  We extended the definition of regular dilation to graph products of
$\mathbb{N}$, which is an important class of quasi-lattice ordered semigroups.
Two important results in dilation theory are unified under our result: namely,
Brehmer's regular dilation on $\mathbb{N}^k$ and Frazho-Bunce-Popescu's
dilation of row contractions. We further show that a representation of a graph
product has an isometric Nica-covariant dilation if and only if it is
$\ast$-regular. A special case of our result was considered by Popescu, and we
studied the connection with Popescu's work.
