D. Chillag et al., ON THE DIAMETER OF A GRAPH RELATED TO CONJUGACY CLASSES OF GROUPS, Bulletin of the London Mathematical Society, 25, 1993, pp. 255-262
Let G be a finite group. Attach to G the following graph F: its vertic
es are the non-central conjugacy classes of G, and two vertices are co
nnected if their cardinalities are not co-prime. Denote by n(GAMMA) th
e number of the connected components of GAMMA. By [1], n(GAMMA) less-t
han-or-equal-to 2 for all finite groups, and if GAMMA is connected. th
e diameter of the graph is at most 4. In this paper we prove that if G
AMMA is connected, then the diameter of the graph is at most 3, and th
is bound is the best possible. Similar results are proved for infinite
FC-groups.