  For all integers $4 \leq r \leq d$, we show that there exists a finite simple
graph $G= G_{r,d}$ with toric ideal $I_G \subset R$ such that $R/I_G$ has
(Castelnuovo-Mumford) regularity $r$ and $h$-polynomial of degree $d$. To
achieve this goal, we identify a family of graphs such that the graded Betti
numbers of the associated toric ideal agree with its initial ideal, and
furthermore, this initial ideal has linear quotients. As a corollary, we can
recover a result of Hibi, Higashitani, Kimura, and O'Keefe that compares the
depth and dimension of toric ideals of graphs.
