{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T04:12:30Z","timestamp":1772770350330,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,9,19]],"date-time":"2021-09-19T00:00:00Z","timestamp":1632009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let R be a finite ring and r\u2208R. The r-noncommuting graph of R, denoted by \u0393Rr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]\u2260r and [x,y]\u2260\u2212r. In this paper, we obtain expressions for vertex degrees and show that \u0393Rr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that \u0393Rr is a tree, in particular a star graph. It is also shown that \u0393R1r and \u0393R2\u03c8(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (\u03d5,\u03c8). Further, we consider the induced subgraph \u0394Rr of \u0393Rr (induced by the non-central elements of R) and obtain results on clique number and diameter of \u0394Rr along with certain characterizations of finite noncommutative rings such that \u0394Rr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n\u22646.<\/jats:p>","DOI":"10.3390\/axioms10030233","type":"journal-article","created":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T07:52:14Z","timestamp":1632124334000},"page":"233","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On r-Noncommuting Graph of Finite Rings"],"prefix":"10.3390","volume":"10","author":[{"given":"Rajat Kanti","family":"Nath","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Tezpur University, Sonitpur 784028, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Monalisha","family":"Sharma","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Tezpur University, Sonitpur 784028, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Parama","family":"Dutta","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lakhimpur Girls\u2019 College, Lakhimpur 787031, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2817-3400","authenticated-orcid":false,"given":"Yilun","family":"Shang","sequence":"additional","affiliation":[{"name":"Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1550027","DOI":"10.1142\/S1793830915500275","article-title":"Non-commuting graphs of rings","volume":"7","author":"Erfanian","year":"2015","journal-title":"Discret. 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