Average Hitting Times and Recurrence Structures I: Powers of Cycle Graphs
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arXiv
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2026
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| author | Miezaki, Tsuyoshi Tamura, Shunya |
| author_facet | Miezaki, Tsuyoshi Tamura, Shunya |
| contents | We investigate the average hitting times of simple random walks on the $k$-th power graph $C_N^k$ of the cycle graph $C_N$. First, we show that the average hitting times are characterized by a difference equation corresponding to the graph Laplacian. Next, by using the cyclic symmetry of $C_N^k$, we derive a spectral representation via Fourier analysis. Furthermore, by applying factorization and partial fraction decomposition of the corresponding difference operator, we obtain an explicit formula for the average hitting times consisting of a quadratic term and finitely many correction terms. These correction terms are described by second-order linear recurrence sequences associated with the characteristic polynomials, and can be regarded as natural generalizations of Fibonacci-type sequences. As a consequence, our formulas recover the known results for cycle graphs and squares of cycle graphs in a unified way. Moreover, from the formulas obtained for average hitting times, we derive explicit formulas for the effective resistances, the numbers of spanning trees, the numbers of two-component spanning forests, and the numbers of spanning trees of vertex-identified graphs. In particular, for the third power graph $C_N^3$ of the cycle graph, all of these quantities are written explicitly in terms of complex conjugate Fibonacci-type sequences. Our results clarify structural relations between random walk quantities and combinatorial quantities on cycle power graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_09229 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Average Hitting Times and Recurrence Structures I: Powers of Cycle Graphs Miezaki, Tsuyoshi Tamura, Shunya Combinatorics 05C30, 05C50, 05A15 We investigate the average hitting times of simple random walks on the $k$-th power graph $C_N^k$ of the cycle graph $C_N$. First, we show that the average hitting times are characterized by a difference equation corresponding to the graph Laplacian. Next, by using the cyclic symmetry of $C_N^k$, we derive a spectral representation via Fourier analysis. Furthermore, by applying factorization and partial fraction decomposition of the corresponding difference operator, we obtain an explicit formula for the average hitting times consisting of a quadratic term and finitely many correction terms. These correction terms are described by second-order linear recurrence sequences associated with the characteristic polynomials, and can be regarded as natural generalizations of Fibonacci-type sequences. As a consequence, our formulas recover the known results for cycle graphs and squares of cycle graphs in a unified way. Moreover, from the formulas obtained for average hitting times, we derive explicit formulas for the effective resistances, the numbers of spanning trees, the numbers of two-component spanning forests, and the numbers of spanning trees of vertex-identified graphs. In particular, for the third power graph $C_N^3$ of the cycle graph, all of these quantities are written explicitly in terms of complex conjugate Fibonacci-type sequences. Our results clarify structural relations between random walk quantities and combinatorial quantities on cycle power graphs. |
| title | Average Hitting Times and Recurrence Structures I: Powers of Cycle Graphs |
| topic | Combinatorics 05C30, 05C50, 05A15 |
| url | https://arxiv.org/abs/2605.09229 |