  There are a variety of ways to associate directed or undirected graphs to a
group. It may be interesting to investigate the relations between the structure
of these graphs and characterizing certain properties of the group in terms of
some properties of the associated graph. The power graph $\mathcal{P}(G)$ of a
group $G$ is a simple graph whose vertex-set is $G$ and two vertices $x$ and
$y$ in $G$ are adjacent if and only if $y=x^m$ or $x=y^m$ for some positive
integer $m$. We also pay attention to the subgraph $\mathcal{P}^\ast(G)$ of
$\mathcal{P}(G)$ which is obtained by deleting the vertex 1 (the identity
element of $G$). In the present paper, we first investigate some properties of
the power graph $\mathcal{P}(G)$ and the subgraph $\mathcal{P}^\ast(G)$. We
next prove that many of finite groups such as finite simple groups, symmetric
groups and the automorphism groups of sporadic simple groups can be uniquely
determined by their power graphs among all finite groups. We have also
determined up to isomorphism the structure of any finite group $G$ such that
the graph $\mathcal{P}^\ast(G)$ is a strongly regular graph, a bipartite graph,
a planar graph or an Eulerian graph. Finally, we obtained some infinite
families of finite groups such that the graph $\mathcal{P}^\ast(G)$ containing
some cut-edges.
