MPI-I-94-121

We study the short term behavior of random walks on graphs, in particular, the rate at which a random walk discovers new vertices and edges. We prove a conjecture by Linial that the expected time to find $\cal N$ distinct vertices is $O({\cal N} ^ 3)$. We also prove an upper bound of $O({\cal M} ^ 2)$ on the expected time to traverse $\cal M$ edges, and $O(\cal M\cal N)$ on the expected time to either visit $\cal N$ vertices or traverse $\cal M$ edges (whichever comes first).