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In this paper, we analyze the spectral properties of the Cayley graphs $\\mathcal{T}_{m,n,k} = \\Gamma(\\mathbb{Z}_m \\ltimes_k \\mathbb{Z}_n, \\{(\\pm 1,0),(0,\\pm 1)\\})$, where $m,n \\geq 3$ and $k^m \\equiv 1 \\pmod{n}$. We show that the adjacency matrix of $\\mathcal{T}_{m,n,k}$, upto relabeling, is a block circulant matrix, and we also obtain an explicit description of these blocks. 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