Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of C*(𝔽+θ). The periodic C*-algebras are characterized, and it is shown that C*(𝔽+θ) ≃ C(𝕋) ⊗ 𝒰 where 𝒰 is a simple C*-algebra.
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Davidson et al. (2009) studied this question.
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