{"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"}