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In this essay we present a recursive formula to determine the \\emph{independence number} $\\alpha(J_n(1)) = |\\Bbb I|$ with, $\\Bbb I = \\{v_{i,j}| v_1 = v_{1,1} \\in \\Bbb I$ and $v_i = v_{i,j} =v_{(d^+(v_{m, (j-1)}) + m +1)}\\}.$ We also prove that for the Jaco Graph, $J_n(1), n \\in \\Bbb N$ with the prime Jaconian vertex $v_i$ the chromatic number, $\\chi(J_n(1))$ is given by: \\begin{equation*} \\chi(J_n(1)) \\begin{cases} = (n-i) + 1, &\\text{if and only if the edge $v_iv_n$ exists,}\\\\ \\\\ = n-i &\\text{otherwise.} \\end{cases} \\end{equation*} We fu"},"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":"1410.8328","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2014-10-30T11:14:30Z","cross_cats_sorted":[],"title_canon_sha256":"ffbc931740ab19b5c3293a18037bd4fd8636a0666b4dce503d101f15697e4a49","abstract_canon_sha256":"e51be8dcebdcc32d115252646ecb29395b7656407a3aac6096e709b588b79179"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:00.134601Z","signature_b64":"M6rKsCLoCC54Zj1EkHqGhckAyWJhcSK5AEoEcoCpAqf74WWZmCYg1hGaOrwX7iK9yzSN1F8zGEeK4WKSm0GyDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"495bb6b1ee07108170716162e23629bd7c93ac8684523411347d915c9a261f56","last_reissued_at":"2026-05-18T02:39:00.134097Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:00.134097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Contemplating some invariants of the Jaco Graph, $J_n(1), n \\in \\Bbb N$","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Johan Kok, Susanth C","submitted_at":"2014-10-30T11:14:30Z","abstract_excerpt":"Kok et.al. [7] introduced Jaco Graphs (\\emph{order 1}). In this essay we present a recursive formula to determine the \\emph{independence number} $\\alpha(J_n(1)) = |\\Bbb I|$ with, $\\Bbb I = \\{v_{i,j}| v_1 = v_{1,1} \\in \\Bbb I$ and $v_i = v_{i,j} =v_{(d^+(v_{m, (j-1)}) + m +1)}\\}.$ We also prove that for the Jaco Graph, $J_n(1), n \\in \\Bbb N$ with the prime Jaconian vertex $v_i$ the chromatic number, $\\chi(J_n(1))$ is given by: \\begin{equation*} \\chi(J_n(1)) \\begin{cases} = (n-i) + 1, &\\text{if and only if the edge $v_iv_n$ exists,}\\\\ \\\\ = n-i &\\text{otherwise.} \\end{cases} \\end{equation*} We fu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8328","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":"1410.8328","created_at":"2026-05-18T02:39:00.134169+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.8328v1","created_at":"2026-05-18T02:39:00.134169+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8328","created_at":"2026-05-18T02:39:00.134169+00:00"},{"alias_kind":"pith_short_12","alias_value":"JFN3NMPOA4II","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"JFN3NMPOA4IIC4DR","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"JFN3NMPO","created_at":"2026-05-18T12:28:33.132498+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/JFN3NMPOA4IIC4DRMFROENRJXV","json":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV.json","graph_json":"https://pith.science/api/pith-number/JFN3NMPOA4IIC4DRMFROENRJXV/graph.json","events_json":"https://pith.science/api/pith-number/JFN3NMPOA4IIC4DRMFROENRJXV/events.json","paper":"https://pith.science/paper/JFN3NMPO"},"agent_actions":{"view_html":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV","download_json":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV.json","view_paper":"https://pith.science/paper/JFN3NMPO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.8328&json=true","fetch_graph":"https://pith.science/api/pith-number/JFN3NMPOA4IIC4DRMFROENRJXV/graph.json","fetch_events":"https://pith.science/api/pith-number/JFN3NMPOA4IIC4DRMFROENRJXV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/action/storage_attestation","attest_author":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/action/author_attestation","sign_citation":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/action/citation_signature","submit_replication":"https://pith.science/pith/JFN3NMPOA4IIC4DRMFROENRJXV/action/replication_record"}},"created_at":"2026-05-18T02:39:00.134169+00:00","updated_at":"2026-05-18T02:39:00.134169+00:00"}