Spectral properties of the Cayley Graphs of split metacyclic groups

TL;DR

This paper investigates the spectral properties of Cayley graphs of split metacyclic groups, revealing their adjacency matrices are block circulant and establishing conditions under which these graphs are not Ramanujan.

Contribution

It provides an explicit block circulant matrix description of these Cayley graphs and extends spectral bounds to determine non-Ramanujan cases.

Findings

Adjacency matrices are block circulant with explicit blocks

Cayley graphs are not Ramanujan for certain parameters

Extension of Walker-Mieghem result to Hermitian matrices

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 $\mathcal{T}_{m,n,k} = \Gamma(\mathbb{Z}_m \ltimes_k \mathbb{Z}_n, \{(\pm 1,0),(0,\pm 1)\})$, 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 $n \geq 400$.