Average Hitting Times and Recurrence Structures I: Powers of Cycle Graphs

TL;DR

This paper derives explicit formulas for average hitting times and related quantities on cycle power graphs using spectral analysis and recurrence sequences, unifying known results and revealing structural relations.

Contribution

It introduces a unified approach to compute hitting times, resistances, and spanning tree counts on cycle power graphs using spectral and recurrence methods.

Findings

Explicit formulas for average hitting times on cycle power graphs.

Connection between hitting times and Fibonacci-type recurrence sequences.

Formulas for resistances and spanning trees expressed via complex Fibonacci sequences.

Abstract

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.