AI 中文总结
该研究针对循环图幂与正则图的笛卡尔积图，利用离散傅里叶分析和拉普拉斯谱分解推导了平均击中时间的递推结构，还给出了生成树与双分量生成森林数量的公式及实例。
AI 中文摘要
在我们之前的研究工作\uc81cite{MiezakiTamura2026}中，我们阐明了循环图的第k次幂图$C_N^k$上平均击中时间所呈现的二阶线性递推结构。在本文中，对于一个具有m个顶点的连通r-正则图G，我们研究了简单随机游走在笛卡尔积图$C_N^k \boxtimes G$上的平均击中时间。通过在$C_N^k$方向上使用离散傅里叶分析，并结合G的拉普拉斯谱分解，我们将平均击中时间分解为与$C_N^k$上平均击中时间成比例的分量，以及源自G的非零拉普拉斯特征空间的修正项。对于每个非零拉普拉斯特征值，我们引入了一种切比雪夫型多项式，当其所有根均为单根时，我们将修正项表示为有限格林型和。此外，对于具有相同G坐标的两个顶点，我们将该表达式转化为形式为$V_\tau V_{N-\tau}/V_N$的二阶线性递推表示。当G是行走正则图时，具有相同G坐标的两个顶点之间的平均击中时间仅取决于G的拉普拉斯特征值及其重数。我们还推导了$C_N^k \boxtimes G$的生成树数量与双分量生成森林数量的公式，并给出了若干显式示例。
英文摘要
In our previous work \cite{MiezakiTamura2026}, we clarified the second-order linear recurrence structures appearing in the average hitting times on the $k$-th power graph $C_N^k$ of the cycle graph. In this paper, for a connected $r$-regular graph $G$ on $m$ vertices, we investigate the average hitting times of the simple random walk on the Cartesian product graph $C_N^k \square G$. By using discrete Fourier analysis in the $C_N^k$ direction and the Laplacian spectral decomposition of $G$, we decompose the average hitting time into a component proportional to the average hitting time on $C_N^k$ and correction terms arising from the nonzero Laplacian eigenspaces of $G$. For each nonzero Laplacian eigenvalue, we introduce a Chebyshev-type polynomial, and when all of its roots are simple, we express the correction term as a finite Green-type sum. Furthermore, for two vertices having the same $G$-coordinate, we transform this expression into a second-order linear recurrence representation of the form $V_\ell V_{N-\ell}/V_N$. When $G$ is a walk-regular graph, the average hitting time between two vertices having the same $G$-coordinate depends only on the Laplacian eigenvalues of $G$ and their multiplicities. We also derive formulas for the number of spanning trees and the number of two-component spanning forests of $C_N^k \square G$, and give several explicit examples.
Comments33 pages
