Semiregularity and connectivity of the non-graph of a finite group

Given a class of finite groups, we consider the graph Σ(G) whose vertices are the elements of G and where two vertices g,h G are adjacent if and only if g,h. Moreover, we denote by (G) the set of the isolated vertices of (G) and by (G) the graph obtained from (G) by deleting the isolated vertices. We address the following question: to what extent the fact that (H) is a subgroup of H for any H ≤ G, implies that the graph (G) is connected