The Enhanced Power Graph and the Commuting Graph

The purpose of this note is to define a graph whose vertex set is a finite group g, whose edge set iscontained in that of the commuting graph of g and contains the enhanced power graph of g. we callthis graph the deep commuting graph of g. two elements of g are joined in the deep commuting graphif and only if their inverse images in every central extension of g commute.we give conditions for the graph to be equal to either of the enhanced power graph and the commutinggraph, and show that the automorphism group of g acts as automorphisms of the deep commuting graph.