Description:
<jats:title>Abstract</jats:title><jats:p>The discrete Green's functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this article, we will give combinatorial interpretations of Green's functions in terms of enumerating trees and forests in a graph that will be used to derive further formulas for several graph invariants. For example, we show that the trace of the Green's function associated with the combinatorial Laplacian of a connected simple graph on vertices satisfies <jats:disp-formula /> where denotes the eigenvalues of the combinatorial Laplacian, denotes the number of spanning trees, and denotes the set of rooted spanning 2‐forests in . We will prove forest formulas for discrete Green's functions for directed and weighted graphs and apply them to study random walks on graphs and digraphs. We derive a forest expression of the hitting time for digraphs, which gives combinatorial proofs to old and new results about hitting times, traces of discrete Green's functions, and other related quantities.</jats:p>