PreprintCC BY 4.0arXiv
Spectral characterization of the uniform theta graph Θ(t,2) and classification of 6-periodic Grover walks
Abstract
We characterize the uniform theta graph Θ(t,2) by the spectrum of its normalized adjacency matrix, or equivalently, by the spectrum of its normalized Laplacian matrix. We also investigate the periodicity of Grover walks on nonregular graphs, which is closely related to the eigenvalues of the normalized adjacency matrix and those of the time evolution matrix of the Grover walk. We show that the Dutch windmill graph Dn(t) is 2n-periodic and that the uniform theta graph Θ(t,n) is (2n+2)-periodic. Furthermore, we completely determine the connected $6$-periodic graphs and prove that they are precisely D3(t) with t ≥ 2 and Θ(t,2) with t ≥ 1.
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BibTeX
@misc{kubota2026spectral,
title = {{Spectral characterization of the uniform theta graph $Θ(t,2)$ and classification of 6-periodic Grover walks}},
author = {Sho Kubota},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08315},
eprint = {2609.08315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08315},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral characterization of the uniform theta graph Θ(t,2) and classification of 6-periodic Grover walks
AU - Sho Kubota
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08315
UR - https://arxiv.org/abs/2609.08315
AB - We characterize the uniform theta graph Θ(t,2) by the spectrum of its normalized adjacency matrix, or equivalently, by the spectrum of its normalized Laplacian matrix. We also investigate the periodicity of Grover walks on nonregular graphs, which is closely related to the eigenvalues of the normalized adjacency matrix and those of the time evolution matrix of the Grover walk. We show that the Dutch windmill graph D_n^(t) is 2n-periodic and that the uniform theta graph Θ(t,n) is (2n+2)-periodic. Furthermore, we completely determine the connected $6$-periodic graphs and prove that they are precisely D_3^(t) with t ≥ 2 and Θ(t,2) with t ≥ 1.
ER -