Using graphs as a tool to encode properties of groups is a well established approach to many problems nowadays.
Given a class of groups $\mathfrak{F}$ and a group $G$, we consider the graph whose vertices are the elements of $G$, and there is an edge between two vertices $g,h \in G$, if $\langle g,h\rangle \not \in \mathfrak F$. The subgraph induced by the non-isolated vertices is the non-$\mathfrak F$ graph of $G$. This object is a generalization of some known graphs  (e.g. the non-commuting graph defined by Paul Erdös) previously studied with ad-hoc techniques which we try to put in a general framework. We investigate mainly the set of isolated vertices, which sometimes is a subgroup with an algebraic meaning and some connectivity properties. We apply these results to various notable classes.