Group nilpotency from a graph point of view

Let ${\mathrm{\Gamma }}_G$ 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 ${\mathrm{\Gamma }}_H$ is isomorphic to ${\mathrm{\Gamma }}_G$ 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.