On the punctured power graph of a finite group

The punctured power graph P (G) of a finite group G is the graph which has as vertex set the nonidentity elements of G, where two distinct elements are adjacent if one is a power of the other. We show that P (G) has diameter at most 2 if and only if G is nilpotent and every Sylow subgroup of G is either a cyclic group or a generalized quaternion 2-group. Also, we show that if G is a finite group and P (G) has diameter 3, then G is not simple. Finally, we show that P (G) is Eulerian if and only if G is a cyclic 2-group or a generalized quaternion 2-group.