Forest formulas of discrete Green's functions

Abstract. The discrete Green’s functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this paper, 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 G associated with the combinatorial Laplacian of a connected simple graph Γ on n vertices satisfies

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