Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$

Abstract

The \textit{transition matrix} of a graph $\Gamma$ with adjacency matrix $A$ is defined by $H(\tau ):=\exp (-\mathbf{i}\tau A)$, where $\tau \in \mathbb{R}$ and $\mathbf{i}=\sqrt{-1}$. The graph $\Gamma$ exhibits \textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there exists ${\tau }_0(>0)\in \mathbb{R}$ such that $|H({\tau }_0{)}_{uv}|=1$. For a positive integer $n$, the group $V_{8n}$ is defined as $V_{8n}:=⟨a,b:a^{2n}=b^4=1,ba=a^{-1}{b}^{-1},b^{-1}a=a^{-1}b⟩$. In this paper, we study the existence of perfect state transfer on Cayley graphs $\text{Cay}(V_{8n},S)$. We present some necessary and sufficient conditions for the existence of perfect state transfer on $\text{Cay}(V_{8n},S)$.