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

• URL: http://arxiv.org/abs/2405.02122v2
• Date: Tue, 25 Jun 2024 15:54:39 GMT
• Title: Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$
• Authors: Akash Kalita, Bikash Bhattacharjya,
• Abstract summary: We study the existence of perfect state transfer on Cayley graphs $textCay(V_8n, S)$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\lvert H(\tau_0)_{uv} \rvert = 1$. For a positive integer $n$, the group $V_{8n}$ is defined as $V_{8n} := \langle a,b \colon a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. 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 $\text{Cay}(V_{8n}, S)$.

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