Abstract Let M be a non-commutative gamma ring and Z Γ M Z (M) denote the center of the gamma ring M. The vertices a and b are consecutive if a ≠ b and aαb = bαa for every α ∈ Γ, with vertices taken from the set M − Z Γ M M-Z (M). This graph is called the commuting graph of the gamma ring M. We show that the complement graph of the commuting graph of M is connected but not a complete bipartite graph. Additionally, if the diameter of the complement graph of the commuting graph is 1, then M M must equal 4. It is also shown that the complement graphs of the commuting graphs for all Γ-rings of order greater than 16 are planar. Furthermore, the commuting graph of a Γ-ring M, where the order is p 2 or p 3 for a prime p, is the disjoint union of some complete graphs.
Okan Arslan (Thu,) studied this question.