On the directed normalizing graph associated with a group

In this paper we investigate the $directed$ $normalizing$ $graph$ associated with a group $G$, defined as the simple directed graph whose vertices are the elements of $G$, with an arrow from $x$ to $y$ whenever the subgroup $\langle x \rangle$ is normal in $\langle x, y \rangle$. Our analysis focuses on the set of bidirectional universal vertices and, in particular, on the induced subgraph obtained by removing them, where the most interesting connectivity phenomena occur. We characterize the groups for which this induced subgraph is strongly connected and determine bounds for its diameter. Finally, we show how properties of this graph reflect algebraic features of the underlying group.