The effective resistance of the $N$-cycle graph with four nearest neighbors

Abstract

The exact expression for the effective resistance between any two vertices of the $N$-cycle graph with four nearest neighbors $C_{N}(1,2)$, is given. It turns out that this expression is written in terms of the effective resistance of the $N$-cycle graph $C_{N}$, the square of the Fibonacci numbers, and the bisected Fibonacci numbers. As a consequence closed form formulas for the total effective resistance, the first passage time (FPT), and the mean first passage time (MFPT) for the simple random walk on the the $N$-cycle graph with four nearest neighbors are obtained. Finally, a closed form formula for the effective resistance of $C_{N}(1,2)$ with all first neighbors removed is obtained.