{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:7A7PONQWIIGFQAPOLASNTISAEG","short_pith_number":"pith:7A7PONQW","canonical_record":{"source":{"id":"1312.7674","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-30T10:50:12Z","cross_cats_sorted":[],"title_canon_sha256":"fc1f02514dac53611133967e110a396ad39417cd39057dab39d48383d1847d54","abstract_canon_sha256":"99da558b0be96ea22a699b224eeb728efc84327739c54bd776ebc7e03249850e"},"schema_version":"1.0"},"canonical_sha256":"f83ef73616420c5801ee5824d9a24021aa5c7ee140494f502f76a3b237b1c0ba","source":{"kind":"arxiv","id":"1312.7674","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.7674","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"arxiv_version","alias_value":"1312.7674v5","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.7674","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"pith_short_12","alias_value":"7A7PONQWIIGF","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7A7PONQWIIGFQAPO","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7A7PONQW","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:7A7PONQWIIGFQAPOLASNTISAEG","target":"record","payload":{"canonical_record":{"source":{"id":"1312.7674","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-30T10:50:12Z","cross_cats_sorted":[],"title_canon_sha256":"fc1f02514dac53611133967e110a396ad39417cd39057dab39d48383d1847d54","abstract_canon_sha256":"99da558b0be96ea22a699b224eeb728efc84327739c54bd776ebc7e03249850e"},"schema_version":"1.0"},"canonical_sha256":"f83ef73616420c5801ee5824d9a24021aa5c7ee140494f502f76a3b237b1c0ba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:48.041979Z","signature_b64":"+AcE8W87qjFpZ0Quf1ZfB98h4c2JuQNI+ClAnq+KjFbtN5aX+yFJPFxvjbb7RUu4beESf7zk7KBDjOc3Q9MwCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f83ef73616420c5801ee5824d9a24021aa5c7ee140494f502f76a3b237b1c0ba","last_reissued_at":"2026-05-18T02:55:48.041280Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:48.041280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.7674","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FmpA3qtVy5zkKpN3H8fU2qJ8sRYqyD1Uul9VgluxpbQXK5W7+mEm7Qs6xMBahiNzAxWoiIg8LV8O/gXuwh3hDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T22:45:24.349641Z"},"content_sha256":"42c99de65001557ada3e62301d0b218834eb8e5bd43ee58e76273ea86c1d89d2","schema_version":"1.0","event_id":"sha256:42c99de65001557ada3e62301d0b218834eb8e5bd43ee58e76273ea86c1d89d2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:7A7PONQWIIGFQAPOLASNTISAEG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Study on Arithmetic Integer Additive Set-Indexers of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"K. A. Germina, N. K. Sudev","submitted_at":"2013-12-30T10:50:12Z","abstract_excerpt":"A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \\rightarrow2^{X}$ such that the function $f^{\\oplus}:E(G)\\rightarrow2^{X}-\\{\\emptyset\\}$ defined by $f^{\\oplus}(uv) = f(u){\\oplus} f(v)$ for every $uv{\\in} E(G)$ is also injective, where $2^{X}$ is the set of all subsets of $X$ and $\\oplus$ is the symmetric difference of sets. An integer additive set-indexer is defined as an injective function $f:V(G)\\rightarrow 2^{\\mathbb{N}_0}$ such that the induced function $f^+:E(G) \\rightarrow 2^{\\mathbb{N}_0}$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.7674","kind":"arxiv","version":5},"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":"2026-05-18T02:55:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ah5uTGaIRamRTkhITKJLkf5UfS4Zu8rgXf8AD7BFMwMACctK5nrjzrHHSWfb/Ab6edeqX4rjjIhm0FaIXKJMAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T22:45:24.350328Z"},"content_sha256":"4c54e59cc16a975217a218ff60416fe0f3235e88be8d8abde42d41c464649075","schema_version":"1.0","event_id":"sha256:4c54e59cc16a975217a218ff60416fe0f3235e88be8d8abde42d41c464649075"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7A7PONQWIIGFQAPOLASNTISAEG/bundle.json","state_url":"https://pith.science/pith/7A7PONQWIIGFQAPOLASNTISAEG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7A7PONQWIIGFQAPOLASNTISAEG/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":"2026-06-11T22:45:24Z","links":{"resolver":"https://pith.science/pith/7A7PONQWIIGFQAPOLASNTISAEG","bundle":"https://pith.science/pith/7A7PONQWIIGFQAPOLASNTISAEG/bundle.json","state":"https://pith.science/pith/7A7PONQWIIGFQAPOLASNTISAEG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7A7PONQWIIGFQAPOLASNTISAEG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7A7PONQWIIGFQAPOLASNTISAEG","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":"99da558b0be96ea22a699b224eeb728efc84327739c54bd776ebc7e03249850e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-30T10:50:12Z","title_canon_sha256":"fc1f02514dac53611133967e110a396ad39417cd39057dab39d48383d1847d54"},"schema_version":"1.0","source":{"id":"1312.7674","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.7674","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"arxiv_version","alias_value":"1312.7674v5","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.7674","created_at":"2026-05-18T02:55:48Z"},{"alias_kind":"pith_short_12","alias_value":"7A7PONQWIIGF","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7A7PONQWIIGFQAPO","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7A7PONQW","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:4c54e59cc16a975217a218ff60416fe0f3235e88be8d8abde42d41c464649075","target":"graph","created_at":"2026-05-18T02:55:48Z","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":"A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \\rightarrow2^{X}$ such that the function $f^{\\oplus}:E(G)\\rightarrow2^{X}-\\{\\emptyset\\}$ defined by $f^{\\oplus}(uv) = f(u){\\oplus} f(v)$ for every $uv{\\in} E(G)$ is also injective, where $2^{X}$ is the set of all subsets of $X$ and $\\oplus$ is the symmetric difference of sets. An integer additive set-indexer is defined as an injective function $f:V(G)\\rightarrow 2^{\\mathbb{N}_0}$ such that the induced function $f^+:E(G) \\rightarrow 2^{\\mathbb{N}_0}$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which a","authors_text":"K. A. Germina, N. K. Sudev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-30T10:50:12Z","title":"A Study on Arithmetic Integer Additive Set-Indexers of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.7674","kind":"arxiv","version":5},"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":"sha256:42c99de65001557ada3e62301d0b218834eb8e5bd43ee58e76273ea86c1d89d2","target":"record","created_at":"2026-05-18T02:55:48Z","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":"99da558b0be96ea22a699b224eeb728efc84327739c54bd776ebc7e03249850e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-30T10:50:12Z","title_canon_sha256":"fc1f02514dac53611133967e110a396ad39417cd39057dab39d48383d1847d54"},"schema_version":"1.0","source":{"id":"1312.7674","kind":"arxiv","version":5}},"canonical_sha256":"f83ef73616420c5801ee5824d9a24021aa5c7ee140494f502f76a3b237b1c0ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f83ef73616420c5801ee5824d9a24021aa5c7ee140494f502f76a3b237b1c0ba","first_computed_at":"2026-05-18T02:55:48.041280Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:48.041280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+AcE8W87qjFpZ0Quf1ZfB98h4c2JuQNI+ClAnq+KjFbtN5aX+yFJPFxvjbb7RUu4beESf7zk7KBDjOc3Q9MwCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:48.041979Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.7674","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42c99de65001557ada3e62301d0b218834eb8e5bd43ee58e76273ea86c1d89d2","sha256:4c54e59cc16a975217a218ff60416fe0f3235e88be8d8abde42d41c464649075"],"state_sha256":"6c08f5e5e1bc2ec56ce2d2150b627e56700793aa67d373bcddb14a3f7885439a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mlFrdEd+Ufsna8FuNAvIMOxB3DCnU72sQDRpQBdeQIAiV1uVez1E5sCnRkSPjcMEYmHAv9yjSi612Rk0U7qbAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T22:45:24.354901Z","bundle_sha256":"ea1e3747bd678291c83444cdb013649fe386ae1ac9464509afbbd2238ad7ce90"}}