{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:BMGARRAQ7AAO2RQDQ3CEQOMXS2","short_pith_number":"pith:BMGARRAQ","schema_version":"1.0","canonical_sha256":"0b0c08c410f800ed460386c4483997969e64f0ceebd02736ae2e45466621f2bc","source":{"kind":"arxiv","id":"1611.01416","version":1},"attestation_state":"computed","paper":{"title":"Chromatic Zagreb indices for graphical embodiment of colour clusters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Johan Kok, Muhammad Kamran Jamil, Naduvath Sudev","submitted_at":"2016-11-01T02:22:44Z","abstract_excerpt":"For a colour cluster $\\mathbb{C} =(\\mathcal{C}_1,\\mathcal{C}_2, \\mathcal{C}_3,\\ldots,\\mathcal{C}_\\ell)$, where $\\mathcal{C}_i$ is a colour class such that $|\\mathcal{C}_i|=r_i$, a positive integer, we investigate two types of simple connected graph structures $G^{\\mathbb{C}}_1$, $G^{\\mathbb{C}}_2$ which represent graphical embodiments of the colour cluster such that the chromatic numbers $\\chi(G^{\\mathbb{C}}_1)=\\chi(G^{\\mathbb{C}}_2)=\\ell$ and $\\min\\{\\varepsilon(G^{\\mathbb{C}}_1)\\}=\\min\\{\\varepsilon(G^{\\mathbb{C}}_2)\\} =\\sum\\limits_{i=1}^{\\ell}r_i-1$. Therefore, the problem is the edge-minimal"},"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":"1611.01416","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2016-11-01T02:22:44Z","cross_cats_sorted":[],"title_canon_sha256":"3f449c2b7595963e49e71f503e6690d3a6da30370964b0210de8c1743b33dd08","abstract_canon_sha256":"33b22b4e55731dfcf7d497a3e01bce7183bb86294d5bf53d024769c34ebe06cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:45.723361Z","signature_b64":"XB5X5pNrzNF9blGzWFXqHe6OGQm9lgUkIl+mueewtF8hQfhKRNI8Fzy5DOLX4FGzmi3XKgTcTL5YoQh7Az1oAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b0c08c410f800ed460386c4483997969e64f0ceebd02736ae2e45466621f2bc","last_reissued_at":"2026-05-18T00:53:45.722924Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:45.722924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chromatic Zagreb indices for graphical embodiment of colour clusters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Johan Kok, Muhammad Kamran Jamil, Naduvath Sudev","submitted_at":"2016-11-01T02:22:44Z","abstract_excerpt":"For a colour cluster $\\mathbb{C} =(\\mathcal{C}_1,\\mathcal{C}_2, \\mathcal{C}_3,\\ldots,\\mathcal{C}_\\ell)$, where $\\mathcal{C}_i$ is a colour class such that $|\\mathcal{C}_i|=r_i$, a positive integer, we investigate two types of simple connected graph structures $G^{\\mathbb{C}}_1$, $G^{\\mathbb{C}}_2$ which represent graphical embodiments of the colour cluster such that the chromatic numbers $\\chi(G^{\\mathbb{C}}_1)=\\chi(G^{\\mathbb{C}}_2)=\\ell$ and $\\min\\{\\varepsilon(G^{\\mathbb{C}}_1)\\}=\\min\\{\\varepsilon(G^{\\mathbb{C}}_2)\\} =\\sum\\limits_{i=1}^{\\ell}r_i-1$. Therefore, the problem is the edge-minimal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01416","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":"1611.01416","created_at":"2026-05-18T00:53:45.722993+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.01416v1","created_at":"2026-05-18T00:53:45.722993+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.01416","created_at":"2026-05-18T00:53:45.722993+00:00"},{"alias_kind":"pith_short_12","alias_value":"BMGARRAQ7AAO","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"BMGARRAQ7AAO2RQD","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"BMGARRAQ","created_at":"2026-05-18T12:30:07.202191+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/BMGARRAQ7AAO2RQDQ3CEQOMXS2","json":"https://pith.science/pith/BMGARRAQ7AAO2RQDQ3CEQOMXS2.json","graph_json":"https://pith.science/api/pith-number/BMGARRAQ7AAO2RQDQ3CEQOMXS2/graph.json","events_json":"https://pith.science/api/pith-number/BMGARRAQ7AAO2RQDQ3CEQOMXS2/events.json","paper":"https://pith.science/paper/BMGARRAQ"},"agent_actions":{"view_html":"https://pith.science/pith/BMGARRAQ7AAO2RQDQ3CEQOMXS2","download_json":"https://pith.science/pith/BMGARRAQ7AAO2RQDQ3CEQOMXS2.json","view_paper":"https://pith.science/paper/BMGARRAQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.01416&json=true","fetch_graph":"https://pith.science/api/pith-number/BMGARRAQ7AAO2RQDQ3CEQOMXS2/graph.json","fetch_events":"https://pith.science/api/pith-number/BMGARRAQ7AAO2RQDQ3CEQOMXS2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BMGARRAQ7AAO2RQDQ3CEQOMXS2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BMGARRAQ7AAO2RQDQ3CEQOMXS2/action/storage_attestation","attest_author":"https://pith.science/pith/BMGARRAQ7AAO2RQDQ3CEQOMXS2/action/author_attestation","sign_citation":"https://pith.science/pith/BMGARRAQ7AAO2RQDQ3CEQOMXS2/action/citation_signature","submit_replication":"https://pith.science/pith/BMGARRAQ7AAO2RQDQ3CEQOMXS2/action/replication_record"}},"created_at":"2026-05-18T00:53:45.722993+00:00","updated_at":"2026-05-18T00:53:45.722993+00:00"}