均匀 theta 图 $\Theta(t,2)$ 的谱刻画与 6 周期 Grover 游走的分类
Spectral characterization of the uniform theta graph $Θ(t,2)$ and classification of 6-periodic Grover walks
- Aichi University of Education（爱知教育大学）
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AI总结：
本文通过谱刻画均匀 theta 图，并证明荷兰风车图与均匀 theta 图的周期性，完全分类了连通的 6 周期图。
AI中文摘要：
我们通过均匀 theta 图 $\Theta(t,2)$ 的归一化邻接矩阵的谱，或等价地，通过其归一化拉普拉斯矩阵的谱来刻画该图。我们还研究了非正则图上 Grover 游走的周期性，这与归一化邻接矩阵的特征值以及 Grover 游走的时间演化矩阵的特征值密切相关。我们证明了荷兰风车图 $D_n^{(t)}$ 是 $2n$ 周期的，并且均匀 theta 图 $\Theta(t,n)$ 是 $(2n+2)$ 周期的。此外，我们完全确定了连通的 6 周期图，并证明它们恰好是 $t \geq 2$ 的 $D_3^{(t)}$ 和 $t \geq 1$ 的 $\Theta(t,2)$。
英文摘要：
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 $t \geq 2$ and $Θ(t,2)$ with $t \geq 1$.
