{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:7LLL3NZDYOD4JRU3NVOSXR5ZWA","short_pith_number":"pith:7LLL3NZD","schema_version":"1.0","canonical_sha256":"fad6bdb723c387c4c69b6d5d2bc7b9b033138a51389839685735f15aff67f57c","source":{"kind":"arxiv","id":"1202.4204","version":1},"attestation_state":"computed","paper":{"title":"Vertex Isoperimetric Inequalities for a Family of Graphs on Z^k","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. J. Radcliffe, Ellen Veomett","submitted_at":"2012-02-20T01:08:20Z","abstract_excerpt":"We consider the family of graphs whose vertex set is Z^k where two vertices are connected by an edge when their l\\infty-distance is 1. We prove the optimal vertex isoperimetric inequality for this family of graphs. That is, given a positive integer n, we find a set A \\subset Z^k of size n such that the number of vertices who share an edge with some vertex in A is minimized. These sets of minimal boundary are nested, and the proof uses the technique of compression.\n  We also show a method of calculating the vertex boundary for certain subsets in this family of graphs. This calculation and the i"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1202.4204","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-20T01:08:20Z","cross_cats_sorted":[],"title_canon_sha256":"33108bb463e9adee62c01c23f26be92586c9b943bdd9d30a77ff570710a4d523","abstract_canon_sha256":"7bfda9a7fa2d860e083e97081433a292041d7a44ea19cd7f711bae39ac04c7cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:52.041025Z","signature_b64":"Xa6ATZ5oQc7RrjwXngqli2O7GETZFz8HmKNyejg6uY1p5Vos0VvQh3VzqUCr53z7CNdu5gan35yVp7dpGSOUDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fad6bdb723c387c4c69b6d5d2bc7b9b033138a51389839685735f15aff67f57c","last_reissued_at":"2026-05-18T04:01:52.040565Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:52.040565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vertex Isoperimetric Inequalities for a Family of Graphs on Z^k","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. J. Radcliffe, Ellen Veomett","submitted_at":"2012-02-20T01:08:20Z","abstract_excerpt":"We consider the family of graphs whose vertex set is Z^k where two vertices are connected by an edge when their l\\infty-distance is 1. We prove the optimal vertex isoperimetric inequality for this family of graphs. That is, given a positive integer n, we find a set A \\subset Z^k of size n such that the number of vertices who share an edge with some vertex in A is minimized. These sets of minimal boundary are nested, and the proof uses the technique of compression.\n  We also show a method of calculating the vertex boundary for certain subsets in this family of graphs. This calculation and the i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4204","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1202.4204","created_at":"2026-05-18T04:01:52.040624+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.4204v1","created_at":"2026-05-18T04:01:52.040624+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4204","created_at":"2026-05-18T04:01:52.040624+00:00"},{"alias_kind":"pith_short_12","alias_value":"7LLL3NZDYOD4","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_16","alias_value":"7LLL3NZDYOD4JRU3","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_8","alias_value":"7LLL3NZD","created_at":"2026-05-18T12:26:58.693483+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA","json":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA.json","graph_json":"https://pith.science/api/pith-number/7LLL3NZDYOD4JRU3NVOSXR5ZWA/graph.json","events_json":"https://pith.science/api/pith-number/7LLL3NZDYOD4JRU3NVOSXR5ZWA/events.json","paper":"https://pith.science/paper/7LLL3NZD"},"agent_actions":{"view_html":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA","download_json":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA.json","view_paper":"https://pith.science/paper/7LLL3NZD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.4204&json=true","fetch_graph":"https://pith.science/api/pith-number/7LLL3NZDYOD4JRU3NVOSXR5ZWA/graph.json","fetch_events":"https://pith.science/api/pith-number/7LLL3NZDYOD4JRU3NVOSXR5ZWA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA/action/storage_attestation","attest_author":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA/action/author_attestation","sign_citation":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA/action/citation_signature","submit_replication":"https://pith.science/pith/7LLL3NZDYOD4JRU3NVOSXR5ZWA/action/replication_record"}},"created_at":"2026-05-18T04:01:52.040624+00:00","updated_at":"2026-05-18T04:01:52.040624+00:00"}