Spectral properties of the Cayley Graphs of split metacyclic groups

Abstract

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 ${\mathcal{T}}_{m,n,k}=\Gamma ({\mathbb{Z}}_m{⋉}_k{\mathbb{Z}}_n,\{(\pm 1,0),(0,\pm 1)\})$, where $m,n\ge 3$ and $k^m\equiv 1(\mathrm{mod}n)$. 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 $n\ge 400$.

arXiv comment18 pages, 1 table

References in corpus (1)

• Diameter 2 Cayley Graphs of Dihedral Groups