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

  • Ghodratollah Aalipour
  • Saieed Akbari
  • Peter J. Cameron
  • Reza Nikandish
  • Farzad Shaveisi
Keywords: Clique number, Chromatic number, Independence number, Group


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.

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