{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:D5YSMGPRJBHSGCIGTHZB5BHI6Y","short_pith_number":"pith:D5YSMGPR","canonical_record":{"source":{"id":"1311.3049","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2013-11-13T08:27:44Z","cross_cats_sorted":[],"title_canon_sha256":"93d4405aa57256f811f77321c65746cfe6344ff1faeeff2c83410ea96e74b284","abstract_canon_sha256":"adf05b1df08c951694c2d16ebbc52ec0509b1e62c949d879f76ded7f1d423240"},"schema_version":"1.0"},"canonical_sha256":"1f712619f1484f23090699f21e84e8f63b3fadeb71bbed6d6cca92124415b2ac","source":{"kind":"arxiv","id":"1311.3049","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.3049","created_at":"2026-05-18T03:07:13Z"},{"alias_kind":"arxiv_version","alias_value":"1311.3049v1","created_at":"2026-05-18T03:07:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.3049","created_at":"2026-05-18T03:07:13Z"},{"alias_kind":"pith_short_12","alias_value":"D5YSMGPRJBHS","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"D5YSMGPRJBHSGCIG","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"D5YSMGPR","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:D5YSMGPRJBHSGCIGTHZB5BHI6Y","target":"record","payload":{"canonical_record":{"source":{"id":"1311.3049","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2013-11-13T08:27:44Z","cross_cats_sorted":[],"title_canon_sha256":"93d4405aa57256f811f77321c65746cfe6344ff1faeeff2c83410ea96e74b284","abstract_canon_sha256":"adf05b1df08c951694c2d16ebbc52ec0509b1e62c949d879f76ded7f1d423240"},"schema_version":"1.0"},"canonical_sha256":"1f712619f1484f23090699f21e84e8f63b3fadeb71bbed6d6cca92124415b2ac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:13.268781Z","signature_b64":"Wuk74VPkKBrHa6RSp/375ijVs7TXX2d7JUPj4ZsicFi/wMo34vA80IsD09OBvJAWO+M5qJOr1xYps/2J6u9pBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f712619f1484f23090699f21e84e8f63b3fadeb71bbed6d6cca92124415b2ac","last_reissued_at":"2026-05-18T03:07:13.268339Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:13.268339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.3049","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":"2026-05-18T03:07:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uQjhl6/9AaUEyCsSKUjYjhYnNuQXH2rVxEtezfGxAIhhCxjladGEzDaSN5HW0tuSEx1Es5/SvmGKakG9VLy1Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:23:23.844510Z"},"content_sha256":"03a1ffb5497052f5a108f9758bcdffa754df7df3e2d462d076f152092a54cce5","schema_version":"1.0","event_id":"sha256:03a1ffb5497052f5a108f9758bcdffa754df7df3e2d462d076f152092a54cce5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:D5YSMGPRJBHSGCIGTHZB5BHI6Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The extremal problems on the inertia of weighted bicyclic graphs","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Feifei Song, Shibing Deng, Shuchao Li","submitted_at":"2013-11-13T08:27:44Z","abstract_excerpt":"Let $G_w$ be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of $G_w$ are called positive inertia index, negative inertia index and nullity of $G_w$, and denoted by $i_{+}(G_w)$, $i_{-}(G_w)$, $i_{0}(G_w)$, respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order $n$ with pendant vertices is obtained. Moreover, all the weighted bicyclic graphs of order $n$ with at most two positive, two negative and at least $n-4$ zero eigenvalues are identified, respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.3049","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":"2026-05-18T03:07:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s6yN+nG5quS7fSmRCfsNg+JxtVsmIYlwMeNc4gUvpO7iutwIJSkprLlvIkJ+6N02zJfJeplXtNbF08SXoxuDDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:23:23.845140Z"},"content_sha256":"778040869554ae1acab41b7f1afb3a7f17b7d4f3e000a601a52853c6a159570f","schema_version":"1.0","event_id":"sha256:778040869554ae1acab41b7f1afb3a7f17b7d4f3e000a601a52853c6a159570f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/D5YSMGPRJBHSGCIGTHZB5BHI6Y/bundle.json","state_url":"https://pith.science/pith/D5YSMGPRJBHSGCIGTHZB5BHI6Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/D5YSMGPRJBHSGCIGTHZB5BHI6Y/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":"2026-06-08T14:23:23Z","links":{"resolver":"https://pith.science/pith/D5YSMGPRJBHSGCIGTHZB5BHI6Y","bundle":"https://pith.science/pith/D5YSMGPRJBHSGCIGTHZB5BHI6Y/bundle.json","state":"https://pith.science/pith/D5YSMGPRJBHSGCIGTHZB5BHI6Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/D5YSMGPRJBHSGCIGTHZB5BHI6Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:D5YSMGPRJBHSGCIGTHZB5BHI6Y","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":"adf05b1df08c951694c2d16ebbc52ec0509b1e62c949d879f76ded7f1d423240","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2013-11-13T08:27:44Z","title_canon_sha256":"93d4405aa57256f811f77321c65746cfe6344ff1faeeff2c83410ea96e74b284"},"schema_version":"1.0","source":{"id":"1311.3049","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.3049","created_at":"2026-05-18T03:07:13Z"},{"alias_kind":"arxiv_version","alias_value":"1311.3049v1","created_at":"2026-05-18T03:07:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.3049","created_at":"2026-05-18T03:07:13Z"},{"alias_kind":"pith_short_12","alias_value":"D5YSMGPRJBHS","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"D5YSMGPRJBHSGCIG","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"D5YSMGPR","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:778040869554ae1acab41b7f1afb3a7f17b7d4f3e000a601a52853c6a159570f","target":"graph","created_at":"2026-05-18T03:07:13Z","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"},"paper":{"abstract_excerpt":"Let $G_w$ be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of $G_w$ are called positive inertia index, negative inertia index and nullity of $G_w$, and denoted by $i_{+}(G_w)$, $i_{-}(G_w)$, $i_{0}(G_w)$, respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order $n$ with pendant vertices is obtained. Moreover, all the weighted bicyclic graphs of order $n$ with at most two positive, two negative and at least $n-4$ zero eigenvalues are identified, respectively.","authors_text":"Feifei Song, Shibing Deng, Shuchao Li","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2013-11-13T08:27:44Z","title":"The extremal problems on the inertia of weighted bicyclic graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.3049","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":"sha256:03a1ffb5497052f5a108f9758bcdffa754df7df3e2d462d076f152092a54cce5","target":"record","created_at":"2026-05-18T03:07:13Z","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":"adf05b1df08c951694c2d16ebbc52ec0509b1e62c949d879f76ded7f1d423240","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2013-11-13T08:27:44Z","title_canon_sha256":"93d4405aa57256f811f77321c65746cfe6344ff1faeeff2c83410ea96e74b284"},"schema_version":"1.0","source":{"id":"1311.3049","kind":"arxiv","version":1}},"canonical_sha256":"1f712619f1484f23090699f21e84e8f63b3fadeb71bbed6d6cca92124415b2ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1f712619f1484f23090699f21e84e8f63b3fadeb71bbed6d6cca92124415b2ac","first_computed_at":"2026-05-18T03:07:13.268339Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:13.268339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Wuk74VPkKBrHa6RSp/375ijVs7TXX2d7JUPj4ZsicFi/wMo34vA80IsD09OBvJAWO+M5qJOr1xYps/2J6u9pBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:13.268781Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.3049","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:03a1ffb5497052f5a108f9758bcdffa754df7df3e2d462d076f152092a54cce5","sha256:778040869554ae1acab41b7f1afb3a7f17b7d4f3e000a601a52853c6a159570f"],"state_sha256":"6197446fe5afaa14d7ef01150022ac6ffa3208969e79f894e323e8ea550ecc41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ws/tx7PdYLHvJVr7qpo/3bToc9AxO4f1tYhFlFfzRCVxcd4WzX6bQNgfoVOeedQ8zMrDKtyCEW19PId+jT5CBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T14:23:23.848434Z","bundle_sha256":"e17cbd83cd0475fd45e327df8007081bf5187509d8a9c3b7450d92c528986cd7"}}