  Let $G$ be a finite simple graph. We give a lower bound for the
Castelnuovo-Mumford regularity of the toric ideal $I_G$ associated to $G$ in
terms of the sizes and number of induced complete bipartite graphs in $G$. When
$G$ is a chordal bipartite graph, we find an upper bound for the regularity of
$I_G$ in terms of the size of the bipartition of $G$. We also give a new proof
for the graded Betti numbers of the toric ideal associated to the complete
bipartite graph $K_{2,n}$.
