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 $n geq 400$.