The Discrete Laplacian Acting on 2-Forms and Application

Abstract

In the current paper, we study the discrete Laplacian acting on 2-forms which was introduced and investigated by Chebbi (Potential Anal 49(2):331–358, 2018). We establish a new criterion of essential self-adjointness using the Nelson lemma. Moreover, we give an upper bound on the infimum of the essential spectrum. Furthermore, we establish a link between the adjacency matrix and the discrete Laplacian on 2-forms.

1 Introduction

Spectral graph theory represents an active area of research. In the last few years, the questions of the essential self-adjointness of discrete Laplacian operators on infinite graphs have attracted a lot of interest, see [11, 16, 17, 21]. There exist other definitions of the discrete Laplacian, e.g., [2, 4, 15, 23]. The one we are studying here is the discrete Laplacian acting on 2-forms and denoted by $\mathcal {L}_{2,\mathrm{skew}}$, where skew stands for skew-symmetric. This operator was introduced by Chebbi in [7]. The author shows the relation between the $\chi$-completeness geometric hypothesis for the graph and essentially self-adjointness for the discrete Laplacian $\mathcal {L}_{2,\mathrm{skew}}$. More specifically, the author has proved that $\mathcal {L}_{2,\mathrm{skew}}$ is essential self-adjoint, when the triangulation (we refer to Sect. 2 for a precise definition) is $\chi$-complete.

The current study has two major aims. It first aims to discuss the question of essential self-adjointness for $\mathcal {L}_{2,\mathrm{skew}}$. It is worth noting that this operator depends on the weight $\mathcal {R}$ on oriented triangular faces and the weight $\mathcal {E}$ on oriented edges, see Sect. 4 for more details. In the setting of electrical networks, the weight $\mathcal {E}$ correspond to the conductance. We establish a hypothesis on the weights and involve essential self-adjointness by using the Nelson commutator theorem. The technique of the proof is inspired from [3]. Moreover, we give an upper bound on the infimum of the essential spectrum $\sigma _\mathrm{ess}(\mathcal {L}_{2,\mathrm{skew}}^{\mathcal {F}})$, where $\mathcal {L}_{2,\mathrm{skew}}^{\mathcal {F}}$ is the Friedrichs extension of $\mathcal {L}_{2,\mathrm{skew}}$. Secondly, the paper aims to identify the link between the adjacency matrix and the discrete Laplacian $\mathcal {L}_{2,\mathrm{skew}}$. To achieve this goal, we analyze the structure of $\mathcal {L}_{2,\mathrm{skew}}$. Note that this discrete Laplacian was introduced on a skew-symmetric statistic on the space of 2-forms. We can define the discrete Laplacian $\mathcal {L}_{2,\mathrm{sym}}$ in the symmetric case by the same expression of $\mathcal {L}_{2,\mathrm{sym}}$. In the case of a tri-partite graph, we prove that the two operators are unitarily equivalent. Furthermore, $\mathcal {L}_{2,\mathrm{sym}}$ is unitarily equivalent to the adjacency matrix of the triangular graph, see Sect. 7 for more details. We recall that the spectral theory of adjacency matrix acting on graphs is useful for the study of some gelling polymers, of some electrical networks, and in number theory, see [12, 13, 22].

As for the rest of this paper, it is structured as follows: The next section is devoted to some definitions and notations for graph. We find the definitions of two different Hilbert structures on the set of faces, in Sect. 3. Both definitions have their own interest. This permits to define two different types of discrete Laplacian associated with faces. The relation between these two operators is clearly presented in Sect. 4.3. In Sect. 5, we discuss the question of essential self-adjointness for the discrete Laplacian $\mathcal {L}_{2,\mathrm{skew}}$. We establish a new criterion of essential self-adjointness using the Nelson lemma. In Sect. 6, we give an upper bound on the infimum of the essential spectrum. The obtained findings from the previous sections are presented in Sect. 7 for the purpose of investigating the questions of boundedness and essential self-adjointness for the adjacency matrix.

2 Generalities About Graphs

We start with some definitions to fix notations for graphs and refer to [9, 10, 19, 23] for surveys on the matter. Let $\mathcal {V}$ be a countable set. We equip $\mathcal {V}$ with the discrete topology. Let $\mathcal {E}:\mathcal {V}\times \mathcal {V}\longrightarrow [0,+\infty )$ and assume that $\mathcal {E}$ is symmetric (i.e., $\mathcal {E}(x,y)=\mathcal {E}(y,x)$, for all $x,y\in \mathcal {V}$). Let $m:\mathcal {V}\longrightarrow (0,+\infty )$. We say that $\mathcal {G}=(\mathcal {V},m,\mathcal {E})$ is a weighted graph with vertices$\mathcal {V}$, weight of vertices m and weight of edges$\mathcal {E}$. In the setting of electrical networks, the weights correspond to the conductances. We say that x, y are neighbors if $\mathcal {E}(x,y)\ne 0$ and we denote it by $x\sim y$. A graph $\mathcal {G}$ is simple if it has no loops (i.e., $\mathcal {E}(x,x)=0$), $m=1$ and $\mathcal {E}$ has values in $\{0,1\}$. The set of neighbors of $x\in \mathcal {V}$ is denoted by

$$\begin{aligned} \mathcal {N}_{\mathcal {G}}(x):=\{y\in \mathcal {V}:\ \mathcal {E}(x,y)\ne 0\}. \end{aligned}$$

A graph is locally finite if $\sharp \mathcal {N}_{\mathcal {G}}(x)$ is finite for all $x\in \mathcal {V}$. The weighted degree of vertices is given by

$$\begin{aligned} d_{\mathcal {V}}(x):=\displaystyle \frac{1}{m(x)}\sum _{y\sim x}\mathcal {E}(x,y). \end{aligned}$$

When $\mathcal {G}$ is simple, $d_{\mathcal {V}}(x)=\sharp \mathcal {N}_{\mathcal {G}}(x)$. A graph $\mathcal {G}$ is connected, if for all $x,~~y\in \mathcal {V}$, there exists an $x-y$-path, i.e., there is a sequence $(x_{1},\ldots ,x_{N+1})\in \mathcal {V}^{N+1}$ such that $x_{1}=x,~~x_{N+1}=y$ and $\mathcal {E}(x_{n},x_{n+1})>0$ for all $n\in \{1,\ldots ,N\}$. If no vertices appear more than once in $(x_{1},\ldots ,x_{N})$, the path $(x_{1},\ldots ,x_{N+1})$ is called a simple path. The path is called a cycle or closed when the origin and the end are identical, i.e., $x_{1}=x_{N+1}$. An n-cycle is a cycle with n vertices.

In the sequel, we shall always consider graphs $\mathcal {G}$, which are locally finite, connected and have no loop.

The set of cyclic permutations of $(x,y,z)\in \mathcal {V}^{3}$ is denoted by

$$\begin{aligned} \circlearrowleft (x,y,z):=\{(x,y,z),\ (y,z,x),\ (z,x,y)\}. \end{aligned}$$

Let Tr the set of all simple 3-cycles

$$\begin{aligned} \mathcal {F}= Tr/ \cong \end{aligned}$$

where $\varpi _{1}\cong \varpi _{2}$ if and only if $\varpi _{1}$ is a cyclic permutation of $\varpi _{2}$. The elements of $\mathcal {F}$ are called oriented triangular faces. Atriangulation is a couple $(\mathcal {G},\mathcal {F})$ where $\mathcal {G}$ is a graph and $\mathcal {F}$ is the set of all oriented triangular faces.

In the sequel, we represent the oriented triangular faces by their vertices. For a oriented triangular face $\varpi =(x,y,z)$, we have

$$\begin{aligned} \varpi =(x,y,z)=(y,z,x)=(z,x,y). \end{aligned}$$

Let $(\mathcal {G}=(\mathcal {V},\mathcal {E},m),\mathcal {F})$ be a triangulation and let $\mathcal {R}:\mathcal {V}\times \mathcal {V}\times \mathcal {V}\longrightarrow [0,+\infty )$ such that

$$\begin{aligned} (x,y,z)\in \mathcal {F}\Longleftrightarrow \mathcal {R}(x,y,z)>0. \end{aligned}$$

We assume that $\mathcal {R}$ is symmetric, i.e.,

$$\begin{aligned} \mathcal {R}(x,y,z)=\mathcal {R}(\sigma (x,y,z)) \end{aligned}$$

for all $(x,y,z)\in \mathcal {V}\times \mathcal {V}\times \mathcal {V}$ and for any permutation $\sigma (x,y,z)$ of (x, y, z). We say that $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ is a weighted triangulation with weight of triangular faces $\mathcal {R}$. We say that $\mathcal {T}$ is simple if $\mathcal {G}:=(\mathcal {V},m,\mathcal {E})$ is simple and the weights of the faces equal 1. Choosing an orientation of triangulation consists of defining a partition of $\mathcal {F}$:

$$\begin{aligned}&\mathcal {F}=\mathcal {F}^{+}\sqcup \mathcal {F}^{-},\\&(x_{1},x_{2},x_{3})\in \mathcal {F}^{+} \Longleftrightarrow (x_{3},x_{2},x_{1})\in \mathcal {F}^{-}. \end{aligned}$$

For $(x_{1},x_{2},x_{3})\in \mathcal {F}$, we denote

$$\begin{aligned} -(x_{1},x_{2},x_{3}):=(x_{3},x_{2},x_{1}). \end{aligned}$$

In this case, we have

$$\begin{aligned} \mathcal {R}(x_{1},x_{2},x_{3})=\mathcal {R}(-(x_{1},x_{2},x_{3})). \end{aligned}$$

The set of neighbors of the edge (x, y) is given by

$$\begin{aligned} \mathcal {F}_{(x,y)}:=\mathcal {N}_{\mathcal {G}}(x)\cap \mathcal {N}_{\mathcal {G}}(y). \end{aligned}$$

The weighted degree of edges is given by:

$$\begin{aligned} d_{\mathcal {E}}(x,y):=\displaystyle \frac{1}{\mathcal {E}(x,y)}\sum _{z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z). \end{aligned}$$

When $\mathcal {T}$ is simple, $d_{\mathcal {E}}(x,y)=\sharp \mathcal {F}_{(x,y)}.$

3 The Symmetric and Skew-Symmetric Spaces

3.1 Hilbert Structures on the Set of Edges

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. Let $\mathcal {E}:=\{(x,y)\in \mathcal {V}\times \mathcal {V}:\ \mathcal {E}(x,y)>0\}$. The set of 1-cochains (or 1-forms) is given by:

$$\begin{aligned} {\mathcal {C}}_{{\mathrm{skew}}}(\mathcal {E}):= \Big \{f:\mathcal {E}\rightarrow \mathbb {C},\, f(x,y)=-f(y,x) \text{ for } \text{ all } x,y\in \mathcal {V}\Big \}, \end{aligned}$$

where skew stands for skew-symmetric. This corresponds to fermionic statistics. The set of functions with finite support is denoted by $\mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {E})$. Concerning bosonic statistics, we define:

$$\begin{aligned} {\mathcal {C}}_{{\mathrm{sym}}}(\mathcal {E}):= \Big \{f:\mathcal {E}\rightarrow \mathbb {C},\, f(x,y)=f(y,x) \text{ for } \text{ all } x,y\in \mathcal {V}\Big \}. \end{aligned}$$

The set of functions with finite support is denoted by ${\mathcal {C}}_{{\mathrm{sym}}}^{c}(\mathcal {E})$.

We turn to the Hilbert structures.

$$\begin{aligned} \ell ^{2}_{{\mathrm{skew}}}({\mathcal {E}}):=\left\{ f\in {\mathcal {C}}_{{\mathrm{skew}}}(\mathcal {E}) \text{ such } \text{ that } \left\| f \right\| ^{2}:\displaystyle =\frac{1}{2}\sum _{x,y\in \mathcal {V}}{\mathcal {E}}(x,y) | f(x,y)|^{2}<\infty \right\} \end{aligned}$$

and

$$\begin{aligned} \ell ^{2}_{\mathrm{sym}}({\mathcal {E}}):=\left\{ f\in {\mathcal {C}}_{\mathrm{sym}}(\mathcal {E}) \text{ such } \text{ that } \left\| f \right\| ^{2}:\displaystyle =\frac{1}{2}\sum _{x,y\in \mathcal {V}}{\mathcal {E}}(x,y) | f(x,y)|^{2}<\infty \right\} . \end{aligned}$$

The associated scalar product is given by

$$\begin{aligned} \langle f,g\rangle :=\frac{1}{2}\sum _{(x,y)\in \mathcal {E}}{\mathcal {E}}(x,y) \overline{f(x,y)}g(x,y), \end{aligned}$$

when f and g are both in $\ell ^{2}_{{\mathrm{skew}}}({\mathcal {E}})$ or in $\ell ^{2}_{\mathrm{sym}}({\mathcal {E}})$.

3.2 Hilbert Structures on the Set of Faces

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. The set of 2-cochains or 2-forms is given by

$$\begin{aligned} \mathcal {C}_{\mathrm{skew}}(\mathcal {R})=\Big \{f:\mathcal {F}\longrightarrow \mathbb {C}:f(x,y,z)=-f(z,y,x)\Big \}. \end{aligned}$$

(1)

The set of functions with finite support is denoted by $\mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$. Concerning the case symmetric, we define

$$\begin{aligned} \mathcal {C}_{\mathrm{sym}}(\mathcal {R})=\Big \{f:\mathcal {F}\longrightarrow \mathbb {C}:f(x,y,z)=f(z,y,x)\Big \}. \end{aligned}$$

The set of functions with finite support is denoted by $\mathcal {C}_\mathrm{sym}^{c}(\mathcal {R})$. Let us define the Hilbert spaces $\ell ^{2}_{\mathrm{skew}}(\mathcal {R})$ and $\ell ^{2}_{\mathrm{sym}}(\mathcal {R})$ as the sets of cochains with finite norm, we have

$$\begin{aligned} \ell ^{2}_{\mathrm{skew}}(\mathcal {R}):=\left\{ f\in \mathcal {C}_{\mathrm{skew}}(\mathcal {R}):~~ \displaystyle \left\| f\right\| ^{2}=\frac{1}{2}\sum _{(x,y,z)\in \mathcal {F}}\mathcal {R}(x,y,z)\mid f(x,y,z)\mid ^{2} <\infty \right\} \end{aligned}$$

and

$$\begin{aligned} \ell ^{2}_{\mathrm{sym}}(\mathcal {R}):=\left\{ f\in \mathcal {C}_{\mathrm{sym}}(\mathcal {R}):~~ \displaystyle \left\| f\right\| ^{2}=\frac{1}{2}\sum _{(x,y,z)\in \mathcal {F}}\mathcal {R}(x,y,z)\mid f(x,y,z)\mid ^{2}<\infty \right\} . \end{aligned}$$

The associated scalar product is given by

$$\begin{aligned} \langle f,g\rangle&:=\displaystyle \frac{1}{2}\sum _{(x,y,z)\in \mathcal {F}}\mathcal {R}(x,y,z)\overline{f(x,y,z)}g(x,y,z)\\&:=\displaystyle \frac{1}{6}\sum _{(x,y)\in \mathcal {E}}\sum _{z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z)\overline{f(x,y,z)}g(x,y,z) \end{aligned}$$

when f and g are both in $\ell ^{2}_{\mathrm{skew}}(\mathcal {R})$ or in $\ell ^{2}_{\mathrm{sym}}(\mathcal {R})$.

4 Operators

In this section, we recall the concept of exterior derivative operator associated with a faces space, we refer to [7, 8] for more details. This permits to define the discrete Laplacian acting on 2-forms.

4.1 Skew-Symmetric Case

We start with defining the operators in the skew-symmetric case. The skew-symmetric exterior operator is the operator $d^{1}_{\mathrm{skew}}: \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {E})\longrightarrow \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$, given by

$$\begin{aligned} d_{\mathrm{skew}}^{1}(f)(x,y,z)=f(x,y)+ f(y,z)+ f(z,x). \end{aligned}$$

The skew-symmetric coexterior derivative operator is the formal adjoint of $d_{\mathrm{skew}}^{1}$, i.e., it is the operator $\delta _{\mathrm{skew}}^{1}: \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})\longrightarrow \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {E})$, given by

$$\begin{aligned} \forall f\in \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R}),~~\delta _{\mathrm{skew}}^{1}(f)(x,y) =\displaystyle \frac{1}{\mathcal {E}(x,y)}\sum _{z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z)f(x,y,z). \end{aligned}$$

Both operators are closable (see [7, Lemma 3.1]). We denote their closure by the same symbol. The skew-symmetric discrete Laplacian operator acting on 2-forms is given by

$$\begin{aligned} {\mathcal {L}}_{2,\mathrm{skew}}(f)(x,y,z)&=d_{\mathrm{skew}}^{1}\delta _{\mathrm{skew}}^{1}(f)(x,y,z)\\&=\displaystyle \frac{1}{\mathcal {E}(x,y)}\sum _{t\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,t)f(x,y,t)\\&\quad +\frac{1}{\mathcal {E}(y,z)}\sum _{t\in \mathcal {F}_{(y,z)}}\mathcal {R}(y,z,t)f(y,z,t)\\&\quad +\frac{1}{\mathcal {E}(~z,x)}\sum _{t\in \mathcal {F}_{(z,x)}}\mathcal {R}(z,x,t)f(z,x,t), \end{aligned}$$

with $f\in \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$.

4.2 Symmetric Case

We turn to the symmetric case. The symmetric exterior operator is the operator $d_{\mathrm{sym}}^{1}: \mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {E}) \longrightarrow \mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {R})$, given by

$$\begin{aligned} \forall f\in \mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {E}),~~d^{1}_\mathrm{sym}(f)(x,y,z)=f(x,y)+f(y,z)+f(z,x). \end{aligned}$$

The symmetric coexterior derivative operator is the formal adjoint of $d_{\mathrm{sym}}^{1}$, i.e., it is the operator $\delta _\mathrm{sym}^{1}:\mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {R})\longrightarrow \mathcal {C}_\mathrm{sym}^{c}(\mathcal {E})$, given by

$$\begin{aligned} \forall f \in \mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {R}),\delta _\mathrm{sym}^{1}(f)(x,y):=\displaystyle \frac{1}{\mathcal {E}(x,y)} \sum _{z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z)f(x,y,z). \end{aligned}$$

Indeed, for $f\in \mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {E})$ and $g\in \mathcal {C}_\mathrm{sym}^{c}(\mathcal {R})$, we get:

$$\begin{aligned} \langle d^{1}_{\mathrm{sym}}f,g\rangle&=\displaystyle \frac{1}{2}\sum _{(x,y,z)\in \mathcal {F}}\mathcal {R}(x,y,z)d_\mathrm{sym}^{1}f(x,y,z)\overline{g(x,y,z)}\\&=\displaystyle \frac{1}{6}\sum _{x\sim y}\sum _{z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z)(f(x,y)+f(y,z)+f(z,x)) \overline{g(x,y,z)}\\&=\displaystyle \frac{1}{2}\sum _{x\sim y}\sum _{z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z)f(x,y)\overline{g(x,y,z)}\\&=\displaystyle \frac{1}{2}\sum _{x\sim y}\mathcal {E}(x,y)f(x,y)\overline{\Big (\frac{1}{\mathcal {E}(x,y)}\sum _{z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z)g(x,y,z)\Big )}\\&=\langle f,\delta _{\mathrm{sym}}^{1}g\rangle . \end{aligned}$$

The operators $d_{\mathrm{sym}}^{1}$ and $\delta _{\mathrm{sym}}^{1}$ are closable. Indeed, since $\delta _{\mathrm{sym}}^{1}:\ell ^{2}_\mathrm{sym}(\mathcal {R})\longrightarrow \ell ^{2}_{\mathrm{sym}} (\mathcal {E})$ (resp. $d_\mathrm{sym}^{1}:\ell ^{2}_{\mathrm{sym}}(\mathcal {E})\longrightarrow \ell ^{2}_\mathrm{sym}(\mathcal {R}))$ is with dense domain then $\delta _{\mathrm{sym}}^{1}$ (resp. $d_{\mathrm{sym}}^{1})$ is closable. We denote their closure by the same symbol. The symmetric discrete Laplacian operator acting on 2-forms is the operator ${\mathcal {L}}_{2, \mathrm{sym}}=d_\mathrm{sym}^{1}\delta _{\mathrm{sym}}^{1}$, given by the same expression of ${\mathcal {L}}_{2, \mathrm{skew}}$, but they do not act on the same space.

4.3 Relationship Between ${\mathcal {L}}_{2,\mathrm{skew}}$ and ${\mathcal {L}}_{2,\mathrm{sym}}$

The two operators $\mathcal {L}_{2,\mathrm{skew}}$ and $\mathcal {L}_{2,\mathrm{sym}}$ have the same expression. However, they do not act on the same spaces. Namely, when $\mathcal {T}$ is tri-partite, we shall prove that the two operators are unitarily equivalent.

Definition 1

A tri-partite graph is a graph whose vertices can be partitioned into 3 disjoint sets so that there are no two vertices within the same set are adjacent. A tri-partite triangulation is a triangulation $\mathcal {T}=(\mathcal {G},\mathcal {F})$ such that $\mathcal {G}$ is tri-partite.

Theorem 1

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E}, \mathcal {R})$ be a tri-partite weighted triangulation. Then, ${\mathcal {L}}_{2,\mathrm{skew}}$ and ${\mathcal {L}}_{2,\mathrm{sym}}$ are unitarily equivalent.

Proof

We consider the tri-partite decomposition $\{\mathcal {V}_{1},\mathcal {V}_{2},\mathcal {V}_{3}\}$. Set

$$\begin{aligned} \circlearrowleft \mathcal {V}_{1}\times \mathcal {V}_{2}\times \mathcal {V}_{3}=\{\circlearrowleft (x,y,z): \ (x,y,z)\in \mathcal {V}_{1}\times \mathcal {V}_{2}\times \mathcal {V}_{3}\}. \end{aligned}$$

Let $\mathcal {U}:\ell _{\mathrm{skew}}^{2}(\mathcal {R})\longrightarrow \ell _\mathrm{sym}^{2}(\mathcal {R})$ be the unitary map given by

$$\begin{aligned} \mathcal {U}(f)(x,y,z)=S(x,y,z)f(x,y,z), \end{aligned}$$

where

$$\begin{aligned} S(x,y,z):=\left\{ \begin{array}{ll} 1, &{} \quad {if } (x,y,z)\in \circlearrowleft \mathcal {V}_{1}\times \mathcal {V}_{2}\times \mathcal {V}_{3} ,\\ ~~~~~\\ -1, &{} \quad {if } (x,y,z)\in \circlearrowleft \mathcal {V}_{3}\times \mathcal {V}_{2}\times \mathcal {V}_{1}. \end{array} \right. \end{aligned}$$

Let $\mathcal {W}$ be the following mapping from $\ell ^{2}_\mathrm{sym}(\mathcal {R})$ into $\ell ^{2}_{\mathrm{skew}}(\mathcal {R}):\ \mathcal {W}(f)(x,y,z)=S(x,y,z)f(x,y,z)$. Then

$$\begin{aligned} \langle \mathcal {U}f,g\rangle =\langle f,\mathcal {W}g\rangle ,\ \mathcal {U}(\mathcal {W}(g))=g \text{ and } \mathcal {W}(\mathcal {U}(f))=f \end{aligned}$$

for all $f\in \ell ^{2}_{\mathrm{skew}}(\mathcal {F})$ and $g\in \ell ^{2}_\mathrm{sym}(\mathcal {F})$. So we have

$$\begin{aligned} \mathcal {W}(f)=\mathcal {U}^{-1}(f)=\mathcal {U}^{*}(f) \end{aligned}$$

for all $f\in \ell ^{2}_{\mathrm{sym}}(\mathcal {R})$. Therefore,

$$\begin{aligned} \mathcal {U}{\mathcal {L}}_{2,\mathrm{skew}}\mathcal {U}^{-1}(f)(x,y,z)&=\displaystyle \frac{S(x,y,z)}{\mathcal {E}(x,y)}\sum _{t\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,t)S(x,y,t)f(x,y,t)\\&\quad +\frac{S(x,y,z)}{\mathcal {E}(y,z)}\sum _{t\in \mathcal {F}_{(y,z)}} \mathcal {R}(y,z,t)S(y,z,t)f(y,z,t)\\&\quad +\frac{S(x,y,z)}{\mathcal {E}(~z,x)}\sum _{t\in \mathcal {F}_{(z,x)}} \mathcal {R}(z,x,t)S(z,x,t)f(z,x,t). \end{aligned}$$

Moreover, noting that

$$\begin{aligned} S(x,y,z)S(x,y,s)=S(x,y,z)S(y,z,t)=S(x,y,z)S(z,x,u)=1 \end{aligned}$$

for all $(x,y,z)\in \mathcal {F}$, $s\in \mathcal {F}_{(x,y)}$, $t\in \mathcal {F}_{(y,z)}$ and $u\in \mathcal {F}_{(z,x)}$, we get $\mathcal {U}{\mathcal {L}}_{2,\mathrm{skew}}\mathcal {U}^{-1}={\mathcal {L}}_{2,\mathrm{sym}}$. $\square$

5 A Nelson Criterium

For the general theory of unbounded Hermitian operators and their extensions, we refer the reader to [19, 24, 26]. Recalling the set $\mathcal {C}_{\mathrm{skew}}(\mathcal {R})$ in (1). Let ${\mathcal {L}}_{\mathrm{skew}}$ be the following mapping from $\mathcal {C}_{\mathrm{skew}}(\mathcal {R})$ into itself:

$$\begin{aligned} {\mathcal {L}}_{\mathrm{skew}}(f)(x,y,z)&=\displaystyle \frac{1}{\mathcal {E}(x,y)}\sum _{t\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,t)f(x,y,t)\\&\quad \displaystyle +\frac{1}{\mathcal {E}(y,z)}\sum _{t\in \mathcal {F}_{(y,z)}}\mathcal {R}(y,z,t)f(y,z,t)\\&\quad \displaystyle +\frac{1}{\mathcal {E}(~z,x)}\sum _{t\in \mathcal {F}_{(z,x)}}\mathcal {R}(z,x,t)f(z,x,t). \end{aligned}$$

Let ${\mathcal {L}}_{2,\mathrm{max},\mathrm{skew}}$ be the restrictions of ${\mathcal {L}}_{\mathrm{skew}}$ to

$$\begin{aligned} \mathcal {D}({\mathcal {L}}_{2,\mathrm{max},\mathrm{skew}}):=\Big \{f\in \ell ^{2}_{\mathrm{skew}}(\mathcal {R}) \text{ such } \text{ that } {\mathcal {L}}_{\mathrm{skew}}f\in \ell ^{2}_{\mathrm{skew}}(\mathcal {R})\Big \}. \end{aligned}$$

Lemma 1

${\mathcal {L}}_{2,\mathrm{skew}}^{*}={\mathcal {L}}_{2,\mathrm{max},\mathrm{skew}}$.

Proof

Let $f\in \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$ and let $g\in \mathcal {C}_{\mathrm{skew}}(\mathcal {R})$. Let $\mathcal {F}_{00}$ the support of f and set

$$\begin{aligned} \mathcal {F}_{0}=\Big \{(x,y,z)\in \mathcal {F}:~~\exists u\in \mathcal {V},\ \{(x,y,u),~~(y,z,u),~~(z,x,u)\}\cap \mathcal {F}_{00}\ne \emptyset \Big \} \end{aligned}$$

which is a finite set. Then, $\mathrm{supp}({\mathcal {L}}_{2,\mathrm{skew}})\subset \mathcal {F}_{0}$ and the following relation holds:

$$\begin{aligned}&\displaystyle \frac{1}{2}\sum _{(x,y,z)\in \mathcal {F}_{0}} \mathcal {R}(x,y,z){\mathcal {L}}_{2,\mathrm{skew}}(f)(x,y,z)\overline{g(x,y,z)}\nonumber \\&\quad =\displaystyle \frac{1}{2}\sum _{(x,y,z)\in \mathcal {F}_{0}} \mathcal {R}(x,y,z)\left( \frac{1}{\mathcal {E}(x,y)}\sum _{t\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,t)f(x,y,t)\right. \nonumber \\&\qquad +\frac{1}{\mathcal {E}(y,z)} \sum _{t\in \mathcal {F}_{(y,z)}}\mathcal {R}(y,z,t)f(y,z,t)\nonumber \\&\qquad \left. +\frac{1}{\mathcal {E}(~z,x)}\sum _{t\in \mathcal {F}_{(z,x)}}\mathcal {R}(z,x,t) f(z,x,t)\right) \overline{g(x,y,z)} \nonumber \\&\quad =\displaystyle \frac{1}{2}\sum _{(x,y,z)\in \mathcal {F}_{00}} \mathcal {R}(x,y,z)f(x,y,z) \left( \overline{\frac{1}{\mathcal {E}(x,y)}\sum _{u\in \mathcal {V}}\mathcal {R}(x,y,u)g(x,y,u)}\right. \nonumber \\&\qquad +\overline{\frac{1}{\mathcal {E}(y,z)}\sum _{u\in \mathcal {V}} \mathcal {R}(y,z,u)g(y,z,u)} \nonumber \\&\qquad \left. +\overline{\frac{1}{\mathcal {E}(z,x)}\sum _{u\in \mathcal {V}}\mathcal {R} (z,x,u)g(z,x,u)}\right) \nonumber \\&\quad =\displaystyle \frac{1}{2}\sum _{x,y,z\in \mathcal {V}} \mathcal {R}(x,y,z)f(x,y,z)\overline{{\mathcal {L}}_{2}g(x,y,z)}. \end{aligned}$$

(2)

Let $g\in \mathcal {D}({\mathcal {L}}_{2,\mathrm{max},\mathrm{skew}})$. It follows from (2) that

$$\begin{aligned} \langle {\mathcal {L}}_{2,\mathrm{skew}}f,g\rangle =\langle f,{\mathcal {L}}_{2,\mathrm{max},\mathrm{skew}}g\rangle \end{aligned}$$

for all $f\in \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$, which implies that $g\in \mathcal {D}({\mathcal {L}}_{2,\mathrm{skew}}^{*})$. Now let $g\in \mathcal {D}({\mathcal {L}}_{2,\mathrm{skew}}^{*})$. Let $(x,y,z)\in \mathcal {F}$ and let

$$\begin{aligned} f=\displaystyle \frac{1}{\mathcal {R}(x,y,z)}\Big (\mathbf{1}_{\circlearrowleft (x,y,z)}- \mathbf{1}_{\circlearrowleft (z,y,x)}\Big ). \end{aligned}$$

Then, $f\in \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$ and we obtain from (2):

$$\begin{aligned} \overline{({\mathcal {L}}_{2,\mathrm{skew}}^{*}g)(x,y,z)}&=\langle f,{\mathcal {L}}_{2,\mathrm{skew}}^{*}g\rangle \\&=\langle {\mathcal {L}}_{2,\mathrm{skew}}f,g \rangle \\&=\displaystyle \frac{1}{2}\sum _{(u,v,w)\in \mathcal {F}}\mathcal {R}(u,v,w){\mathcal {L}}_{2, \mathrm{skew}}f(u,v,w)\overline{g(u,v,w)}\\&=\displaystyle \frac{1}{2}\sum _{(u,v,w)\in \mathcal {F}}\mathcal {R}(u,v,w)f(u,v,w)\overline{{\mathcal {L}}_{2,\mathrm max,\mathrm{skew}}g(u,v,w)}\\&=\overline{({\mathcal {L}}_{2,\mathrm{max},\mathrm{skew}}g)(x,y,z)} \end{aligned}$$

which implies that ${\mathcal {L}}_{2,\mathrm{skew}}g={\mathcal {L}}_{2,\mathrm max,\mathrm{skew}}g\in \ell ^{2}_{\mathrm{skew}}(\mathcal {R})$ by the definition of the adjoint, it follows that $g\in \mathcal {D}({\mathcal {L}}_{2,\mathrm max, \mathrm{skew}})$. Hence, ${\mathcal {L}}_{2,\mathrm{skew}}^{*}={\mathcal {L}}_{2,\mathrm{max},\mathrm{skew}}$. $\square$

Remark 1

Let $\mathcal {L}_{\mathrm{sym}}$ be the mapping from $\mathcal {C}_{sym}(\mathcal {R})$ into itself given by the same expression of $\mathcal {L}_{2, \mathrm{sym}}$. Then, ${\mathcal {L}}_{2,\mathrm{sym}}^{*}={\mathcal {L}}_{2,\mathrm{max},\mathrm{sym}}$ where ${\mathcal {L}}_{2,\mathrm{max},\mathrm{sym}}$ is the restrictions of $\mathcal {L}_\mathrm{sym}$ to

$$\begin{aligned} \mathcal {D}({\mathcal {L}}_{2,\mathrm{max},\mathrm sym}):=\Big \{f\in \ell ^{2}_{\mathrm{sym}}(\mathcal {R}) \text{ such } \text{ that } {\mathcal {L}}_{\mathrm{sym}}f\in \ell ^{2}_{\mathrm{sym}}(\mathcal {R})\Big \}. \end{aligned}$$

Using the Nelson commutator theorem, we prove the criterium of essential self-adjointness for $\mathcal {L}_{2,\mathrm{skew}}$ and $\mathcal {L}_{2,\mathrm sym}$.

Theorem 2

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. Set

$$\begin{aligned} \mathcal {N}(x,y,z)=1+d_{\mathcal {E}}(x,y)+d_{\mathcal {E}}(y,z)+d_{\mathcal {E}}(z,x). \end{aligned}$$

Suppose that

$$\begin{aligned} \sup _{ x\sim y,\ z\in \mathcal {F}_{(x,y)}}\displaystyle \sum _{r\in \mathcal {F}_{(x,y)}}\frac{1}{\mathcal {E}(x,y)}\mathcal {R}(x,y,r) \mid \mathcal {N}(x,y,r)-\mathcal {N}(x,y,z)\mid ^{2}<\infty . \end{aligned}$$

Then, $\mathcal {L}_{2,\mathrm{skew}}$ is essentially self-adjoint on $\mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$ and $\mathcal {L}_{2,\mathrm{sym}}$ is essentially self-adjoint on $\mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {R})$.

Proof

Let $\mathcal {N}$ be the operator of multiplication by $\mathcal {N}(.,.,.)$ ant take $f\in \mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {F})$. Going over the same techniques of the proof of [3, Theorem 5.13], we obtain:

$$\begin{aligned} \Vert \mathcal {L}_{2,\mathrm{skew}}f\Vert ^{2}\displaystyle&\displaystyle \le \frac{2}{3}\sum _{x\sim y,z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z)\left( \frac{1}{\mathcal {E}^{2}(x,y)} \left| \sum _{t\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,t)f(x,y,t)\right| ^{2}\right. \\&\quad +\displaystyle \frac{1}{\mathcal {E}^{2}(y,z)}\left| \sum _{t\in \mathcal {F}_{(y,z)}}\mathcal {R}(y,z,t)f(y,z,t)\right| ^{2}\\&\quad \left. +\displaystyle \frac{1}{\mathcal {E}^{2}(z,x)}\left| \sum _{t\in \mathcal {F}_{(z,x)}}\mathcal {R}(z,x,t)f(z,x,t)\right| ^{2}\right) \\&\le \displaystyle 2\sum _{x\sim y,z\in \mathcal {F}_{(x,y)}} \mathcal {R}(x,y,z)\frac{1}{\mathcal {E}^{2}(x,y)}\left| \sum _{t\in \mathcal {F}_{(x,y)}} \mathcal {R}(x,y,t)f(x,y,t)\right| ^{2}\\&\le 2 \displaystyle \sum _{x\sim y,z\in \mathcal {F}_{(x,y)}} \mathcal {R}(x,y,z)\frac{1}{\mathcal {E}^{2}(x,y)}\left( \sum _{r\in \mathcal {F}_{(x,y)}} \mathcal {R}(x,y,r)\right) \\&\quad \times \, \left( \sum _{t\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,t)\mid f(x,y,t)\mid ^{2}\right) \\&=\displaystyle 2\sum _{x\sim y,z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z) \left( \frac{1}{\mathcal {E}(x,y)}\sum _{t\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,t)\right) ^{2} \mid f(x,y,z)\mid ^{2}\\&\displaystyle \le 12 \Vert \mathcal {N}(f)\Vert ^{2}. \end{aligned}$$

Moreover, we notice that $\mathcal {N}(.,.,.)$ is symmetric and $f(x,y,z)=-f(z,y,x)$ and let $J=\mid \langle f,[\mathcal {L}_{2,\mathrm{skew}},\mathcal {N}]f\rangle \mid$. We get:

$$\begin{aligned}&\le \displaystyle \frac{1}{12}\sum _{x\sim y,z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z)\Big (\mid f(x,y,z)\mid ^{2}+\mid [\mathcal {L}_{2,\mathrm{skew}},\mathcal {N}](f)(x,y,z)\mid ^{2}\Big )\\&\le \displaystyle \frac{1}{2}\Vert \mathcal {N}^{\frac{1}{2}}(f)\Vert ^{2}+\sum _{x\sim y,z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z) \left| \sum _{t\in \mathcal {F}_{(x,y)}}\frac{1}{\mathcal {E}(x,y)}\mathcal {R}(x,y,t)\right. \\&\quad \left. \times \, \Big (\mathcal {N}(x,y,t)-\mathcal {N}(x,y,z)\Big )f(x,y,t)\right| ^{2}\\&\displaystyle \le \frac{1}{2}\Vert \mathcal {N}^{\frac{1}{2}}(f)\Vert ^{2}+\sum _{x\sim y,z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z) \left( \sum _{t\in \mathcal {F}_{(x,y)}}\frac{1}{\mathcal {E}(x,y)}\mathcal {R}(x,y,t)\right) \\&\quad \times \, \left( \sum _{r\in \mathcal {F}_{(x,y)}}\frac{1}{\mathcal {E}(x,y)}\mathcal {R}(x,y,r)\mid \mathcal {N}(x,y,r)-\mathcal {N}(x,y,z)\mid ^{2}\mid f(x,y,r)\mid ^{2}\right) \\&=\frac{1}{2}\Vert \mathcal {N}^{\frac{1}{2}}(f)\Vert ^{2}+\sum _{x\sim y,z\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,z) \left( \sum _{t\in \mathcal {F}_{(x,y)}}\frac{1}{\mathcal {E}(x,y)}\mathcal {R}(x,y,t)\right) \\&\quad \times \, \displaystyle \underbrace{\sum _{r\in \mathcal {V}}\frac{1}{\mathcal {E}(x,y)}\mathcal {F}(x,y,r)\mid \mathcal {N}(x,y,r)-\mathcal {N}(x,y,z)\mid ^{2}}_{\le C}\mid f(x,y,z)\mid ^{2}\\&\displaystyle \le \left( \frac{1+12C}{2}\right) \Vert \mathcal {N}^{\frac{1}{2}}(f)\Vert . \end{aligned}$$

Applying [25, Theorem X.37], the result follows. The proof of $\mathcal {L}_{2,\mathrm{sym}}$ may be checked in the same way as the proof of $\mathcal {L}_{2,\mathrm{skew}}$. $\square$

Corollary 1

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a simple triangulation. Assume that

$$\begin{aligned} \sup _{x\sim y,z\in \mathcal {F}_{(x,y)}}\sum _{r\in \mathcal {F}_{(x,y)}}|\sharp \mathcal {F}_{(y,z)}+\sharp \mathcal {F}_{(z,x)} -\sharp \mathcal {F}_{(y,r)}-\sharp \mathcal {F}_{(r,x)}|^{2}<\infty . \end{aligned}$$

Then, $\mathcal {L}_{2,\mathrm{skew}}$ is essentially self-adjoint on $\mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$ and $\mathcal {L}_{2,\mathrm{sym}}$ is essentially self-adjoint on $\mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {R})$.

6 Essential Spectrum

Let A be a closed, densely defined linear operator on a Banach space X, and let $\sigma (A)$ denote the spectrum of A. We denote by $\mathcal {K}(X)$ the set of compact operators on X to itself. We define the essential spectrum of the operator A by

$$\begin{aligned} \sigma _\mathrm{ess}(A)=\bigcap _{K\in \mathcal {K}(X)}\sigma (A+K). \end{aligned}$$

It is well known that if A is a self-adjoint operator on a Hilbert space, the essential spectrum of A is the set of limit points of the spectrum of A, i.e., all points of the spectrum except isolated eigenvalues of finite multiplicity, see [27]. Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. Note that ${\mathcal {L}}_{2,\mathrm{skew}}$ is nonnegative symmetric operator on $\mathcal {C}^{c}_{\mathrm{skew}}(\mathcal {R})$. We consider the quadratic form

$$\begin{aligned} q(f,g)=\langle f,{\mathcal {L}}_{2,\mathrm{skew}}g\rangle +\langle f,g\rangle \end{aligned}$$

on ${\mathcal {C}}_{\mathrm{skew}}^{c}(\mathcal {R})\times {\mathcal {C}}_{ skew}^{c}(\mathcal {R})$. Let $\mathcal {H}_{1}$ be the completion of ${\mathcal {C}}_{\mathrm{skew}}^{c}(\mathcal {R})$ under the norm

$$\begin{aligned} \left\| f\right\| _{q}=\displaystyle \sqrt{\langle {\mathcal {L}}_{2,\mathrm{skew}}f,f \rangle +\left\| f\right\| ^{2}}. \end{aligned}$$

We define the Friedrichs extension ${\mathcal {L}}_{2,\mathrm{skew}}^{\mathcal {F}}$ of ${\mathcal {L}}_{2,\mathrm{skew}}$ by:

(i):

A vector f is in domain $\mathcal {D}({\mathcal {L}}_{2,\mathrm{skew}}^{\mathcal {F}})$ if and only if $f\in \mathcal {H}_{1}$ and $\mathcal {C}^{c}_{\mathrm{skew}}(\mathcal {R}) \ni g\longmapsto \langle f,{\mathcal {L}}_{2,\mathrm{skew}}g\rangle +\langle f,g\rangle$ extends to a norm continuous function on $\ell ^{2}_{\mathrm{skew} }(\mathcal {R})$.

(ii):

For each $f\in \mathcal {D}({\mathcal {L}}_{2,\mathrm{skew}}^{\mathcal {F}})$, there is a unique $u_{f}$ such that $\langle f,{\mathcal {L}}_{2,\mathrm{skew}}g\rangle +\langle f,g\rangle =\langle u_{f},g\rangle$ by Riesz’ Theorem. The Friedrichs extension of ${\mathcal {L}}_{2,\mathrm{skew}}$, is given by ${\mathcal {L}}_{2,\mathrm{skew}}^{\mathcal {F}}f=u_{f}-f$. It is a self-adjoint extension of ${\mathcal {L}}_{2,\mathrm{skew}}$, e.g., see [25, Theorem X.23]. Note that ${\mathcal {L}}_{2,\mathrm{skew}}^{\mathcal {F}}$ is bounded if and only if $d_{\mathcal {E}}(.)$ is bounded, e.g., see [8].

Theorem 3

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E}, \mathcal {R})$ be a weighted triangulation and let $\mathcal {F}_{0}=\{\mathcal {K}\subset \mathcal {F}:\ \mathcal {K}\ \mathrm{finite}\}$. Then,

$$\begin{aligned} \inf \sigma \left( {\mathcal {L}}_{2,\mathrm{skew}}^{\mathcal {F}}\right) \le \inf _{(x,y,z) \in \mathcal {F}}\mathcal {R}(x,y,z)\left( \frac{1}{\mathcal {E}(x,y)}+\frac{1}{\mathcal {E}(y,z)} +\frac{1}{\mathcal {E}(z,x)}\right) . \end{aligned}$$

and

$$\begin{aligned} \inf \sigma _\mathrm{ess}\left( {\mathcal {L}}_{2,\mathrm{skew}}^{\mathcal {F}}\right) \le \sup _{\mathcal {K} \subset \mathcal {F}_{0}}\inf _{(x,y,z)\in \mathcal {K}^{c}}\mathcal {R}(x,y,z)\left( \frac{1}{\mathcal {E}(x,y)}+\frac{1}{\mathcal {E}(y,z)} +\frac{1}{\mathcal {E}(z,x)}\right) . \end{aligned}$$

In particular, if $\mathcal {T}$ is a simple triangulation then ${\mathcal {L}}_{2,\mathrm{skew}}^{\mathcal {F}}$ is not with compact resolvent.

Proof

Let $(x_{0},y_{0},z_{0})\in \mathcal {F}$ and let

$$\begin{aligned} f=\displaystyle \frac{\mathbf{1}_{\circlearrowleft (x_{0},y_{0},z_{0})} -\mathbf{1}_{\circlearrowleft (z_{0},y_{0},x_{0})}}{\sqrt{\mathcal {R}(x_{0},y_{0},z_{0})}} \end{aligned}$$

where $\mathbf{1}_{\circlearrowleft (x_{0},y_{0},z_{0})}$ denotes the indicator function of $\circlearrowleft (x_{0},y_{0},z_{0})$. Then, $\Vert f\Vert =1$ and

$$\begin{aligned} \langle f,{\mathcal {L}}_{2,\mathrm{skew}}f\rangle =\displaystyle \mathcal {R}(x_{0},y_{0},z_{0})\left( \frac{1}{\mathcal {E}(x_{0},y_{0})}+\frac{1}{\mathcal {E}(y_{0},z_{0})}+\frac{1}{\mathcal {E}(z_{0},x_{0})}\right) \end{aligned}$$

Applying [20, Proposition 3], the result follows. $\square$

7 Application to the Study of the Adjacency Matrix

7.1 Adjacency Matrix

The adjacency matrix has important implications. For example, it uses the semi-boundedness in order to give meaning to the heat equation, see [5]. Let $\mathcal {G}=(\mathcal {V},m,\mathcal {E})$ be a weighted graph. We define the set of 0-cochains on $\mathcal {V}$ by

$$\begin{aligned} \mathcal {C}(\mathcal {V})=\{f:\mathcal {V}\longrightarrow \mathbb {C}\}. \end{aligned}$$

We denote by $\mathcal {C}^{c}(\mathcal {V})$ the 0-cochains with finite support in $\mathcal {V}$. We associate a Hilbert space to $\mathcal {V}:$

$$\begin{aligned} \ell ^{2}(\mathcal {V})=\Big \{f\in \mathcal {C}(\mathcal {V}) \text{ such } \text{ that } \Vert f\Vert ^{2}=\sum _{x\in \mathcal {V}}m(x)|f(x)|^{2}<\infty \Big \}. \end{aligned}$$

The associated scalar product is given by

$$\begin{aligned} \langle f,g \rangle =\displaystyle \sum _{x\in \mathcal {V}}m(x)\overline{f(x)}g(x),\ \quad \text{ for } f,\ g\in \ell ^{2}(\mathcal {V}). \end{aligned}$$

We define the adjacency matrix:

$$\begin{aligned} \mathcal {A}_{\mathcal {G}}(f)(x)=\displaystyle \frac{1}{m(x)}\sum _{y\in \mathcal {V}}\mathcal {E}(x,y)f(y),\ f\in \mathcal {C}^{c}(\mathcal {V}). \end{aligned}$$

It is symmetric and thus closable. We denote its closure by the same symbol. When $\mathcal {G}$ is simple, we have that $\mathcal {A}_{\mathcal {G}}$ is unbounded if and only if it is unbounded from above and if and only if the degree is unbounded, see [14].

7.2 Triangular Graph

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. Set $\widehat{\mathcal {V}}=\mathcal {F}{ /} \sim$, where $\varpi \sim -\varpi$.

Definition 2

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. Set $\widehat{\mathcal {V}}=\mathcal {F}{/} \sim$, where $\varpi \sim -\varpi$. The triangular graph of $\mathcal {T}$ is the graph $\widehat{\mathcal {G}}=(\widehat{\mathcal {V}},\widehat{m},\widehat{\mathcal {E}})$ where $\widehat{m}=1$ and

$$\begin{aligned} \widehat{\mathcal {E}}((x_{0},y_{0},z_{0}),(x,y,z))&=\displaystyle \sqrt{\mathcal {R}(x_{0},y_{0},z_{0})}\left( \frac{\sqrt{\mathcal {R}(x,y,z)}}{\mathcal {E}(x,y)}1_{x=x_{0},\ y=y_{0}}\right. \\&\left. \quad +\frac{\sqrt{\mathcal {R}(x,y,z)}}{\mathcal {E}(y,z)}1_{y=y_{0},\ z=z_{0}}+\frac{\sqrt{\mathcal {R}(x,y,z)}}{\mathcal {E}(z,x)}1_{z=z_{0},\ x=x_{0}}\right) \end{aligned}$$

if $(x_{0},y_{0},z_{0})\ne (x,y,z)$ and 0 otherwise.

Remark 2

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a triangulation. The adjacency matrix on $\widehat{\mathcal {G}}$ is given by

$$\begin{aligned} \mathcal {A}_{\widehat{\mathcal {G}}}(f)(x_{0},y_{0},z_{0})&:=\displaystyle \sum _{z\in \mathcal {V},\ z\ne z_{0}} \frac{\sqrt{\mathcal {R}(x_{0},y_{0},z_{0})\mathcal {R}(x_{0},y_{0},z)}}{\mathcal {E}(x_{0},y_{0})}f(x_{0},y_{0},z)\\&\quad + \sum _{x\in \mathcal {V},\ x\ne x_{0}}\frac{\sqrt{\mathcal {R}(x_{0},y_{0},z_{0})\mathcal {R}(x,y_{0},z_{0})}}{\mathcal {E}(y_{0},z_{0})}f(x,y_{0},z_{0}) \\&\quad + \sum _{y\in \mathcal {V},\ y\ne y_{0}}\frac{\sqrt{\mathcal {R}(x_{0},y_{0},z_{0})\mathcal {R}(x_{0},y,z_{0})}}{\mathcal {E}(z_{0},x_{0})}f(x_{0},y,z_{0}) \end{aligned}$$

for all $f\in \mathcal {C}^{c}(\widehat{\mathcal {V}})$.

Proposition 1

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. Then, ${\mathcal {L}}_{2,\mathrm{sym}}$ is unitarily equivalent to

$$\begin{aligned} \mathcal {A}_{\widehat{\mathcal {G}}}+\mathcal {Q}(V) \end{aligned}$$

where

$$\begin{aligned} V(x_{0},y_{0},z_{0})=\displaystyle \frac{\mathcal {R}(x_{0},y_{0},z_{0})}{\mathcal {R}(x_{0},y_{0})}+\frac{\mathcal {R}(x_{0},y_{0},z_{0})}{\mathcal {E}(y_{0},z_{0})}+\frac{\mathcal {R}(x_{0},y_{0},z_{0})}{\mathcal {E}(z_{0},x_{0})} \end{aligned}$$

and $\mathcal {Q}(V)$ be the operator of multiplication by V.

Proof

Set $U:\ell ^{2}_{\mathrm{sym}}(\mathcal {R})\longrightarrow \ell ^{2}(\widehat{\mathcal {V}})$ as the operator given by

$$\begin{aligned} U(f)(x,y,z)=\sqrt{\mathcal {R}(x,y,z)}f(x,y,z). \end{aligned}$$

Notice that

$$\begin{aligned} U^{-1}(f)(x,y,z)=U^{*}(f)(x,y,z)=\displaystyle \frac{1}{\sqrt{\mathcal {R}(x,y,z)}}f(x,y,z) \end{aligned}$$

for all $f\in \ell ^{2}(\widehat{\mathcal {V}})$. Notice now that on $\mathcal {C}_{c}(\widehat{\mathcal {V}})$

$$\begin{aligned} U{\mathcal {L}}_{2,\mathrm{sym}}U^{-1}(f)(x_{0},y_{0},z_{0})&=\displaystyle \sum _{t\in \mathcal {V}}\frac{\sqrt{\mathcal {R}(x_{0},y_{0},z_{0}) \mathcal {R}(x_{0},y_{0},t)}}{\mathcal {E}(x_{0},y_{0})}f(x_{0},y_{0},t)\\&\quad +\sum _{t\in \mathcal {V}}\frac{\sqrt{\mathcal {R}(x_{0},y_{0},z_{0}) \mathcal {R}(y_{0},z_{0},t)}}{\mathcal {E}(y_{0},z_{0})}f(y_{0},z_{0},t)\\&\quad + \sum _{t\in \mathcal {V}}\frac{\sqrt{\mathcal {R}(x_{0},y_{0},z_{0}) \mathcal {R}(z_{0},x_{0},t)}}{\mathcal {E}(z_{0},x_{0})}f(z_{0},x_{0},t). \end{aligned}$$

Using Remark 2, we obtain the result. $\square$

Corollary 2

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted tri-partite triangulation. If V is bounded on $\mathcal {F}$, then

$$\begin{aligned} \mathcal {A}_{\widehat{\mathcal {G}}}\ge \sup _{(x, y,z)\in \mathcal {F}}V(x,y,z). \end{aligned}$$

In particular, $\mathcal {A}_{\widehat{\mathcal {G}}}\ge -3$ when $\mathcal {T}$ is simple.

Proof

Since ${\mathcal {L}}_{2,\mathrm{sym}}$ is nonnegative operator, then

$$\begin{aligned} \langle \mathcal {A}_{\widehat{\mathcal {G}}}f,f\rangle \ge -3\left\| f\right\| ^{2} \end{aligned}$$

for all $f\in \mathcal {C}^{c}(\widehat{\mathcal {V}})$. $\square$

Corollary 3

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a tri-partite weighted triangulation. Set

$$\begin{aligned} \mathcal {N}(x,y,z)=1+\displaystyle \sum _{r\in \mathcal {V}}\Big (d_{\mathcal {E}}(x,y)+d_{\mathcal {E}}(y,z)+d_{\mathcal {E}}(z,x)\Big ). \end{aligned}$$

Suppose that

$$\begin{aligned} \sup _{x\sim y,z\in \mathcal {F}_{(x,y)}}\displaystyle \sum _{r\in \mathcal {F}_{(x,y)}}\frac{1}{\mathcal {E}(x,y)}\mathcal {R}(x,y,r)\mid \mathcal {N}(x,y,r)-\mathcal {N}(x,y,z)\mid ^{2}<\infty . \end{aligned}$$

and V is bounded. Then, $\mathcal {A}_{\widehat{\mathcal {G}}}$ is essentially self-adjoint on $\mathcal {C}^{c}(\widehat{\mathcal {V}})$.

Proof

Combine Proposition 1, Theorems 2 and 1. $\square$

7.3 Geometric Hypothesis

We recall the two following definitions:

Definition 3

[1, Definition 8] The graph $\mathcal {G}:=(\mathcal {V}, m, \mathcal {E})$ is $\chi$-complete if there exists an increasing sequence of finite set $(\mathcal {V}_{n})_{n}$ such that $\mathcal {V}=\cup _{n}\mathcal {V}_{n}$ and there exist related functions $\chi _{n}$ satisfying the following three conditions:

• (1) $\chi _{n}\in \mathcal {C}^{c}(\mathcal {V}),~0\le \chi _{n} \le 1$,

• (2) $\chi _{n}(x)=1$ if $x\in \mathcal {V}_{n},$

• (3) $\exists C>0,~\forall n\in \mathbb {N},~x\in \mathcal {V}$,

  $$\begin{aligned}\displaystyle \frac{1}{m(x)}\sum _{y\in \mathcal {V}}\mathcal {E}(x,y) |\chi _{n}(x)-\chi _{n}(y)|^{2}\le C. \end{aligned}$$

Definition 4

[7, Definition 4.2] A weighted triangulation $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ is $\chi$-complete, if

• (1) $\mathcal {G}=(\mathcal {V},m,\mathcal {E})$ is $\chi$-complete.

• (2) $\exists M>0$, $\forall n\in \mathbb {N},\ (x,y)\in \mathcal {E}$

  $$\begin{aligned} \displaystyle \frac{1}{\mathcal {E}(x,y)}\sum _{t\in \mathcal {F}_{(x,y)}}\mathcal {R}(x,y,t)|2\chi _{n}(t)-\chi _{n}(x)-\chi _{n}(y)|^{2}\le M. \end{aligned}$$

We recall the criterion obtained in [7].

Theorem 4

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a $\chi$-complete weighted triangulation then $\mathcal {L}_{2,\mathrm{skew}}$ is essentially self-adjoint on $\mathcal {C}_{\mathrm{skew}}^{c}(\mathcal {R})$.

Corollary 4

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a $\chi$-complete weighted triangulation. If $\mathcal {T}$ is tri-partite then $\mathcal {A}_{\widehat{\mathcal {G}}}$ is essentially self-adjoint on $\mathcal {C}^{c}(\widehat{\mathcal {V}})$.

Proof

Combine Proposition 1, Theorems 1 and 4. $\square$

7.4 Book-Like Triangulation

We recall the definition of one-dimensional decomposition given in [6] for the case of graphs.

Definition 5

[6] A one-dimensional decomposition of the graph $\mathcal {G}=(\mathcal {V},m,\mathcal {E})$ is a family of finite sets $(S_{n})_{n\in \mathbb {N}}$ which forms a partitions of $\mathcal {V}$, that is $\mathcal {V}=\cup _{n\in \mathbb {N}}S_{n}$, and such that for all $x\in S_{n}$, $y\in S_{m}$,

$$\begin{aligned} \mathcal {E}(x,y)>0\Longrightarrow |n-m|\le 1. \end{aligned}$$

The following definition is introduced in [7].

Definition 6

Let $\mathcal {T}:=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation and $(S_{n})_{n\in \mathbb {N}}$ a one-dimensional decomposition of the graph $\mathcal {G}=(\mathcal {V},m,\mathcal {E})$. We say that $\mathcal {T}$ is a book-like triangulation if

• (1) $\sharp S_{0}=1,\ \sharp S_{2n+1}=2$ and $\sharp (S_{2n+1}^{2}\cap \mathcal {E})=1$, for all $n\in \mathbb {N}$.

• (2) $x,\ y\in S_{2n+2}\Longrightarrow \mathcal {E}(x,y)=0$,

• (3) $\forall x\in S_{2n+1}$, $\mathcal {N}_{\mathcal {G}}(x)=S_{2n}\cup S_{2n+2}.$

We recall [7, Proposition 6.5]:

Proposition 2

Let $\mathcal {T}$ be a simple book-like triangulation. Assume that

$$\begin{aligned} n\longmapsto \displaystyle \frac{\sharp S_{2n}}{\sharp S_{2(n+1)}}\in \ell ^{1}(\mathbb {N}). \end{aligned}$$

(3)

Then, $\mathcal {L}_{2,\mathrm{skew}}$ is not essentially self-adjoint on $\mathcal {C}^{c}_{\mathrm{skew}}(\mathcal {R})$.

Corollary 5

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a simple book-like triangulation satisfying (3). Then, $\mathcal {A}_{\widehat{\mathcal {G}}}$ is not essentially self-adjoint on $\mathcal {C}^{c}(\widehat{\mathcal {V}})$.

Proof

Set $S_{0}=\{x_{0}^{0}\}$, $S_{2n+1}=\{x_{n}^{0},x_{n}^{1}\}$, $\mathcal {V}_{0}=\cup _{n}S_{2n}$, $\mathcal {V}_{1}=\cup _{n}\{x_{n}^{0}:\ n\in \mathbb {N}\}$ and $\mathcal {V}_{2}=\cup _{n}\{x_{n}^{1}:\ n\in \mathbb {N}\}$. Then,

$$\begin{aligned} \mathcal {V}=\mathcal {V}_{0}\cup \mathcal {V}_{1}\cup \mathcal {V}_{2} \text{ and } \mathcal {E}\cap (\mathcal {V}_{0}^{2}\cup \mathcal {V}_{1}^{2}\cup \mathcal {V}_{2}^{2})=\emptyset . \end{aligned}$$

So, $\mathcal {T}$ is tri-partite. Using Theorem 1, Propositions 1 and 2, the result holds. $\square$

7.5 Triangular Anti-tree

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. The sphere of radius $n\in \mathbb {N}$ around a vertex $v\in \mathcal {V}$ is the set

$$\begin{aligned} \mathcal {S}_{n}(v):=\Big \{w\in \mathcal {V}:\ d_{\mathcal {V}}(v,w)=n\Big \}. \end{aligned}$$

We recall that $\mathcal {C}_{n}$ denotes the n-cycle graph, i.e., $\mathcal {V}=\mathbb {Z}/n\mathbb {Z}$, where $\mathcal {E}(x,y)>0$ if and only if $|x-y|=1$. Let $\mathcal {G}(\mathcal {S}_{n}(o))=(\mathcal {S}_{n}(v),\mathcal {E}')$ where $\mathcal {E}'=\displaystyle \mathcal {E}\mid _{\mathcal {S}_{n}(v)\times \mathcal {S}_{n}(v)}$.

Definition 7

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a weighted triangulation. We say that $\mathcal {T}$ is anti-tree if there exists a vertex $o\in \mathcal {V}$ such that

• (1) For all $n\in \mathbb {N}^{*}$ and $v\in \mathcal {S}_{n}(o)$, we have

  $$\begin{aligned} \mathcal {N}_{\mathcal {G}}(v)\backslash \mathcal {S}_{n}(o)=\mathcal {S}_{n-1}(o)\cup \mathcal {S}_{n+1}(o). \end{aligned}$$

• (2) For all $n\in \mathbb {N}^{*},\mathcal {G}(\mathcal {S}_{n}(o))\simeq \mathcal {C}_{\sharp \mathcal {S}_{n}(o)}$.

Theorem 5

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a simple triangular anti-tree whose root in o. Set $s_{n}=\sharp \mathcal {S}_{n}(o)$ and assume that

$$\begin{aligned} n\longmapsto \displaystyle \frac{s_{n}^{2}}{s_{n+2}}\in \ell ^{1}(\mathbb {N}). \end{aligned}$$

(4)

Then, $\mathcal {L}_{2,\mathrm{sym}}$ does not essentially self-adjoint on $\mathcal {C}_{\mathrm{sym}}^{c}(\mathcal {R})$.

Proof

Set $f\in \ell _{\mathrm{sym}}^{2}(\mathcal {F})\backslash \{0\}$ such that $f\in \ker (\mathcal {L}_{2,\mathrm{sym}}^{*}+i)$ and such that f is constant on $\mathcal {S}_{n}\times \mathcal {S}_{n+1}^{2}\cup \mathcal {S}_{n}^{2}\times \mathcal {S}_{n+1}$, $n\in \mathbb {N}^{*}$. We denote the constant value by $C_{n}$. It takes the value 0 on $\mathcal {S}_{n}^{2}$. We have the following equation:

$$\begin{aligned} (s_{n}+4+i)C_{n}+s_{n+2}C_{n+1}=0. \end{aligned}$$

Therefore,

$$\begin{aligned} \displaystyle \Vert f|_{\mathcal {S}_{n+1}\times \mathcal {S}_{n+2}^{2}}\Vert ^{2}&=\Vert f|_{\mathcal {S}_{n+1}^{2}\times \mathcal {S}_{n+2}}\Vert ^{2}\\&=\displaystyle \frac{1}{2}s_{n+2}s_{n+1}|C_{n+1}|^{2}\\&\le \displaystyle \frac{|s_{n}+4+i|^{2}}{s_{n}s_{n+2}}\Vert f|_{\mathcal {S}_{n}\times \mathcal {S}_{n+1}^{2}}\Vert ^{2}. \end{aligned}$$

Since $\displaystyle \lim _{n\rightarrow \infty }\frac{s_{n}^{2}}{s_{n+2}}=0$, we get by induction:

$$\begin{aligned} M:=\displaystyle \sup _{n\in \mathbb {N}^{*}}\Vert f|_{\mathcal {S}_{n}\times \mathcal {S}_{n+1}^{2}}\Vert ^{2}<\infty . \end{aligned}$$

Then, we have

$$\begin{aligned} \displaystyle \Vert f|_{\mathcal {S}_{n+1}\times \mathcal {S}_{n+2}^{2}}\Vert ^{2}\le M\frac{|i+4+s_{n}|}{s_{n}s_{n+2}}. \end{aligned}$$

From Eq. (4), we infer that $f\in \ell _{\mathrm{sym}}^{2}(\mathcal {F})$. Using [25, Theorem X.36], we conclude that $\mathcal {L}_{2,\mathrm{sym}}$ is not essentially self-adjoint. $\square$

Corollary 6

Let $\mathcal {T}=(\mathcal {V},m,\mathcal {E},\mathcal {R})$ be a simple triangular anti-tree satisfying (4). Then, $\mathcal {A}_{\widehat{\mathcal {G}}}$ is not essentially self-adjoint on $\mathcal {C}^{c}(\widehat{\mathcal {V}})$.