## Enhanced Power Graphs of Finite Groups. (arXiv:1810.07627v1 [math.CO])

The enhanced power graph $\mathcal G_e(\mathbf G)$ of a group $\mathbf G$ is
the graph with vertex set $G$ such that two vertices $x$ and $y$ are adjacent
if they are contained in a same cyclic subgroup. We prove that finite groups
with isomorphic enhanced power graphs have isomorphic directed power graphs. We
show that any isomorphism between power graphs of finite groups is an
isomorhism between enhanced power graphs of these groups, and we find all
finite groups $\mathbf G$ for which $\mathrm{Aut}(\mathcal G_e(\mathbf G)$ is
abelian, all finite groups $\mathbf G$ with $\lvert\mathrm{Aut}(\mathcal
G_e(\mathbf G)\rvert$ being prime power, and all finite groups $\mathbf G$ with
$\lvert\mathrm{Aut}(\mathcal G_e(\mathbf G)\rvert$ being square free. Also we
describe enhanced power graphs of finite abelian groups. Finally, we give a
characterization of finite nilpotent groups whose enhanced power graphs are
perfect, and we present a sufficient condition for a finite group to have
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