{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:E45EZBE523V7NEYVHMYPPDDWEY","short_pith_number":"pith:E45EZBE5","canonical_record":{"source":{"id":"1307.0059","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-29T01:24:25Z","cross_cats_sorted":[],"title_canon_sha256":"81e3ee61683595ddc10603e7b763a05549db7e86aa5e8721232f2f46d0fe15b8","abstract_canon_sha256":"67534310b8c0baf1bb403bebb04be18bb424c5df479d590e51fdf383b027e169"},"schema_version":"1.0"},"canonical_sha256":"273a4c849dd6ebf693153b30f78c7626249008f8531dfa503e7262f1763e0571","source":{"kind":"arxiv","id":"1307.0059","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0059","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0059v1","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0059","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"pith_short_12","alias_value":"E45EZBE523V7","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"E45EZBE523V7NEYV","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"E45EZBE5","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:E45EZBE523V7NEYVHMYPPDDWEY","target":"record","payload":{"canonical_record":{"source":{"id":"1307.0059","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-29T01:24:25Z","cross_cats_sorted":[],"title_canon_sha256":"81e3ee61683595ddc10603e7b763a05549db7e86aa5e8721232f2f46d0fe15b8","abstract_canon_sha256":"67534310b8c0baf1bb403bebb04be18bb424c5df479d590e51fdf383b027e169"},"schema_version":"1.0"},"canonical_sha256":"273a4c849dd6ebf693153b30f78c7626249008f8531dfa503e7262f1763e0571","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:34.361871Z","signature_b64":"LmLdbYjKpergXfteUwUV9dmC26ePQxpf5ZrWZwD30AOIKT9tBx6xHojAEuVqRM4HBCAgo/mkfAR/zGFc5xfMDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"273a4c849dd6ebf693153b30f78c7626249008f8531dfa503e7262f1763e0571","last_reissued_at":"2026-05-18T03:19:34.361513Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:34.361513Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.0059","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":"2026-05-18T03:19:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"koMf+xSSdLWTMjm/GrBQ3QQQFdevFXHT8FH12IdTMZ+QBf8aDFQoXgGi5vE4gATidNr8yJbASF9ZEJ61dU9SAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T08:55:30.821486Z"},"content_sha256":"1f78ebbad4c90a24756211f6ab9901e888547f8933724b615bf3bbe868d383f7","schema_version":"1.0","event_id":"sha256:1f78ebbad4c90a24756211f6ab9901e888547f8933724b615bf3bbe868d383f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:E45EZBE523V7NEYVHMYPPDDWEY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The inertia of weighted unicyclic graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guihai Yu, Lihua Feng, Xiao-Dong Zhang","submitted_at":"2013-06-29T01:24:25Z","abstract_excerpt":"Let $G_w$ be a weighted graph. The \\textit{inertia} of $G_w$ is the triple $In(G_w)=\\big(i_+(G_w),i_-(G_w), $ $ i_0(G_w)\\big)$, where $i_+(G_w),i_-(G_w),i_0(G_w)$ are the number of the positive, negative and zero eigenvalues of the adjacency matrix $A(G_w)$ of $G_w$ including their multiplicities, respectively. $i_+(G_w)$, $i_-(G_w)$ is called the \\textit{positive, negative index of inertia} of $G_w$, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order $n$ with fixed girth and characterize all weighted unicyclic graphs att"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0059","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":"2026-05-18T03:19:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KS9ZypTz2xM91RdIn1mcyXBcmLayWj7n8Cgln/b5B3lz4uY1z5aSaUXh4dRxCkxRv07wY/iC8l1IHyUcUTwzCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T08:55:30.822227Z"},"content_sha256":"e897b1bec77958947dbd9236194a745489d31c9743033c7d5f183e61245c8fd6","schema_version":"1.0","event_id":"sha256:e897b1bec77958947dbd9236194a745489d31c9743033c7d5f183e61245c8fd6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E45EZBE523V7NEYVHMYPPDDWEY/bundle.json","state_url":"https://pith.science/pith/E45EZBE523V7NEYVHMYPPDDWEY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E45EZBE523V7NEYVHMYPPDDWEY/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":"2026-06-09T08:55:30Z","links":{"resolver":"https://pith.science/pith/E45EZBE523V7NEYVHMYPPDDWEY","bundle":"https://pith.science/pith/E45EZBE523V7NEYVHMYPPDDWEY/bundle.json","state":"https://pith.science/pith/E45EZBE523V7NEYVHMYPPDDWEY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E45EZBE523V7NEYVHMYPPDDWEY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:E45EZBE523V7NEYVHMYPPDDWEY","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":"67534310b8c0baf1bb403bebb04be18bb424c5df479d590e51fdf383b027e169","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-29T01:24:25Z","title_canon_sha256":"81e3ee61683595ddc10603e7b763a05549db7e86aa5e8721232f2f46d0fe15b8"},"schema_version":"1.0","source":{"id":"1307.0059","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0059","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0059v1","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0059","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"pith_short_12","alias_value":"E45EZBE523V7","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"E45EZBE523V7NEYV","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"E45EZBE5","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:e897b1bec77958947dbd9236194a745489d31c9743033c7d5f183e61245c8fd6","target":"graph","created_at":"2026-05-18T03:19:34Z","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 \\textit{inertia} of $G_w$ is the triple $In(G_w)=\\big(i_+(G_w),i_-(G_w), $ $ i_0(G_w)\\big)$, where $i_+(G_w),i_-(G_w),i_0(G_w)$ are the number of the positive, negative and zero eigenvalues of the adjacency matrix $A(G_w)$ of $G_w$ including their multiplicities, respectively. $i_+(G_w)$, $i_-(G_w)$ is called the \\textit{positive, negative index of inertia} of $G_w$, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order $n$ with fixed girth and characterize all weighted unicyclic graphs att","authors_text":"Guihai Yu, Lihua Feng, Xiao-Dong Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-29T01:24:25Z","title":"The inertia of weighted unicyclic graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0059","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":"sha256:1f78ebbad4c90a24756211f6ab9901e888547f8933724b615bf3bbe868d383f7","target":"record","created_at":"2026-05-18T03:19:34Z","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":"67534310b8c0baf1bb403bebb04be18bb424c5df479d590e51fdf383b027e169","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-29T01:24:25Z","title_canon_sha256":"81e3ee61683595ddc10603e7b763a05549db7e86aa5e8721232f2f46d0fe15b8"},"schema_version":"1.0","source":{"id":"1307.0059","kind":"arxiv","version":1}},"canonical_sha256":"273a4c849dd6ebf693153b30f78c7626249008f8531dfa503e7262f1763e0571","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"273a4c849dd6ebf693153b30f78c7626249008f8531dfa503e7262f1763e0571","first_computed_at":"2026-05-18T03:19:34.361513Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:34.361513Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LmLdbYjKpergXfteUwUV9dmC26ePQxpf5ZrWZwD30AOIKT9tBx6xHojAEuVqRM4HBCAgo/mkfAR/zGFc5xfMDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:34.361871Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.0059","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1f78ebbad4c90a24756211f6ab9901e888547f8933724b615bf3bbe868d383f7","sha256:e897b1bec77958947dbd9236194a745489d31c9743033c7d5f183e61245c8fd6"],"state_sha256":"ea1c514cdf1580087caebaec06fd1b38c94ac6686783015b58dfd11df5bcc01b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"czcJ2y0ebQFMF+OKSHDmIBoutSkc91AOAo2+4vg2kIapYfU8hTuAONWau3oFra+GDLvIyUo+bljbymzK8HKnAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T08:55:30.826308Z","bundle_sha256":"6cc2e99fd7dd740024fdcc6cc5b10dc8ece4ea224ca666539a3018dfe256c2d5"}}