### On the Structure of the Power Graph and the Enhanced Power Graph of a Group

#### Abstract

Let $G$ be a group. The *power graph* of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the *commuting graph* by introducing a new graph which lies in between. We call this new graph as the *enhanced power graph*. For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal.