On r-Noncommuting Graph of Finite Rings

Nath, Rajat Kanti, Sharma, Monalisha, Dutta, Parama and Shang, Yilun (2021) On r-Noncommuting Graph of Finite Rings. Axioms, 10 (3). p. 223. ISSN 2075-1680

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Official URL: https://doi.org/10.3390/axioms10030233


Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if x,y≠r and x,y≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr 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≤6.

Item Type: Article
Uncontrolled Keywords: finite ring; noncommuting graph; isoclinism
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Computer and Information Sciences
Depositing User: Elena Carlaw
Date Deposited: 20 Sep 2021 10:10
Last Modified: 20 Sep 2021 10:15
URI: http://nrl.northumbria.ac.uk/id/eprint/47260

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