  The discrete Green's function (without boundary) $\mathbb{G}$ is a
pseudo-inverse of the combinatorial Laplace operator of a graph $G=(V,E)$. We
reveal the intimate connection between Green's function and the theory of exact
stopping rules for random walks on graphs. We give an elementary formula for
Green's function in terms of state-to-state hitting times of the underlying
graph. Namely, $\mathbb{G}(i,j) = \pi_j \left( \sum_{k \in V} \pi_k H(k,j) -
H(i,j) \right)$ where $\pi_i$ is the stationary distribution at vertex $i$ and
$H(i,j)$ is the expected hitting time for a random walk starting from vertex
$i$ to first reach vertex $j$. This formula also holds for the digraph Laplace
operator.
  The most important characteristics of a stopping rule are its exit
frequencies, which are the expected number of exits of a given vertex before
the rule halts the walk. We show that Green's function is, in fact, a matrix of
exit frequencies plus a rank one matrix. In the undirected case, we derive
spectral formulas for Green's function and for some mixing measures arising
from stopping rules. Finally, we further explore the exit frequency matrix
point-of-view, and discuss a natural generalization of Green's function for any
distribution $\tau$ defined on the vertex set of the graph.

Combinatorics;Probability