Expected hitting times for a random walk on a connected graph

Abstract A random walk on a graph is defined in which a particle moves from one vertex to any adjoining vertex, each with equal probability. The expected number of steps to get from one point to another is considered. It is shown that the maximum expectation for a graph with N vertices is O( N 3 ). It is also shown that for all graphs whose maximal valence is bounded by a constant K the maximum expectation is O (N 2 ).