{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:5BG5PH7ZCHYDHLICH42AY4O4AE","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":"2b0e37a49c5f2bcd285979ddf7a81ebe08cb23b6f72a131f4792cbb0b45504fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2015-05-11T10:44:04Z","title_canon_sha256":"7dbba6c90219fd86e347de91793f99f9b908583117b970826f76cb7c3b3adb4b"},"schema_version":"1.0","source":{"id":"1505.03841","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.03841","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"arxiv_version","alias_value":"1505.03841v1","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03841","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"pith_short_12","alias_value":"5BG5PH7ZCHYD","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"5BG5PH7ZCHYDHLIC","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"5BG5PH7Z","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:1f6bca9862cc79bdbac558209421f26b9a50b5fd5f1c222c4ed65fe9bbdf33f8","target":"graph","created_at":"2026-05-18T02:09:52Z","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 $\\mathbb{N}_0$ denote the set of all non-negative integers and $\\mathcal{P}(\\mathbb{N}_0)$ be its power set. An integer additive set-indexer (IASI) of a given graph $G$ is an injective function $f:V(G)\\to \\mathcal{P}(\\mathbb{N}_0)$ such that the induced function $f^+:E(G) \\to \\mathcal{P}(\\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ of a graph $G$ is said to be a weak IASI of $G$ if $|f^+(uv)|=\\max(|f(u)|,|f(v)|)$ for all $u,v\\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The sparing number of a graph $G$ is the minimum numbe","authors_text":"Augustine Germina, Naduvath Sudev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2015-05-11T10:44:04Z","title":"A Note on the Sparing Number of the Sieve Graphs of Certain Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03841","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":"sha256:65b1a0e5aefc2eec975fa385ead4278ad0fd8bfcc3a6571dec934ed09155c1fe","target":"record","created_at":"2026-05-18T02:09:52Z","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":"2b0e37a49c5f2bcd285979ddf7a81ebe08cb23b6f72a131f4792cbb0b45504fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2015-05-11T10:44:04Z","title_canon_sha256":"7dbba6c90219fd86e347de91793f99f9b908583117b970826f76cb7c3b3adb4b"},"schema_version":"1.0","source":{"id":"1505.03841","kind":"arxiv","version":1}},"canonical_sha256":"e84dd79ff911f033ad023f340c71dc012ecca03683b8a2bb9b7a8aa5890b8588","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e84dd79ff911f033ad023f340c71dc012ecca03683b8a2bb9b7a8aa5890b8588","first_computed_at":"2026-05-18T02:09:52.873840Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:09:52.873840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6O/ae0Y6+JIHzde+G/vNK1cizd2E0gpQu0H5WAPnkpn01gFVJPUKbWtbMhpbKhfBjWN7xtmbh8yMQUpje4hODw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:09:52.874606Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.03841","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65b1a0e5aefc2eec975fa385ead4278ad0fd8bfcc3a6571dec934ed09155c1fe","sha256:1f6bca9862cc79bdbac558209421f26b9a50b5fd5f1c222c4ed65fe9bbdf33f8"],"state_sha256":"d853d3b4c4a622b10e17cd2167fa2f79c9441d74584b48183636316d4d17c92c"}