Forest formulas of discrete Green's functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909118517739520 |
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| author | Chung, Fan Zeng, Ji |
| author_facet | Chung, Fan Zeng, Ji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_01324 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Forest formulas of discrete Green's functions Chung, Fan Zeng, Ji Combinatorics 05C50 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. |
| title | Forest formulas of discrete Green's functions |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2109.01324 |