Spectral characterization of the uniform theta graph $\Theta(t,2)$ and classification of 6-periodic Grover walks

Abstract:We characterize the uniform theta graph $\Theta(t,2)$ 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 $\Theta(t,n)$ is $(2n+2)$-periodic. Furthermore, we completely determine the connected $6$-periodic graphs and prove that they are precisely $D_3^{(t)}$ with $t \geq 2$ and $\Theta(t,2)$ with $t \geq 1$.

Combinatorics (math.CO); Quantum Physics (quant-ph)

arXiv:2609.08315 [math.CO]

(or arXiv:2609.08315v1 [math.CO] for this version)

https://doi.org/10.48550/arXiv.2609.08315 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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