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We describe the primitive ideal space of the $C^{\ast}$-algebra of arow-finite $k$-graph with no sources when every ideal is gauge invariant. Wecharacterize which spectral spaces can occur, and compute the primitive idealspace of two examples. In order to do this we prove some new results onaperiodicity. Our computations indicate that when every ideal is gaugeinvariant, the primitive ideal space only depends on the 1-skeleton of the$k$-graph in question.

Aperiodicity and the primitive ideal space of a row-finite $k$-graph $C^*$-algebra