The non-$\mathfrak F$-graph of a finite group

Abstract

Given a formation $\mathfrak F$, we consider the graph whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h \rangle \notin\mathfrak F$. We are interested in the two following questions. Is the set of the isolated vertices of this graph a subgroup of $G$? Is the subgraph obtained by deleting the isolated vertices a connected graph?