Let G be a finite connected simple graph and IG​ the toric ideal of the
edge ring K[G] of G. In the present paper, we study finite graphs G with
the property that IG​ is generated by quadratic binomials and IG​
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 IG​ 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.Comment: 11 pages, 17 figures, Typos corrected, Reference adde