Let G be a finite graph allowing loops, having no multiple edge and no isolated vertex. The edge polytope P_G is the convex hull of all columns of the vertex-edge incidence matrix of G. Let I_G denote the toric ideal of P_G. It is known that several properties of P_G and I_G are characterized by cycles, closed walks etc. of G.

By classifying graphs whose edge polytope is simple, it is proved that the toric ideals I_G of G possesses a quadratic Groebner basis (i.e., P_G possesses a quadratic triangulation) if the edge polytope P_G of G is simple. This result is related with a conjecture on toric ideals of smooth polytopes and the conjecture is positive for edge polytopes. If possible, I would like to explain how to compute the Ehrhart polynomial and the normalized volume of simple edge polytopes by using Groebner bases.

This is a joint work with Takayuki Hibi.