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.

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No abstract

The relative commutativity degree of a subgroup H of a finite group G, denoted by Pr(H, G), is the probability that an element of G commutes with an element of H. In this article we obtain some lower and upper bounds for Pr(H, G) and their consequences. We also study an invariance property of Pr(H, G) and its generalizations, under isoclinism of pairs of groups.