Toric ideals and diagonal 2-minors

Let $G$ be a simple graph on the vertex set ${1,\ldots,n}$. An algebraic object attached to $G$ is the ideal $P_G$ generated by diagonal 2-minors of an $n \times n$ matrix of variables. In this paper we first provide some general results concerning the ideal $P_G$. It is also proved that if $G$ is bipartite, then every initial ideal of $P_G$ is generated by squarefree monomials. Furthermore, we completely characterize all graphs $G$ for which $P_G$ is the toric ideal associated to a finite simple graph. As a byproduct we obtain classes of toric ideals associated to non-bipartite graphs which have quadratic Gröbner bases. Finally, we provide information in certain cases about the universal Gröbner basis of $P_G$.