On the average hitting times of the squares of cycles

TL;DR

This paper presents a simplified, unified combinatorial formula for the average hitting times of simple random walks on the square of cycle graphs, revealing connections to Fibonacci numbers and avoiding spectral graph theory.

Contribution

It introduces a new elementary and shorter proof for the average hitting times on $C^2_N$, unifying the even and odd cases and linking results to Fibonacci numbers.

Findings

Unified formula for all N cases

Shorter, combinatorial proof method

Explicit relation to Fibonacci numbers

Abstract

The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks from one vertex to any other vertex on the square $C^2_N$ of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of Statistical Physics}, \textbf{154} (2014) 1177-1190]. In that paper, the author gives the expression for the even $N$ case and the expression for the odd $N$ case separately. In this paper, by using an elementary method different from Chair (2014), we give a much simpler single formula for the HT's of simple random walks on $C^2_N$. Our proof is considerably short and fully combinatorial, in particular, has no-need of any spectral graph theoretical arguments. Not only the formula itself but also intermediate results through the process of our proof describe clear relations between the HT's of simple random walks on $C^2_N$ and the Fibonacci numbers.