Average hitting times and recurrence structures II: Cartesian products of powers of cycles and regular graphs

Abstract

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.