On r-Noncommuting Graph of Finite Rings

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