Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914848581877760 |
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| author | Kalita, Akash Bhattacharjya, Bikash |
| author_facet | Kalita, Akash Bhattacharjya, Bikash |
| contents | The \textit{transition matrix} of a graph $Γ$ with adjacency matrix $A$ is defined by $H(τ) := \exp(-\mathbf{i}τA)$, where $τ\in \mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$. The graph $Γ$ exhibits \textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there exists $τ_0(>0)\in \mathbb{R}$ such that $\lvert H(τ_0)_{uv} \rvert = 1$. 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 $\text{Cay}(V_{8n}, S)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02122 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$ Kalita, Akash Bhattacharjya, Bikash Quantum Physics Combinatorics 05C25, 81P45, 81Q35 The \textit{transition matrix} of a graph $Γ$ with adjacency matrix $A$ is defined by $H(τ) := \exp(-\mathbf{i}τA)$, where $τ\in \mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$. The graph $Γ$ exhibits \textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there exists $τ_0(>0)\in \mathbb{R}$ such that $\lvert H(τ_0)_{uv} \rvert = 1$. 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 $\text{Cay}(V_{8n}, S)$. |
| title | Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$ |
| topic | Quantum Physics Combinatorics 05C25, 81P45, 81Q35 |
| url | https://arxiv.org/abs/2405.02122 |