Let R be a finite ring and r ∈ R. The aim of this paper is to study the probability that the commutator of a randomly chosen pair of elements of R equals r.

In this paper, we compute Laplacian energy of the non-commuting graphs of some classes of finite non-abelian groups.

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.

The commuting graph of a non-commutative ring R with center Z(R) is a simple undirected graph whose vertex set is R \ Z(R) and two vertices x, y are adjacent if and only if xy = yx. In this paper, we compute the spectrum and genus of commuting graphs of some classes of finite rings.

A finite or infinite group is called an n-centralizer group if it has n numbers of distinct centralizers. In this paper, we prove that a finite or infinite group G is a 4-centralizer group if and only if G/Z(G) is isomorphic to C2×C2. This extends a result of Belcastro and Sherman.