Assume that $G$ is a finite graph and we have a simple random walk starting at some vertex $v$ of $G$. We fix $n$, and consider the probability that the random walk does not return to $v$ after $n$ steps.
Does this probability increase if we add finitely many edges and vertices to the graph? What if we add cycles?
This might have to do with Rayleigh's monotonicity but I'm not sure how prove it. If we add edges, then the effective resistance between any two vertices of $G$ will decrease. But we are also adding other vertices so the random walk can escape more easily.