On Generalized Non-commuting Graph of a Finite Ring

Abstract

The non-commuting graph Γ R of a finite ring R with center Z(R) is a simple undirected graph whose vertex set is R \ Z(R) and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we show that Γ R is not isomorphic to certain graphs of any finite non-commutative ring R. Some connections between Γ R and commuting probability of R are also obtained. Further, it is shown that the non-commuting graphs of two Z-isoclinic rings are isomorphic if the centers of the rings have same order.

A generalization of commuting probability of finite rings

“…Recently, Buckley, MacHale and Ní Shé [2] have introduced Zisoclinism between two rings and showed that the commuting probabilities of two Zisoclinic finite rings are same. Dutta, Basnet and Nath have generalized the concept of Z-isoclinism between two rings in Definition 5.1 of [6]. Note that Definition 5.1 of [6] can be generalized in the following way.…”

“…Dutta, Basnet and Nath have generalized the concept of Z-isoclinism between two rings in Definition 5.1 of [6]. Note that Definition 5.1 of [6] can be generalized in the following way. Definition 4.1.…”

“…It may be mentioned here that some connections between Pr(S, K) (where S, K are subrings of R) and generalized non-commuting graph of R can be found in [6].…”

“…[4] and the references therein), but somehow people have neglected the commuting probability of finite rings. At this moment, we have very few papers [1,2,5,6,12] in the literature on the commuting probability of finite rings. Let S and K be two additive subgroups of R and r ∈ R. We define Pr r (S, K) in the following way:…”

On commuting probability of finite rings

“…Therefore, |Y | = |X 1 ||X 2 | and hence the result follows. Now we deduce a relation between Pr r (R) and the number of edges in the r-noncommuting graph of R. The r-noncommuting graph of a finite ring R, denoted by Γ r R , was introduced and studied in [6]. Recall that Γ r R is a graph whose vertex set is R and two distinct vertices x and y are adjacent if [x, y] = r and [y, x] = r. Let E(Γ r R ) denote the set of edges of the graph Γ r R .…”