# Covering Graphs, Magnetic Spectral Gaps and Applications to Polymers and Nanoribbons

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## Abstract

**:**

## 1. Introduction

## 2. Weighted Graphs and Discrete Magnetic Laplacians

**Definition**

**1.**

- (a)
- If ${E}_{0}\cap E(V\backslash {V}_{0})=\varnothing $, we say that $\mathbf{H}$ is a partial subgraph in $\mathbf{G}$. We call$$\begin{array}{cc}\hfill B(\mathbf{H},\mathbf{G}):=& E({V}_{0},V\backslash {V}_{0})\hfill \\ \hfill =& \left\{e\in E\mid {\partial}_{-}e\in {V}_{0},{\partial}_{+}e\in V\backslash {V}_{0}\phantom{\rule{4.pt}{0ex}}or\phantom{\rule{4.pt}{0ex}}{\partial}_{+}e\in {V}_{0},{\partial}_{-}e\in V\backslash {V}_{0}\right\}\hfill \end{array}$$
- (b)
- If ${E}_{0}\subset E({V}_{0})$, then $\mathbf{H}$ is a subgraph of $\mathbf{G}$

- Standard weight:$m(v)=\mathrm{deg}(v)$, $v\in V$, and ${m}_{e}=1$, $e\in E$, so that $\rho (v)={\rho}_{\infty}=1$.
- Combinatorial weight:$m(v)={m}_{e}=1$, $v\in V$, $e\in E$ hence $\rho (v)=\mathrm{deg}(v)$ and ${\rho}_{\infty}={\mathrm{sup}}_{v\in V}\mathrm{deg}(v)$.

**Definition**

**2.**

## 3. Spectral Ordering on Finite Graphs and Magnetic Spectral Gaps

**Definition**

**3.**

- (a)
- We say that ${\mathbf{W}}^{-}$ is spectrally smaller than ${\mathbf{W}}^{+}$ (denoted by ${\mathbf{W}}^{-}\preccurlyeq {\mathbf{W}}^{+}$), if$${n}^{-}\ge {n}^{+}\phantom{\rule{2.em}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{2.em}{0ex}}{\lambda}_{k}({\Delta}_{{\alpha}^{-}}^{{\mathbf{W}}^{-}})\le {\lambda}_{k}({\Delta}_{{\alpha}^{+}}^{{\mathbf{W}}^{+}})\phantom{\rule{1.em}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{1.em}{0ex}}1\le k\le {n}^{-}\phantom{\rule{0.277778em}{0ex}},$$
- (b)
- Consider ${\mathbf{W}}^{\pm}$ as above with ${\mathbf{W}}^{-}\preccurlyeq {\mathbf{W}}^{+}$. We define the associated k-th bracketing interval ${J}_{k}={J}_{k}({\mathbf{W}}^{-},{\mathbf{W}}^{+})$ by$${J}_{k}:=\left[{\lambda}_{k}({\Delta}_{{\alpha}^{-}}^{{\mathbf{W}}^{-}}),{\lambda}_{k}({\Delta}_{{\alpha}^{+}}^{{\mathbf{W}}^{+}})\right]$$

**Definition**

**4**

**.**Let $\mathbf{W}=(\mathbf{G},m)$ be a weighted graph with magnetic potential α and ${E}_{0}\subset E(\mathbf{G})$. We denote by ${\mathbf{W}}^{-}=({\mathbf{G}}^{-},{m}^{-})$ the weighted subgraph with magnetic potential ${\alpha}^{-}$ defined as follows:

- (a)
- $V({\mathbf{G}}^{-})=V(\mathbf{G})$ with ${m}^{-}(v):=m(v)$ for all $v\in V(\mathbf{G})$;
- (b)
- $E({\mathbf{G}}^{-})=E(\mathbf{G})\backslash {E}_{0}$ with ${m}_{e}^{-}:={m}_{e}$ and ${\partial}_{\pm}^{{\mathbf{G}}^{-}}e={\partial}_{\pm}^{\mathbf{G}}e$ for all $e\in E({\mathbf{G}}^{-})$;
- (c)
- ${\alpha}_{e}^{-}={\alpha}_{e}$, $e\in E({\mathbf{G}}^{-})$.

**Definition**

**5**

**.**Let $\mathbf{W}=(\mathbf{G},m)$ be a weighted graph with magnetic potential α and ${V}_{0}\subset V(\mathbf{G})$. We denote by ${\mathbf{W}}^{+}=({\mathbf{G}}^{+},{m}^{+})$ the weighted partial subgraph with magnetic potential ${\alpha}^{+}$ defined as follows:

- (a)
- $V({\mathbf{G}}^{+})=V(\mathbf{G})\backslash {V}_{0}$ with ${m}^{+}(v):=m(v)$ for all $v\in V({\mathbf{G}}^{+})$;
- (b)
- $E({\mathbf{G}}^{+})=E(\mathbf{G})\backslash {\bigcup}_{{v}_{0}\in {V}_{0}}E({v}_{0})$ with ${m}_{e}^{+}={m}_{e}$ for all $e\in E(\mathbf{G})$;
- (c)
- ${\alpha}_{e}^{+}={\alpha}_{e}$, $e\in E({\mathbf{G}}^{+})$.

**Definition**

**6.**

**Theorem**

**1.**

**Definition**

**7.**

- (a)
- The spectral gaps set of $\mathbf{W}$ is defined by$${\mathcal{S}}^{\mathbf{W}}=[0,2{\rho}_{\infty}]\backslash \sigma ({\Delta}^{\mathbf{W}})=[0,2{\rho}_{\infty}]\cap \rho ({\Delta}^{\mathbf{W}})\phantom{\rule{0.277778em}{0ex}},$$
- (b)
- The magnetic spectral gaps set of $\mathbf{W}$ is defined by$${\mathcal{MS}}^{\mathbf{W}}=[0,2{\rho}_{\infty}]\backslash \bigcup _{\alpha \in \mathcal{A}(\mathbf{G})}\sigma ({\Delta}_{\alpha}^{\mathbf{W}})=\bigcap _{\alpha \in \mathcal{A}(\mathbf{G})}\rho ({\Delta}_{\alpha}^{\mathbf{W}})\cap [0,2{\rho}_{\infty}].$$

## 4. Periodic Graphs and Spectral Gaps

#### 4.1. Periodic Graphs and Fundamental Domains

**Definition**

**8.**

- (a)
- a vertex, respectively arc fundamental domain on a Γ-covering graph is given by two subsets ${D}^{V}\subset \tilde{V}$ and ${D}^{E}\subset \tilde{E}$ satisfying$$\begin{array}{cc}\hfill \tilde{V}& =\bigcup _{\gamma \in \Gamma}\gamma {D}^{V}\phantom{\rule{1.em}{0ex}}\mathit{and}\phantom{\rule{1.em}{0ex}}{\gamma}_{1}{D}^{V}\cap {\gamma}_{2}{D}^{V}=\varnothing \phantom{\rule{1.em}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}{\gamma}_{1}\ne {\gamma}_{2},\hfill \\ \hfill \tilde{E}& =\bigcup _{\gamma \in \Gamma}\gamma {D}^{E}\phantom{\rule{1.em}{0ex}}\mathit{and}\phantom{\rule{1.em}{0ex}}{\gamma}_{1}{D}^{E}\cap {\gamma}_{2}{D}^{E}=\varnothing \phantom{\rule{1.em}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}{\gamma}_{1}\ne {\gamma}_{2}\hfill \end{array}$$
- (b)
- a (graph) fundamental domain of a covering graph $\tilde{\mathbf{G}}$ is a partial subgraph (cf., Definition 1)$$\mathbf{H}=({D}^{V},{D}^{E},\partial {\upharpoonright}_{{D}^{E}}),$$$$B(\mathbf{H},\tilde{\mathbf{G}}):=E({D}^{V},V\backslash {D}^{V})$$

**Remark**

**1.**

- (a)
- Fixing a fundamental domain on the covering graph and the group Γ will be used to give coordinates (to the arcs and vertices) on the covering graph $\tilde{\mathbf{G}}$.In fact, given a specific fundamental domain ${D}^{V}$ in a Γ-covering graph $\tilde{\mathbf{G}}$, we can write any $v\in V(\tilde{\mathbf{G}})$ uniquely as $v=\xi (v){v}_{0}$ for a unique pair $(\xi (v),{v}_{0})\in \Gamma \times {D}^{V}$. This observation follows from the fact that the action is free and transitive. We call $\xi (v)$ the Γ-coordinate of v (with respect to the fundamental domain ${D}^{V}$). Similarly, we can define the coordinates for the arcs: any $e\in E(\tilde{\mathbf{G}})$ can be written as $e=\xi (e){e}_{0}$ for a unique pair $(\xi (e),{e}_{0})\in \Gamma \times {D}^{E}$. In particular, we have$$\xi (\gamma v)=\gamma \xi (v)\phantom{\rule{2.em}{0ex}}\mathit{and}\phantom{\rule{2.em}{0ex}}\xi (\gamma e)=\gamma \xi (e),\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}\gamma \in \Gamma .$$
- (b)
- Once we have chosen a fundamental domain $\mathbf{H}=({D}^{V},{D}^{E},\partial )$, we can embed $\mathbf{H}$ into the quotient $\mathbf{G}=\tilde{\mathbf{G}}/\Gamma $ of the covering $\pi :\tilde{\mathbf{G}}\to \mathbf{G}=\tilde{\mathbf{G}}/\Gamma $ by$${D}^{V}\to V(\mathbf{G})=V/\Gamma ,\phantom{\rule{1.em}{0ex}}v\mapsto \left[v\right]\phantom{\rule{2.em}{0ex}}\mathit{and}\phantom{\rule{2.em}{0ex}}{D}^{E}\to E(\mathbf{G})=E/\Gamma ,\phantom{\rule{1.em}{0ex}}e\mapsto \left[e\right],$$

**Definition**

**9.**

#### 4.2. Discrete Floquet Theory

**Proposition**

**1.**

**Proof.**

#### 4.3. Vector Potential as a Floquet Parameter

**Definition**

**10.**

**the lifting property**if there exists $\chi \in \widehat{\Gamma}$ such that:

**Proposition**

**2.**

**Proof.**

#### 4.4. Spectral Localization for the DML on a Covering Graph

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Step 1: Trace of ${\Delta}_{{\beta}^{-}}^{{\mathbf{G}}^{-}}$.**We define ${\mathbf{W}}^{-}=({\mathbf{G}}^{-},{m}^{-})$ where ${\mathbf{G}}^{-}=\mathbf{G}-[B(\mathbf{H},\tilde{\mathbf{G}})]$. Recall that $V({\mathbf{G}}^{-})=V(\mathbf{G})$, $E({\mathbf{G}}^{-})=E(\mathbf{G})\backslash [B(\mathbf{H},\tilde{\mathbf{G}})]$; the weights on $V({\mathbf{G}}^{-})$ and $E({\mathbf{G}}^{-})$ coincide with the corresponding weights on $\mathbf{W}$. The relative weights of ${\mathbf{W}}^{-}$ are

**Step 2: Trace of ${\Delta}^{{\mathbf{W}}^{+}}$.**Let ${\mathbf{W}}^{+}=\left({\mathbf{G}}^{+},{m}^{+}\right)$, then the trace of ${\Delta}_{{\beta}^{+}}^{{\mathbf{W}}^{+}}$ is given by

**Remark**

**2.**

- (a)
- If the graph has the standard weights, the condition becomes:$$\delta =1-\sum _{e\in [B(\mathbf{H},\tilde{\mathbf{G}})]}{\displaystyle \frac{1}{\mathrm{deg}({({v}_{0})}_{e})}}-{\displaystyle \frac{|[B(\mathbf{H},\tilde{\mathbf{G}})]|}{\mathrm{deg}({v}_{0})}}-{\lambda}_{1}({\Delta}_{{\beta}^{-}}^{{W}^{-}})>0\phantom{\rule{0.277778em}{0ex}},$$
- (b)
- If we have the combinatorial weights, the condition becomes simply:$$\delta =\mathrm{deg}({v}_{0})-2\phantom{\rule{0.277778em}{0ex}}|[B(\mathbf{H},\tilde{\mathbf{G}})]|-{\lambda}_{1}({\Delta}_{{\beta}^{-}}^{{W}^{-}})>0\phantom{\rule{0.277778em}{0ex}}.$$

## 5. Examples

#### 5.1. Polyacetylene with Magnetic Field

- Fact 1. Let $\tilde{m}$ be the standard weights and $\tilde{\beta}$ a constant periodic magnetic potential. We show how to apply the bracketing technique to localize the spectrum for a specific value of the magnetic potential (equal to $s=\pi /2$) and then, how the bracketing intervals change as a function of $\tilde{\beta}$. We will show the existence of spectral gaps.
- Fact 2. Let $\tilde{m}$ be the combinatorial weights and $\tilde{\beta}$ a periodic magnetic potential (not necessarily constant). Using the condition on $\delta $ in Equation (17), we show the existence of spectral gaps.
- Fact 3. Let $\tilde{m}$ be the standard weights, we show the existence of periodic magnetic spectral gaps, i.e., a spectral gap which is stable under any perturbation of the constant periodic magnetic field.

#### 5.2. Graphene Nanoribbons

- (i)
- The first variant is called armchair nanoribbon with a width equal to ${N}_{a}$ and denoted as ${N}_{a}$-aGNR (see Figure 5). Consider for example the case of a 3-aGNR which has a similar structure as the poly-para-phenylene (PPP), one of the most important conductive polymers. Let $\mathbf{W}=(\tilde{\mathbf{G}},\tilde{m})$ be the MW-graph with standard weights where $\tilde{\mathbf{G}}$ is the $\mathbb{Z}$-covering graph representing the 3-aGNR and $\tilde{\beta}$ is a constant (periodic) magnetic potential, the idea is to use the bracketing technique to localize $\sigma ({\Delta}_{\tilde{\beta}}^{\tilde{\mathbf{W}}})$ and we proceed as in the previous examples. Figure 6a is the finite quotient graph $\mathbf{G}=\tilde{\mathbf{G}}/\mathbb{Z}$. Define in this case ${E}_{1}=\left\{{e}_{1}\right\}$ and ${V}_{1}=\left\{{v}_{1}\right\}$ so that ${V}_{1}$ is a neighborhood of ${E}_{1}$ (see Definition 6). We construct ${\mathbf{W}}_{1}^{+}$ and ${\mathbf{W}}_{1}^{-}$ as before: ${\mathbf{G}}_{1}^{+}=\mathbf{G}-{E}_{1}$ and ${\mathbf{G}}_{1}^{-}=\mathbf{G}-{V}_{1}$ (cf., Figure 6b). The weights are induced as in Definitions 4 and 5. Using again the notation of the Theorem 1 and Proposition 2 we obtain now a spectral localization J that depends on $\tilde{\beta}$. Finally, in Figure 6c, we plot the spectral bands and gaps specified by J for the different values of the magnetic field within the interval $[0,2\pi ]$. Observe that in this case, we do not have a spectral gap common to all values of $\tilde{\beta}$ (as we had for the polyacetylene).A similar analysis could be done for any ${N}_{a}$-aGNR under the action of any periodic magnetic potential, and the bracketing technique will give good estimates of the intervals where the spectrum lies.Also, observe that for the combinatorial weights, we can show the existence of spectral gaps using the condition of Equation (17) as in Fact 2 in the polyacetylene example. We have in this case,$$\delta =\mathrm{deg}({v}_{1})-2\phantom{\rule{0.277778em}{0ex}}|[B(\mathbf{H},\tilde{\mathbf{G}})]|-{\lambda}_{1}({\Delta}_{{\beta}^{-}}^{{W}^{-}})>3-2-1=0.$$
- (ii)
- The second variant is the so-called zigzag nanoribbon with a width equal to ${N}_{z}$ are denoted as ${N}_{z}$-zGNR (see Figure 5). Consider $\mathbf{W}=(\tilde{\mathbf{G}},\tilde{m})$ the MW-graph with standard weights and $\tilde{\mathbf{G}}$ is the graph given by the zigzag nanoribbons for a fixed ${N}_{z}$, and $\tilde{\beta}\sim 0$ acting on $\tilde{\mathbf{G}}$. In this case, our spectral localization method does not specify spectral gaps (i.e., the spectral bands overlap). The reason is that for any width ${N}_{z}$ the spectrum of the zigzag nanoribbons satisfy $\sigma ({\Delta}_{0}^{\tilde{\mathbf{W}}})=[0,2]$, i.e., in this case there are no spectral gaps. This fact is also confirmed by our method.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The structure of the bracketing intervals J is represented with gray bands and the spectral gaps with white bands. Both bands depend on the constant (periodic) magnetic potential $\tilde{\beta}$ acting on 3-aGNR.

**Figure 2.**Spectral gaps of the polyacetylene graph for a constant magnetic potential $\beta =s$. Here, J is the spectral localization for the pair $\mathbf{G}-\left\{{e}_{1}\right\}$ and $\mathbf{G}-\left\{{v}_{1}\right\}$. Bipartiteness gives a finer localization $J\cap \kappa (J)$. In this case, we obtain the spectrum almost exactly, except for the spectral value 1.

**Figure 3.**The horizontal axis represents the values of the magnetic potential $\tilde{\beta}\in [0,2\pi )$ acting on the polyacetylene polymer with standard weights. For any fixed $\tilde{\beta}$ we obtain the intervals J given by the bracketing technique as we did in the case $\tilde{\beta}=\pi /2$ in Figure 2 (and also using the symmetry given by the bipartiteness). In the vertical axis, we represent the spectral bands and gaps for each constant value $\tilde{\beta}$.

**Figure 4.**Using this graph ${\mathbf{G}}_{1}^{-}$ and ${\mathbf{G}}_{1}^{+}$, we can find spectral gaps in common for all periodic magnetic potential $\tilde{\beta}$ acting on the polyethylene, represented by the covering graph $\mathbf{G}$.

**Figure 5.**Two structures of the graphene nanoribbons: armchair and zigzag. These structures are covering graphs only in one direction.

**Figure 6.**Spectral structure in bands/gaps of the magnetic Laplacian on the nanoribbons 3-aGNR. A constant magnetic potential is acting on the graph with value $\tilde{\beta}=s$. Here, the bracketing intervals J gives a localization set of the spectrum, and this localization is given by the pair ${\mathbf{G}}^{-}=\mathbf{G}-\left\{{e}_{1}\right\}$ and ${\mathbf{G}}^{+}=\mathbf{G}-\left\{{v}_{1}\right\}$, together with the bipartitness and interlacing.

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**MDPI and ACS Style**

Fabila-Carrasco, J.S.; Lledó, F.
Covering Graphs, Magnetic Spectral Gaps and Applications to Polymers and Nanoribbons. *Symmetry* **2019**, *11*, 1163.
https://doi.org/10.3390/sym11091163

**AMA Style**

Fabila-Carrasco JS, Lledó F.
Covering Graphs, Magnetic Spectral Gaps and Applications to Polymers and Nanoribbons. *Symmetry*. 2019; 11(9):1163.
https://doi.org/10.3390/sym11091163

**Chicago/Turabian Style**

Fabila-Carrasco, John Stewart, and Fernando Lledó.
2019. "Covering Graphs, Magnetic Spectral Gaps and Applications to Polymers and Nanoribbons" *Symmetry* 11, no. 9: 1163.
https://doi.org/10.3390/sym11091163