# Approximations of Metric Graphs by Thick Graphs and Their Laplacians

## Abstract

**:**

## 1. Introduction

## 2. Metric Graphs and Their Laplacians

## 3. Thick Graphs and Their Laplacians

## 4. Convergence of the Resolvents

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

## 5. Generalized Norm Resolvent Convergence

**Definition**

**1.**

**Theorem**

**2.**

**Proof.**

**Lemma**

**3.**

**Lemma**

**4.**

**Corollary**

**1.**

## 6. Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The decomposition of a graph-like space of thickness of order $\epsilon $ into vertex neighborhoods ${X}_{\epsilon ,v}^{\diamond}$ (dark grey) and edge neighborhoods ${X}_{\epsilon ,e}^{\diamond}$ (light grey) according to a metric graph ${X}_{0}$ embedded in ${\mathbb{R}}^{2}$.

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**MDPI and ACS Style**

Post, O.
Approximations of Metric Graphs by Thick Graphs and Their Laplacians. *Symmetry* **2019**, *11*, 369.
https://doi.org/10.3390/sym11030369

**AMA Style**

Post O.
Approximations of Metric Graphs by Thick Graphs and Their Laplacians. *Symmetry*. 2019; 11(3):369.
https://doi.org/10.3390/sym11030369

**Chicago/Turabian Style**

Post, Olaf.
2019. "Approximations of Metric Graphs by Thick Graphs and Their Laplacians" *Symmetry* 11, no. 3: 369.
https://doi.org/10.3390/sym11030369