  Let $G$ be a finite connected simple graph and $I_{G}$ the toric ideal of the
edge ring $K[G]$ of $G$. In the present paper, we study finite graphs $G$ with
the property that $I_{G}$ is generated by quadratic binomials and $I_{G}$
possesses no quadratic Gr\"obner basis. First, we give a nontrivial infinite
series of finite graphs with the above property. Second, we implement a
combinatorial characterization for $I_{G}$ to be generated by quadratic
binomials and, by means of the computer search, we classify the finite graphs
$G$ with the above property, up to 8 vertices.
