  Let $\Gamma_G$ denote a graph associated with a group $G$. A compelling
question about finite groups asks whether or not a finite group $H$ must be
nilpotent provided $\Gamma_H$ is isomorphic to $\Gamma_G$ for a finite
nilpotent group $G$. In the present work we analyze the problem for different
graphs that one can associate with a finite group, both reporting on existing
answers and contributing to new ones.
