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Characterization of Green's function of discrete Schrödinger operator on a finite graph by its spanning subgraphs

Yusuke Higuchi, Etsuo Segawa

Abstract

The Green's function of the discrete Schödinger operator on a finite graph is considered. This setting reproduces Laplacian and signless Laplacian by adjusting appropriate potentials. We show two ways of the expression for the Green's function by using graph structures. The first way is based on the factor of the graph by subtrees which have uni-self-loops; the second way is based on that by odd unicycle graphs.

Characterization of Green's function of discrete Schrödinger operator on a finite graph by its spanning subgraphs

Abstract

The Green's function of the discrete Schödinger operator on a finite graph is considered. This setting reproduces Laplacian and signless Laplacian by adjusting appropriate potentials. We show two ways of the expression for the Green's function by using graph structures. The first way is based on the factor of the graph by subtrees which have uni-self-loops; the second way is based on that by odd unicycle graphs.
Paper Structure (11 sections, 5 theorems, 67 equations, 4 figures)

This paper contains 11 sections, 5 theorems, 67 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Graph notation
  4. Adjacency matrix with potential
  5. Main theorem and related results
  6. Main theorem
  7. Related results
  8. Proof of main theorem
  9. Proof of (\ref{['eq:main1']})
  10. Proof of (\ref{['eq:main2']})
  11. Example

Key Result

Theorem 3.1

Let $\Gamma=(X,A)$ be a connected symmetric graph with the vertex set $X=\{u_1,\dots,u_N\}$ and the boundary $\delta X=\{u_{N-|\delta X|+1},\cdots,u_N\}$. Then the resolvent of $\mathcal{M}_V$, $G_z:=G(z;\Gamma;\delta X;V)\in \mathbb{C}^{(X\setminus \delta X)\times (X\setminus \delta X)}$, is formed for any $u_\ell,u_m\in X\setminus\delta X$.

Figures (4)

  • Figure 1: The graph for the example $\Gamma=C_3+ P_2$ with the boundary vertex set $\delta X=\{4\}$. The deformed graph $\stackrel{\circ}{\Gamma}$ has self-loop to every vertex of $X\setminus \delta X=\{1,2,3\}$.
  • Figure 2: $\mathcal{H}_J(\Gamma;\delta X)$ and $\mathcal{H}_J(\Gamma;\delta X;\ell,m)$: Since $\Gamma$ has only one cycle, whose length is odd, then we have $\mathcal{H}_Q(\Gamma;\delta X)=\mathcal{H}_L(\Gamma;\delta X) \cup \{ (C_3\cup K_1) \}$ and $\mathcal{H}_L(\Gamma;\delta X;\ell,m)=\mathcal{H}_Q(\Gamma;\delta X;\ell,m)$ for any $\ell,m=1,2,3$.
  • Figure 3: Computation of $\iota_1$
  • Figure 4: Computation of $\iota_2$

Theorems & Definitions (15)

  • Definition 2.1: The graph factor for describing the resolvent
  • Definition 3.1: Weights of spanning subgraph and families of spanning subgraphs
  • Theorem 3.1
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 5 more