  Let $G$ be a finite graph allowing loops, having no multiple edge and no
isolated vertex. We associate $G$ with the edge polytope ${\cal P}_G$ and the
toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is
proved that the toric ideals $I_G$ of $G$ possesses a quadratic Gr\"obner basis
if the edge polytope ${\cal P}_G$ of $G$ is simple. It is also shown that, for
a finite graph $G$, the edge polytope is simple but not a simplex if and only
if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the
normalized volume of simple edge polytopes are computed.
