Expander property of the Cayley Graphs of $\mathbb{Z}_m \ltimes \mathbb{Z}_n$

Kashyap Rajeevsarathy, S. Lakshmivarahan, Pawan Kumar Arora

Published 2016-09-20Version 1

Let $\mathcal{T}_{m,n,k}$ denote the Cayley Graph of the split metacyclic group $\mathbb{Z}_m \ltimes_k \mathbb{Z}_n$ with respect to the symmetric generating set $S = \{(\pm,0),(0,\pm 1)\}$. In this paper, we derive conditions for $\mathcal{T}_{m,n,k}$ to be a Ramanujan graph. In particular, we establish that $\mathcal{T}_{m,n,k}$ is not Ramanujan when $\min(m,n) > 8$, and hence, there are only finitely many Ramanujan graphs in the family $\{\mathcal{T}_{m,n,k}\}$. We explicitly list all possible $(m,n,k)$ for which $\mathcal{T}_{m,n,k}$ is Ramanujan. We conclude that the graphs in the family $\{\mathcal{T}_{m,n,k}\}$ are not good spectral expanders for large $m\text{ and } n$, which make them unsuitable for adaptation into large communication networks. Finally, we draw similar conclusions for the Cayley graphs of finite abelian groups.