Consider an arbitrary undirected simple connected graph having vertex set $V$ and edge set $E$, and a lazy simple random walk on it.
How many distinct vertices does the random walk visit until it mixes? (We define the mixing time to be the time the random walk needs to get close to stationarity ($\forall v\in V, \pi(v)=deg(v)/|E|)$).
When a random walk traverses the vertices of a graph it may occur that goes back to already visited vertices. This gets more likely when the walk is trapped in a subset of $V$ with low conductance.
It seems to me that the number of distinct vertices visited by a random walk should be related to the graph's conductance.
The question seems important and preliminary but I can not find anything about it.