The non-commuting, non-generating graph of a nilpotent group

Abstract

For a nilpotent group G, let Ξ(G) be the difference between the complement of the generating graph of G and the commuting graph of G, with vertices corresponding to central elements of G removed. That is, Ξ(G) has vertex set G\ Z(G), with two vertices adjacent if and only if they do not commute and do not generate G. Additionally, let Ξ + (G) be the subgraph of Ξ(G) induced by its non-isolated vertices. We show that if Ξ(G) has an edge, then Ξ + (G) is connected with diameter 2 or 3, with Ξ(G) = Ξ + (G) in the diameter 3 case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When G is finite, we explore the relationship between the structures of G and Ξ(G) in more detail.