In the first part of the talk (joint with Anna Ben Hamou and Roberto Oliviera, https://arxiv.org/abs/1709.00869) I will discuss estimating (up to a factor 1+epsilon) the number of vertices n of a regular graph using random walks on that graph. We show that order n^{1/2} t_rel^{3/4} random walk steps suffice for estimation, and this upper bound is sharp for all values of the relaxation time t_rel. The key to the algorithm is refinement of a bound obtained with Sauerwald, Sousi and Stauffer, Electronic Journal of Probability 22 (2017) on the intersection time for two random walks. I will also present a related bound of order n t_rel^{1/2} on hitting times in regular graphs. In the second part (joint work with Perla Sousi and Jeff Steif, https://arxiv.org/abs/1707.07632 ), I will discuss mixing times on supercritical dynamical percolation, a random subgraph of the discrete torus that changes over time; I will emphasize the method of evolving sets (Morris and P., PTRF 2005) used in the analysis, and the open problem of determining the mixing time when the percolation probability p is near critical.