  Let $G$ be a simple graph on the vertex set $\{1,\ldots,n\}$ with $m$ edges.
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 prove that if
$G$ is bipartite, then every initial ideal of $P_{G}$ is generated by
squarefree monomials of degree at most $\left \lfloor{\frac{m+n+1}{2}} \right
\rfloor$. Furthermore, we completely characterize all connected graphs $G$ for
which $P_{G}$ is the toric ideal associated to a finite simple graph. Finally
we compute in certain cases the universal Gr{\"o}bner basis of $P_{G}$.
