  We exhibit a close connection between hitting times of the simple random walk
on a graph, the Wiener index, and related graph invariants. In the case of
trees we obtain a simple identity relating hitting times to the Wiener index.
  It is well known that the vertices of any graph can be put in a linear
preorder so that vertices appearing earlier in the preorder are "easier to
reach" by a random walk, but "more difficult to get out of". We define various
other natural preorders and study their relationships. These preorders coincide
when the graph is a tree, but not necessarily otherwise.
  Our treatise is self-contained, and puts some known results relating the
behaviour or random walk on a graph to its eigenvalues in a new perspective.
