Hitting Times, Cover Cost, and the Wiener Index of a Tree

It is well known that the vertices of any graph can be put in a linear preorder so that for random walk on the graph vertices appearing earlier in the preorder are “easier to reach but more difficult to get out of”. We exhibit further such preorders corresponding to various functions related to random walk and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. In the case of trees, we prove a simple identity relating hitting times to the Wiener index.