Short random walks on graphs

The short-term behavior of random walks on graphs is studied, in particular, the rate at which a random walk discovers new vertices and edges. A conjecture by Linial that the expected time to find $\cal N$ distinct vertices is $O({\cal N}^{3})$ is proved. In addition, upper bounds of $O({\cal M}^{2})$ on the expected time to traverse $\cal M$ edges and of $O(\cal M \cal N)$ on the expected time to either visit $\cal N$ vertices or traverse $\cal M$ edges (whichever comes first) are proved.