Groups and graphs in probability theory - Sarmin, N. H. (American Institute of Physics Inc., 2016)
In this paper, G denotes a dihedral group of order 2n and Ω denotes the set of all subsets of all commuting elements of size two in the form of (a,b), where a and b commute and |a| = |b| = 2. By extending the concept of commutativity degree, the probability that an element of a group fixes a set can be acquired using the group actions on set. In this paper, the probability that an element of G fixes the set Ω under regular action is computed. The results obtained are then applied to graph theory, more precisely to generalized conjugacy class graph and orbit graph.