{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:D3AZQ4GE3PUJ7RQ5YKQKHBZDW4","short_pith_number":"pith:D3AZQ4GE","schema_version":"1.0","canonical_sha256":"1ec19870c4dbe89fc61dc2a0a38723b71b1a6088db16abe994861fa6e1cb4555","source":{"kind":"arxiv","id":"1505.05811","version":1},"attestation_state":"computed","paper":{"title":"The Metric Dimension of The Tensor Product of Cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. Seify, A. Zaeembashi, H. Amraei, H.R. Maimani","submitted_at":"2015-05-21T18:01:43Z","abstract_excerpt":"Let $G$ be a connected graph and $W=\\{ w_1, w_2, \\ldots, w_k \\} \\subseteq V(G)$ be an ordered set. For every vertex $v$, the metric representation of $v$ with respect to $W$ is an ordered $k$-vector defined as $r(v|W):=(d(v,w_1), d(v,w_2), \\ldots, d(v,w_k))$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension and is denoted by $dim(G)$. In this paper, we study the metric dimension of"},"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":"1505.05811","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-21T18:01:43Z","cross_cats_sorted":[],"title_canon_sha256":"80914f98fa626019f7e867eab06ec55f61e0299f97c3e95100e7915c3be0c98c","abstract_canon_sha256":"a6d549215a7ad88bce91246fe9421e79d7d72c10756ace3fd86de65ee61c09a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:53.418234Z","signature_b64":"LLTsWkHwpt8OjULosHQZGsbmBORJ4RoRwo1k6Se2cbxgtlshaXhSRWo6Bhj7pY4X+zfDL+YaZoQ5CcUxG+TYBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ec19870c4dbe89fc61dc2a0a38723b71b1a6088db16abe994861fa6e1cb4555","last_reissued_at":"2026-05-18T02:03:53.417639Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:53.417639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Metric Dimension of The Tensor Product of Cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. Seify, A. Zaeembashi, H. Amraei, H.R. Maimani","submitted_at":"2015-05-21T18:01:43Z","abstract_excerpt":"Let $G$ be a connected graph and $W=\\{ w_1, w_2, \\ldots, w_k \\} \\subseteq V(G)$ be an ordered set. For every vertex $v$, the metric representation of $v$ with respect to $W$ is an ordered $k$-vector defined as $r(v|W):=(d(v,w_1), d(v,w_2), \\ldots, d(v,w_k))$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension and is denoted by $dim(G)$. In this paper, we study the metric dimension of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.05811","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":"1505.05811","created_at":"2026-05-18T02:03:53.417737+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.05811v1","created_at":"2026-05-18T02:03:53.417737+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.05811","created_at":"2026-05-18T02:03:53.417737+00:00"},{"alias_kind":"pith_short_12","alias_value":"D3AZQ4GE3PUJ","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"D3AZQ4GE3PUJ7RQ5","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"D3AZQ4GE","created_at":"2026-05-18T12:29:17.054201+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/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4","json":"https://pith.science/pith/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4.json","graph_json":"https://pith.science/api/pith-number/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/graph.json","events_json":"https://pith.science/api/pith-number/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/events.json","paper":"https://pith.science/paper/D3AZQ4GE"},"agent_actions":{"view_html":"https://pith.science/pith/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4","download_json":"https://pith.science/pith/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4.json","view_paper":"https://pith.science/paper/D3AZQ4GE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.05811&json=true","fetch_graph":"https://pith.science/api/pith-number/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/graph.json","fetch_events":"https://pith.science/api/pith-number/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/action/storage_attestation","attest_author":"https://pith.science/pith/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/action/author_attestation","sign_citation":"https://pith.science/pith/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/action/citation_signature","submit_replication":"https://pith.science/pith/D3AZQ4GE3PUJ7RQ5YKQKHBZDW4/action/replication_record"}},"created_at":"2026-05-18T02:03:53.417737+00:00","updated_at":"2026-05-18T02:03:53.417737+00:00"}