On generalized non-commuting graph of a finite ring

2016/04/13 by Jutirekha Dutta, Dhiren Kumar Basnet, Dutta, Jutirekha +3
Mathematics · #05C25 #16U70 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:05C25 #msc:16U70

paper · pdf · doi:10.48550/arxiv.1604.03671

13 pages

arxiv created 2016/04/13 · arxiv updated 2016/04/14

Abstract

Let S, K be two subrings of a finite ring R. Then the generalized non-commuting graph of subrings S, K of R, denoted by Γ_{S, K}, is a simple graph whose vertex set is (S ∪ K) ∖ (C_K(S) ∪ C_S(K)) and two distinct vertices a, b are adjacent if and only if a ∈ S or b ∈ S and ab ≠ ba. We determine the diameter, girth and some dominating sets for Γ_{S, K}. Some connections between the Γ_{S, K} and Pr(S, K) are also obtained. Further, \Z-isoclinism between two pairs of finite rings is defined and showed that the generalized non-commuting graphs of two \Z-isoclinic pairs are isomorphic under some condition.

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