  With a simple graph $G$ on $[n]$, we associate a binomial ideal $P_G$
generated by diagonal minors of an $n \times n$ matrix $X=(x_{ij})$ of
variables. We show that for any graph $G$, $P_G$ is a prime complete
intersection ideal and determine the divisor class group of $K[X]/ P_G$. By
using these ideals, one may find a normal domain with free divisor class group
of any given rank.
