{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:TPDQGFOBSKHFKJMPOFK44OBPUI","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":"c9037f028e686874596da97c9d1610b9b743f2f16dc1d309e8ca87efef05165b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-19T17:13:36Z","title_canon_sha256":"b33cb496476acbea4d458b8872df8f75cb1390fb24a9cb0b1efc999bdebc32ed"},"schema_version":"1.0","source":{"id":"1803.07042","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.07042","created_at":"2026-05-18T00:07:17Z"},{"alias_kind":"arxiv_version","alias_value":"1803.07042v2","created_at":"2026-05-18T00:07:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.07042","created_at":"2026-05-18T00:07:17Z"},{"alias_kind":"pith_short_12","alias_value":"TPDQGFOBSKHF","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"TPDQGFOBSKHFKJMP","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"TPDQGFOB","created_at":"2026-05-18T12:32:53Z"}],"graph_snapshots":[{"event_id":"sha256:e989d9913b6de9e8f353360aa2a1d4c5e04c6e0886fe76f6c700ffb439874fd1","target":"graph","created_at":"2026-05-18T00:07:17Z","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":"This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature follow as a corollary of the main results in this work. We show that for most cases our bounds outperform the previous known bounds. Some infinite graphs where the bounds are tight are also presented. Finally, as a byproduct, we derive some lower spectral bounds for the diameter of a graph.","authors_text":"A. Abiad, G. Coutinho, M. A. Fiol","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-19T17:13:36Z","title":"On the $k$-independence number of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07042","kind":"arxiv","version":2},"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:e64bd2ce7c860cb1b93878054a2ac1262093131ce698c1017e67aa73e8da1e45","target":"record","created_at":"2026-05-18T00:07:17Z","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":"c9037f028e686874596da97c9d1610b9b743f2f16dc1d309e8ca87efef05165b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-19T17:13:36Z","title_canon_sha256":"b33cb496476acbea4d458b8872df8f75cb1390fb24a9cb0b1efc999bdebc32ed"},"schema_version":"1.0","source":{"id":"1803.07042","kind":"arxiv","version":2}},"canonical_sha256":"9bc70315c1928e55258f7155ce382fa21836bd264caa1dc1b0d6b354a1c2aae1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9bc70315c1928e55258f7155ce382fa21836bd264caa1dc1b0d6b354a1c2aae1","first_computed_at":"2026-05-18T00:07:17.895724Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:17.895724Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6xt9Jrr3D0m0zlBQY0UJmXAt+xIrHuJn6V4fmlhJth9ArtSA5L+2OVbzr4N1x2qZRXcr4EnW1MWGZ7LEQdMECw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:17.896412Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.07042","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e64bd2ce7c860cb1b93878054a2ac1262093131ce698c1017e67aa73e8da1e45","sha256:e989d9913b6de9e8f353360aa2a1d4c5e04c6e0886fe76f6c700ffb439874fd1"],"state_sha256":"595236a7f443c6e3609fa89ba241879b5a1af66ab1f0a5661da1177902d84b24"}