Group nilpotency from a graph point of view

Let ΓG 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 ΓH is isomorphic to ΓG 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.