  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.
