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 $\mathbf{G}$ associated with the combinatorial Laplacian of a connected simple graph $Γ$ on $n$ vertices satisfies $\text{Tr}(\mathbf{G})=\sum_{λ_i \neq 0} \frac 1 {λ_i}= \frac{1}{nτ}|\mathbb{F}^*_2|$, where $λ_i$ denotes the eigenvalues of the combinatorial Laplacian, $τ$ denotes the number of spanning trees and $\mathbb{F}^*_2$ denotes the set of rooted spanning $2$-forests in $Γ$. 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.

Figures (3)

• Figure 1: (i) A graph $\Gamma_1$; (ii) A spanning tree of $\Gamma_1$; (iii) A spanning $2$-forest of $\Gamma_1$. (iv) A digraph $\Gamma_2$; (v) A rooted spanning tree of $\Gamma_2$; (vi) A rooted spanning $2$-forest of $\Gamma_2$. (Roots pictured as solid nodes.)
• Figure 2: All rooted $2$-forests of $\Gamma_2$ with one root being $b$.
• Figure 3: $\Gamma(5)$.

Theorems & Definitions (37)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Corollary 1.7
• Theorem 2.1: chaiken1982combinatorial
• Lemma 2.2: horn2012matrix0.8.12.3
• Lemma 3.1