Set [a,g]=a -1 g -1 ag and [a,G]={[a,g] ? g ? G}. Let C G (a)={g? G ? ga=ag} be the centralizer of a in G. One can check that [ab] G is a subset of [a] G [b] G and thus if [a] G [b] G is a conjugacy then [a] G [b] G =[ab] G . Attach to a finite group G the following graph ?(G); its vertices are the non-central conjugacy vertices are connected if their cardinalities are not coprime. Denote by ?(G) the number of the connected components of ?(G). For an arbitrary finite grou G we show that n(?(G)) ? 2. Also a finite group G atisfies n(?(G)) ? 2 if and only if G is quasi-Frobenius with abelian kernel and complement. Furthermore the following list is the complete list of all G uch that (?(G)) contains no triangles: Symmetric grou S 3 , the dihedral group D 5 , the three pairwise non-isomorphic non-abelian group of order 12 and the non-abelian grou T 21 of order 21.