  Let $\Delta$ be a 1-dimensional simplicial complex. Then $\Delta$ may be
identified with a finite simple graph $G$. In this article, we investigate the
toric ring $R_G$ of $G$. All graphs $G$ such that $R_G$ is a normal domain are
classified. For such a graph, we determine the set $\mathcal{P}_G$ of height
one monomial prime ideals of $R_G$. In the bipartite case, and in the case of
whiskered cycles, this set is explicitly described. As a consequence, we
determine the canonical class $[\omega_{R_G}]$ and characterize the Gorenstein
property of $R_G$. For a bipartite graph $G$, we show that $R_G$ is Gorenstein
if and only if $G$ is unmixed. For a subclass of non-bipartite graphs $G$,
which includes whiskered cycles, $R_G$ is Gorenstein if and only if $G$ is
unmixed and has an odd number of vertices. Finally, it is proved that $R_G$ is
a pseudo-Gorenstein ring if $G$ is an odd cycle.

Commutative Algebra