  An analysis is given of $*$-representations of rank 2 single vertex graphs.
We develop dilation theory for the non-selfadjoint algebras $\A_\theta$ and
$\A_u$ which are associated with the commutation relation permutation $\theta$
of a 2 graph and, more generally, with commutation relations determined by a
unitary matrix $u$ in $M_m(\bC) \otimes M_n(\bC)$. We show that a defect free
row contractive representation has a unique minimal dilation to a
$*$-representation and we provide a new simpler proof of Solel's row isometric
dilation of two $u$-commuting row contractions. Furthermore it is shown that
the C*-envelope of $\A_u$ is the generalised Cuntz algebra $\O_{X_u}$ for the
product system $X_u$ of $u$; that for $m\geq 2 $ and $n \geq 2 $ contractive
representations of $\Ath$ need not be completely contractive; and that the
universal tensor algebra $\T_+(X_u)$ need not be isometrically isomorphic to
$\A_u$.
