On a commuting graph on conjugacy classes of groups

There are many possible ways for associating a graph with a group or with a ring, for the purpose of investigating these algebraic structures using properties of the associated graph. For example, many authors studied the so-called commuting graph of a group, whose vertices are the nontrivial elements of G and two vertices a, b are connected if ab = ba. This graph has been recently used in order to show that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. There are also many results concerning graphs associated with conjugacy classes in groups. In this article we consider the following graph Gamma = Gamma(G) associated with the conjugacy classes of a nontrivial group G. The vertices V(Gamma) of Gamma are the nontrivial conjugacy classes of G, and we join two different vertices C, D, whenever there exist x in C and y in D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Gamma(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Gamma(G) has at most two components, each of diameter at most 9. More generally, if G is any locally finite group, then Gamma(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Gamma(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes.