Given a binary relation on the elements (or subgroups) of a group, it is natural to study the properties of the graph encoding that relation. A well-known example is the generating graph, whose vertices are the non-identity elements of the group, and whose edges are its generating pairs. Famous results here include the fact that the generating graph of a non-abelian finite simple group is connected with diameter 2 (Breuer, Guralnick and Kantor, 2008), and more generally, if the generating graph of a finite group has no isolated vertices, then its diameter is at most 2 (Burness, Guralnick and Harper, 2021).
Consider now the non-commuting, non-generating graph of a group, obtained by taking the complement of the generating graph, removing edges between elements that commute, and finally removing all vertices corresponding to central elements. We will explore the connectedness and diameter of this graph for various families of (finite and infinite) groups. We will also discuss the diameter of a related graph: the intersection graph of a finite simple group. Here, the vertices are the proper nontrivial subgroups, with edges corresponding to pairs of subgroups that intersect nontrivially.