Spectral properties of the Cayley Graphs of split metacyclic groups

Let $\Gamma(G,S)$ denote the Cayley graph of a group $G$ with respect to a set $S \subset G$. 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. By extending a result due to Walker-Mieghem to Hermitian matrices, we show that $\mathcal{T}_{m,n,k}$ is not Ramanujan, when either $m > 8$, or $n \geq 400$.