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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.02530 |
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| _version_ | 1866910009256837120 |
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| author | Kubota, Sho Sekido, Hiroto Yata, Harunobu Yoshino, Kiyoto |
| author_facet | Kubota, Sho Sekido, Hiroto Yata, Harunobu Yoshino, Kiyoto |
| contents | We study perfect state transfer in Grover walks on two important classes of graphs: strongly regular graphs and strongly walk-regular graphs. The latter class is a generalization of the former. We first give a complete classification of strongly regular graphs that admit perfect state transfer. The only such graphs are the complete bipartite graph $K_{2,2}$ and the complete tripartite graph $K_{2,2,2}$. We then show that, if a connected strongly walk-regular graph that is not a strongly regular graph admits perfect state transfer, then its spectrum must be of the form $\{[k]^1, [\frac{k}{2}]^α, [0]^β, [-\frac{k}{2}]^γ\}$, and we enumerate all feasible spectra of this form up to $k=20$ with the help of a computer. These results are obtained using techniques from algebraic number theory and spectral graph theory, particularly through the analysis of eigenvalues and eigenprojections of a normalized adjacency matrix. While the setting is in quantum walks, the core discussion is developed entirely within the framework of spectral graph theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02530 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly regular and strongly walk-regular graphs that admit perfect state transfer Kubota, Sho Sekido, Hiroto Yata, Harunobu Yoshino, Kiyoto Combinatorics Quantum Physics 05C50, 81Q99 We study perfect state transfer in Grover walks on two important classes of graphs: strongly regular graphs and strongly walk-regular graphs. The latter class is a generalization of the former. We first give a complete classification of strongly regular graphs that admit perfect state transfer. The only such graphs are the complete bipartite graph $K_{2,2}$ and the complete tripartite graph $K_{2,2,2}$. We then show that, if a connected strongly walk-regular graph that is not a strongly regular graph admits perfect state transfer, then its spectrum must be of the form $\{[k]^1, [\frac{k}{2}]^α, [0]^β, [-\frac{k}{2}]^γ\}$, and we enumerate all feasible spectra of this form up to $k=20$ with the help of a computer. These results are obtained using techniques from algebraic number theory and spectral graph theory, particularly through the analysis of eigenvalues and eigenprojections of a normalized adjacency matrix. While the setting is in quantum walks, the core discussion is developed entirely within the framework of spectral graph theory. |
| title | Strongly regular and strongly walk-regular graphs that admit perfect state transfer |
| topic | Combinatorics Quantum Physics 05C50, 81Q99 |
| url | https://arxiv.org/abs/2506.02530 |