2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159663Let $\Fth$ 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 $H_θ$ of $\bZ^\Bk$. If this has rank $s$, then $\ca(\Fth) \cong \rC(\bT^s) \otimes \fA$ for some simple C*-algebra $\fA$.30 pagesOperator AlgebrasFunctional Analysis47L55; 47L30; 47L75; 46L05text