Summary: An analysis is given of \(\ast\)-representations of rank 2 single vertex graphs. We develop dilation theory for the non-selfadjoint algebras \(\mathcal A_\theta\) and \(\mathcal A_u\) 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 \(M_m(\mathbb C)\otimes M_n(\mathbb C)\). We show that a defect free row contractive representation has a unique minimal dilation to a \(\ast\)-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 \(\mathcal A_u\) 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 \(\mathcal A_\theta\) need not be completely contractive; and that the universal tensor algebra \(\mathcal T_+(X_u)\) need not be isometrically isomorphic to \(\mathcal A_u\).