5d376336bb719f7dbf613320be6ce93d58fd776f2cfbe282878fd1c1b4b70b9","schema_version":"1.0","event_id":"sha256:e5d376336bb719f7dbf613320be6ce93d58fd776f2cfbe282878fd1c1b4b70b9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NJKB573PNWVEZMGLX3554FOATF/bundle.json","state_url":"https://pith.science/pith/NJKB573PNWVEZMGLX3554FOATF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NJKB573PNWVEZMGLX3554FOATF/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":"2026-08-23T16:07:31Z","links":{"resolver":"https://pith.science/pith/NJKB573PNWVEZMGLX3554FOATF","bundle":"https://pith.science/pith/NJKB573PNWVEZMGLX3554FOATF/bundle.json","state":"https://pith.science/pith/NJKB573PNWVEZMGLX3554FOATF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NJKB573PNWVEZMGLX3554FOATF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:NJKB573PNWVEZMGLX3554FOATF","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":"32b7d0098eaccbab5c649acfc09a4d0d15891ec134c6ea829477784ec20881cf","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-06-29T06:31:42Z","title_canon_sha256":"9f0246ad2fe55291c79442b17e983be2fa45ab6e75ffdd0eb109d46573b18c83"},"schema_version":"1.0","source":{"id":"2506.23112","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.23112","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"arxiv_version","alias_value":"2506.23112v1","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.23112","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"pith_short_12","alias_value":"NJKB573PNWVE","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"pith_short_16","alias_value":"NJKB573PNWVEZMGL","created_at":"2026-07-05T11:29:15Z"},{"alias_kind":"pith_short_8","alias_value":"NJKB573P","created_at":"2026-07-05T11:29:15Z"}],"graph_snapshots":[{"event_id":"sha256:e5d376336bb719f7dbf613320be6ce93d58fd776f2cfbe282878fd1c1b4b70b9","target":"graph","created_at":"2026-07-05T11:29:15Z","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/2506.23112/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\Gamma=(G, \\sigma)$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $\\sigma: E(G)\\rightarrow \\{+, -\\}$. Denoted by $i_+(\\Gamma)$, $\\theta(\\Gamma)$ and $p(\\Gamma)$ the positive inertia index, the cyclomatic number and the number of pendant vertices of $\\Gamma$, respectively. In this article, we prove that $i_+(\\Gamma)$, $\\theta(\\Gamma)$ and $p(\\Gamma)$ are related by the inequality $i_+(\\Gamma)\\geq \\frac{n-p(\\Gamma)}{2}-\\theta(\\Gamma)$. Furthermore, we completely characterize the signed graph $\\Gamma$ for which $i_+(\\Gamma)=\\frac{n-p(\\Gamma)}{2}-\\theta(\\Gamma)$","authors_text":"Fang Duan, Jie Pu","cross_cats":["math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-06-29T06:31:42Z","title":"Inertia indices of signed graphs with given cyclomatic number and given number of pendant vertices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.23112","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":"sha256:717865cfcf064fa6589b6475024681df6336d47659833533675fd4a5093b027d","target":"record","created_at":"2026-07-05T11:29:15Z","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":"32b7d0098eaccbab5c649acfc09a4d0d15891ec134c6ea829477784ec20881cf","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-06-29T06:31:42Z","title_canon_sha256":"9f0246ad2fe55291c79442b17e983be2fa45ab6e75ffdd0eb109d46573b18c83"},"schema_version":"1.0","source":{"id":"2506.23112","kind":"arxiv","version":1}},"canonical_sha256":"6a541eff6f6daa4cb0cbbefbde15c09945ff63e21460d7e84ae5425438c1add8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a541eff6f6daa4cb0cbbefbde15c09945ff63e21460d7e84ae5425438c1add8","first_computed_at":"2026-07-05T11:29:15.783034Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:29:15.783034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AV4QhLsG/eUPC22iUAgyPdn8z76KMsTMMld0omxP05aH0QdAKwhbtHHdrSHS4Pihihp2zuoIH3DyegOGfs1bDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:29:15.783523Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.23112","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:717865cfcf064fa6589b6475024681df6336d47659833533675fd4a5093b027d","sha256:e5d376336bb719f7dbf613320be6ce93d58fd776f2cfbe282878fd1c1b4b70b9"],"state_sha256":"8aed8545b12c9daf663d1671e186a7441d2372524fef3128eff2c1961541024b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+HPMWHRiGx4uED5dy80NAWhPJSAnkUoZVBm3l6kiBi3XQXyAuLpVIgK5EeHKZiHTAoe8H+Mc+LnZ57IorfkSCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T16:07:31.477306Z","bundle_sha256":"2b37bf48ca5a63a181f6dd5120a509ec71bcfe547e0fbb159279fdf9e4ffffc2"}}