Grover walks on unitary Cayley graphs and integral regular graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911930832125952 |
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| author | Bhakta, Koushik Bhattacharjya, Bikash |
| author_facet | Bhakta, Koushik Bhattacharjya, Bikash |
| contents | The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices $u$ and $v$ are adjacent if $\gcd(u - v, n) = 1$. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01020 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Grover walks on unitary Cayley graphs and integral regular graphs Bhakta, Koushik Bhattacharjya, Bikash Combinatorics 05C50 The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices $u$ and $v$ are adjacent if $\gcd(u - v, n) = 1$. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs. |
| title | Grover walks on unitary Cayley graphs and integral regular graphs |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2405.01020 |