{"paper":{"title":"Relative non-commuting graph of a finite ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Dhiren Kumar Basnet, Jutirekha Dutta","submitted_at":"2017-05-05T10:31:33Z","abstract_excerpt":"Let $S$ be a subring of a finite ring $R$ and $C_R(S) = \\{r \\in R : rs = sr \\;\\forall\\; s \\in S\\}$. The relative non-commuting graph of the subring $S$ in $R$, denoted by $\\Gamma_{S, R}$, is a simple undirected graph whose vertex set is $R \\setminus C_R(S)$ and two distinct vertices $a, b$ are adjacent if and only if $a$ or $b \\in S$ and $ab \\neq ba$. In this paper, we discuss some properties of $\\Gamma_{S, R}$, determine diameter, girth, some dominating sets and chromatic index for $\\Gamma_{S, R}$. Also, we derive some connections between $\\Gamma_{S, R}$ and the relative commuting probability"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02161","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"}