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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 $n geq 400$.
Let $G_{m,n,k} = mathbb{Z}_m ltimes_k mathbb{Z}_n$ be the split metacyclic group, where $k$ is a unit modulo $n$. We derive an upper bound for the diameter of $G_{m,n,k}$ using an arithmetic parameter called the textit{weight}, which depends on $n$,
Let $m,n,r$ be positive integers, and let $G=langle arangle: langle brangle cong mathbb{Z}_n: mathbb{Z}_m$ be a split metacyclic group such that $b^{-1}ab=a^r$. We say that $G$ is {em absolutely split with respect to $langle arangle$} provided that f
Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X subseteq mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ b
Let $X$ be a connected Cayley graph on an abelian group of odd order, such that no two distinct vertices of $X$ have exactly the same neighbours. We show that the direct product $X times K_2$ (also called the canonical double cover of $X$) has only t
Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an i