  Let $\Fth$ be a $\Bk$-graph on a single vertex. We show that every
irreducible atomic $*$-representation is the minimal $*$-dilation of a group
construction representation. It follows that every atomic representation
decomposes as a direct sum or integral of such representations. We characterize
periodicity of $\Fth$ and identify a symmetry subgroup $H_\theta$ of $\bZ^\Bk$.
If this has rank $s$, then $\ca(\Fth) \cong \rC(\bT^s) \otimes \fA$ for some
simple C*-algebra $\fA$.
