On some properties of enhanced power graph

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Given a group G, the enhanced power graph of G denoted by Gₑ(G), is the graph with vertex set G and two distinct vertices x, y are edge connected in Gₑ(G) if there exists z∈ G such that x=zᵐ and y=zⁿ, for some m, n∈ N. In this article, we characterize the enhanced power graph Gₑ(G) of G. The graph Gₑ(G) is complete if and only if G is cyclic, and Gₑ(G) is Eulerian if and only if |G| is odd. We classify all abelian groups and also all non-abelian p-groups G for which Gₑ(G) satisfies the cone property.