THE DISCRETE MAGNETIC LAPLACIAN ACTING on PSEUDO-COCHAINS Nassim Athmouni, Hatem Baloudi, Mondher Damak, Marwa Ennaceur

THE DISCRETE MAGNETIC LAPLACIAN ACTING ON PSEUDO-COCHAINS Nassim Athmouni, Hatem Baloudi, Mondher Damak, Marwa Ennaceur

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Nassim Athmouni, Hatem Baloudi, Mondher Damak, Marwa Ennaceur. THE DISCRETE MAG- NETIC LAPLACIAN ACTING ON PSEUDO-COCHAINS. 2019. hal-02064271v2

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NASSIM ATHMOUNI, HATEM BALOUDI, MONDHER DAMAK, AND MARWA ENNACEUR

Abstract. We consider the notion of χ−completness of the locally ﬁnite graph and we approach this geometric hypothesis for the weighted magnetic graph. Moreover, we establish a link between the magnetic adjacency matrix on line graph and the magnetic discrete Laplacian on 1-forms.

Contents 1. Introduction 1 2. Generalities about graphs 2 3. The discrete magnetic Laplacian 3 4. Essential self-adjointness 5 4.1. Geometric Hypothesis 5 4.2. Case of the class of anti-tree 8 4.3. Case of increasing arithmetic graph 9 4.4. Some remarks on the Stieltjes and Nelson criterium 11 4.5. Some remarks on the spectrum 11 5. Application to the magnetic adjacency matrix 12 References 17

1. Introduction Spectral graph theory represents an active area of research. In the past few year, the question of essential self-adjointness of the discrete magnetic Laplacian on magnetic graph has attracted a lot of interests. For the reference to the literature on this topic, see [17, 26, 31, 27] and reference therein. In [1], in the case of non-magnetic graph, the authors have addressed this question using the notion of the χ completeness. This notion is related to [10, 11] and to the one of intrinsic metric, e.g.g, [1, 21]. In particular,− they show that the Hodge Laplacian is essentially self-adjoint, when the graph is χ complete. This method was more recently adapted in the case of weighted triangulation in [6]. In the present− study, the graphs are considered as one-dimensional simplicial complexes. If the graph contains cycles of length 3, we can consider it as two dimensional simplicial complex, see [13, 16].

2010 Mathematics Subject Classiﬁcation. 05C12, 05C63, 47B25, 81Q35. Key words and phrases. spectral graph theory, inﬁnite graphs, line graph, adjacency matrix. 1 2 NASSIM ATHMOUNI, HATEM BALOUDI, MONDHER DAMAK, AND MARWA ENNACEUR

The current study has two major aims. The first aim is to consider the notion of χ completeness and we approach this geometric hypothesis for the weighted magnetic graph. More precisely,− we introduce the notion of χθ completeness where θ stands for the notion for the magnetic potential acting on edges. We start with showing− that this notion covers many situations, we turn to see Theorem 4.4. In particular, in the case of a simple increasing arithmetic graph, we refer to Section 4.3 for a precise definition, we prove 1 that is χ complete. In contrast, if we define the magnetic potential to be θx,y := , where G − d(x) d(y) − d(.) is the degree of vertices we refer to Section 2 for the definition, we have is not χθ complete. Then, we study the question of essential self-adjointness of the discrete magneticG Laplacian− via the notion of χθ completeness. More precisely, we prove that the discrete magnetic Laplacian is essentially self-adjoint − when the magnetic graph is χθ complete. The technique of the proof is inspired from [1]. At the end of this part, we deal with the question− of essentially self-adjointness via Nelson’ Lemma and quasi-analytic vectors approach. These two techniques are explained in [4] in the case of non magnetic graphs. Our remark is that these two criteria remain applicable for a magnetic graphs and are independent of the magnetic factor on the graph.

The second aim of this paper is to study the magnetic adjacency matrix acting on graph. First, we define the difference magnetic operators dθ acting on 0-forms. We denote by δθ it formal adjoint. This permits to define two different types of discrete operators associated to a locally finite magnetic graph: ∆0,θ = δθdθ and ∆1,θ = dθδθ. In the case of non-magnetic graphs (i.e., θ = 0), the expression of ∆1,0 is different from that obtained in [4]. Indeed, the discrete operator obtained in [4] was built by choosing a skew-symmetric statistic, namely fermionic one. Second, we study the link between the magnetic ad- θ skew jacency matrix acting on line graph and the operator ∆ = ∆ ,θ where (∆ ,θ) A 1,θ 1 |D(∆1,θ)∩Cskew(E) D 1 denotes the domain of ∆1,θ and skew( ) denotes the set of 1 cochains corresponds to fermionic statis- C E − skew tics. In the case of bi-partite graph, by analysing the structure of ∆1,θ , we prove the existence of a multiplication operator ( ) such that θ + ( ) and ∆skew are unitarily equivalent. Q R A Q R 1,θ The paper is structured as follows: The next Section is devoted to some definitions and notations for graph theory. In the third section, we define two discrete magnetic Laplacians operators ∆0,θ associated to vertices and ∆1,θ associated to edges. In the fourth section, we discuss the notion of χθ completeness and we give some examples. In the last section, we establish a link between the magnetic− adjacency matrix acting on vertices on line graph and the discrete magnetic Laplacian action on edges. Acknowledgment. The authors thank Echi Nadhem for the useful discussions and comments on the paper.

2. Generalities about graphs We start with some deﬁnitions and ﬁx our notations for graphs. We refer to [7, 29, 22] for surveys on the matter. A weighted graph is a triple ( , ,m) where is a set at most countable (vertices), : [0, + ) symmetric (Gedges) and mV: E (0, + ).V We say that two vertices x, y are EneighborsV × V −→if (x, y)∞= 0. In this case, we write x Vy. −→ We say that∞ there is a loop in x , if x ∈Vx. The set of neighborsE of x6 is denoted by ∼ ∈V ∼ ∈V (x) := y : x y . NG { ∈V ∼ } THE DISCRETE LAPLACIAN ASSOCIATED TO EDGES WITH APPLICATION TO MAGNETIC ADJACENCY MATRIX3

A graph is locally finite if ♯ (x) for all x . The weighted degree of vertices is given by NG ∈V 1 d (x) := (x, y). V m(x) E yX∈V A graph is connected, if for all x, y , there exists an x y path, i.e., there is a finite set G ∈V − − n (x , ..., xn) such that x = x, xn = y and (xi, xi ) > 0 0 ∈V 0 E +1 for all i 0, ..., n 1 . The minimal possible n is denoted by ρ (x, y) and called the distance between ∈{ − } V x and y. If no edges appear more than once in (x1, ..., xn), the path is called a simple path. The path is called a cycle or closed when the origin and the end are identical, i.e., x1 = xn. An n cycle is a cycle with n vertices. Finally, as we are dealing with magnetic fields, we fix the magnetic potential:−

θ : [ π, π], such that θ(x, y)= θ(x, y). V × V −→ − − We denote θ(x, y) by θx,y. We say that := ( , ,m,θ) is an unoriented weighted magnetic graph. A G V E magnetic graph is simple if m = 1, 0, 1 and θ = 0. When is simple, we have dV (x) = ♯ (x) for all x . G E∈{ } G N ∈V In the sequel, we assume that all graphs are locally ﬁnite, connected and have no loop.

3. The discrete magnetic Laplacian In this section, we will always consider a weighted magnetic graph = ( , ,m,θ). We define the set of 0 cochains (or 0-forms) G V E − ( ) := f : C . C V n V −→ o 2 We denote by c( ) the 0 cochains with finite support in . Let us define the Hilbert space ℓ ( ) as the sets of 0 cochainsC V with− finite norm, i.e., V V − ℓ2( ) := f ( ): f 2 := m(x) f(x) 2 < . V ∈C V k k | | ∞ n xX∈V o The associated scalar product is given by f,g := m(x)f(x)g(x) h i xX∈V for all f,g ℓ2( ). Concerning the set of edges, we define: ∈ V ( ) := f : C . C E n V × V −→ o Set

sym( ) := f : C : f(x, y)= f(y, x) C E n V × V −→ o and

skew( ) := f : C : f(x, y)= f(y, x) . C E n V × V −→ − o 4 NASSIM ATHMOUNI, HATEM BALOUDI, MONDHER DAMAK, AND MARWA ENNACEUR where ( ) corresponds to fermionic statistics and ( ) corresponds to bosanic statistics. The set Cskew E Csym E of functions in ( ) with ﬁnite support is denoted by c( ). We turn to the Hilbert structure on the set of edges: C E C E 1 ℓ2( ) := f ( ): (x, y) f(x, y) 2 < . E ∈C E 2 E | | ∞ n x,yX∈V o The associated scalar product is given by 1 f,g := (x, y)f(x, y)g(x, y) h i 2 E x,yX∈V for all f,g ℓ2( ). The diﬀerence magnetic operator is the operator ∈ E dθ : c( ) c( ) C V −→ C E given by

iθx,y dθf(x, y) := f(y) e f(x). − The magnetic coboundary operator is the formal adjoint of dθ. It is the operator

δθ : c( ) c( ) C E −→ C V given by

1 iθy,x δθg(x) := (x, y)(g(y, x) e g(x, y)) 2m(x) E − Xy for all g c( ) and x . Thus it satisﬁes ∈C E ∈V dθf,g = f,δθg h i h i 2 2 for all f c( ) and g c( ). The operators δθ and dδ are closable. Indeed, δθ : ℓ ( ) ℓ ( ) (resp. 2 ∈C V 2 ∈C E E −→ V dθ : ℓ ( ) ℓ ( )) is with dense domain then δθ ( resp. d) is closable. The magneticV −→ discreteE Laplacian operator acting on 0 form is given by − 1 iθy,x ∆ ,θ(f)(x) := δθdθ(f)(x)= (x, y)(f(x) e f(y)), 0 m(x) E − Xy for all f c( ) and x . The case ∆ , is so-called physical Laplacian and ∆ ,π is the the singless ∈ C V ∈ V 0 0 0 Laplacian, see [12]. If is bi-partite, i.e, is a graph whose vertices can be divided into disjoint sets 1 G 2 2 V and such that = and (x, y) = 0 for all (x, y) , then ∆ , and ∆ ,π are unitarily V2 V V1 ∪V2 E ∈ V1 ∪V2 0 0 0 equivalent, see [4]. If is simple, the discrete Laplacian ∆0,0 is essentially self-adjoint on c( ), see [35]. The magnetic discreteG Laplacian operator acting on c( ) is given by C V C E ∆1,θ(f)(x, y) := dθδθ(f)(x, y) 1 = (z,y)(f(z,y) eiθz,y f(y,z)) 2m(y) E − Xz eiθxy (x, z)(f(z, x) eiθz,x f(x, z)), − 2m(x) E − Xz for all f c( ) and x y. Both operators are symmetric and thus closable. We denote the closure by ∈C E ∼ ∗ the same symbol ∆ ,θ (resp. ∆ ,θ, its domain by (∆ ,θ) (resp. (∆ ,θ), and its adjoint by ∆ (resp. 0 1 D 0 D 0 0,θ THE DISCRETE LAPLACIAN ASSOCIATED TO EDGES WITH APPLICATION TO MAGNETIC ADJACENCY MATRIX5

∗ ∆1,θ). In this paper, the graphs are considered as one-dimensional simplicial complexes. If the graph contains cycles of length 3, we can consider it as two dimensional simplicial complex. In this case, the expression of ∆1,0 is different because contains a part acting on the triangular faces, see [3, 6, 24]. The magnetic Gauβ-operator is defined on c( ) c( ) by: C V ⊕C E 0 δθ Dθ := dθ + δθ = . ∼ dθ 0 This operator is of Dirac type and is motived by the Hodge Laplacian: 2 ∆θ, := ∆θ := (dθ + δθ) ∆ ,θ ∆ ,θ. HL ≃ 0 ⊕ 1 We refer to [1] for the case of non magnetic field. Notation: We denote by N the non-negative integer. In particular, 0 N. For x , we set d(x) := ♯ (x) and by N∗ the one of positive integers. ∈ ∈ V NG 4. Essential self-adjointness In this section, we discuss the question of essential self-adjointness via the notion of χ completeness which was introduced and investigated by Torki in [1]. We approach this geometric notion− and we discuss some examples. 4.1. Geometric Hypothesis. We first recall the notion of χ completeness in the case of non-magnetic graph obtained in [1]: − Definition 4.1. The graph := ( , ,m) is χ complete if there exists an increasing sequence of finite G V E − set n such that = n n and there exist related functions χn satisfying: S c V ∪ S 1) χn ( ), 0 χn 1, ∈C V ≤ ≤ 2) χn(x)=1 for all x n. 3) M > 0, n N, x∈ S , such that ∃ ∀ ∈ ∈V 1 2 (x, y) dχn(x, y) M. m(x) E | | ≤ Xy In [1], in the case of non-magnetic graph, the authors schow that the two discrete Laplacian operators are essentially self-adjoint, when the graph is χ complete. In the case of magnetic graph, we do no − know that if the χ completeness hypothesis implis that ∆0,θ and ∆1,θ are essentially self-adjoint. We introduce the following− definition:

Deﬁnition 4.2. Let := ( , ,m,θ) be a magnetic weighted graph and let ( n)n be an increasing G V E O sequence of ﬁnite sets such that = n n and there exists (ηn)n and φn with V ∪ O i) ηn c( ), 0 ηn 1 and ηn(x)=1 for all x n, ∈C V ≤ ≤ ∈ O ii) φn ( ) such that φn converge to 0, ∈C V iφ Let χn = ηne n . We say that is χθ complete if the exist is a non-negative c such that for all x and n N we have G − ∈ V ∈ 1 2 (1) (x, y) dθχn(x, y) < c. m(x) E | | Xy

Remark 4.3. Let := ( , ,m,θ) be a weighted magnetic graph. If is χθ complete then G V E G − 1 θ (2) (x, y) sin xy 2< c. m(x) E | 2 | Xy 6 NASSIM ATHMOUNI, HATEM BALOUDI, MONDHER DAMAK, AND MARWA ENNACEUR

Now, we ask the following question: Is there a graph that must χ complete if it is without magnetic − potential and not χθ complete if with nonzero magnetic potential? We recall that the graph is assumed connected. For x , x− and n N, we set 0 ∈V ∈

Bn(x ) := y : ρ (x ,y) n , x := ρ (x , x) and d := d . 0 { ∈V V 0 ≤ } | | V 0 Bn(x0) V |Bn(x0) Theorem 4.4. Let := ( , ,m,θ) be a magnetic graph with the phase G V E 1 , if d(x) = d(y), d2(x) d2(y) 6 θx,y =  −  0, otherwise.  Let x . Suppose that 0 ∈V

(x, y) 1 sup E < and d = O(n 4 ). m(x)d(x)d(y) ∞ Bn(x0) x∈V Xy

Then is χθ complete. G − Proof. Let n N∗, x and set ∈ ∈V

x φ (x) := | | . n n

We note that n4 + 1 satisﬁes the following equation

(3) nX2 + n2X + n9 n6 +2n5 n2 + n =0. − − −

1 2 Let J(x)= (x, y) dθχn,θ(x, y) . m(x) E | | Xy The ﬁst case: If x Bn(x ), we take into account the equation (3), then for some c, C> 0 independent ∈ 0 THE DISCRETE LAPLACIAN ASSOCIATED TO EDGES WITH APPLICATION TO MAGNETIC ADJACENCY MATRIX7 from x and n, we obtain

4 φn y φn(x)+ θx,y J(x) (x, y) sin ( ) − 2 ≤ m(x) E | 2 | y∈BXn(x0) 2 n8 n5 +2n4 + n2 +1 2 + (x, y) − m(x) E (n + 1)(n8 n5 +2n4 n + 1) |y|X=n+1 − − 9 6 5 2 8 n n +2n 2n φn(y) φn(x)+ θy,x 2 + (x, y) − − sin( − ) m(x) E (n + 1)(n8 n5 +2n4 n + 1) 2 |y|X=n+1 − − c 1 1 (x, y) + ≤ m(x) E n2 d(x)d(y) y∈BXn+1(x0) C. ≤ The second case: x = n4 + 1, we take into account the equation (3) then, for some C > 0 independent from x and n, we| obtain|

2n4 n +1 J(x)= (x, y) − E n11 n8 +2n7 n4 + n3 |yX|=n4 − − C (x, y) E . ≤ m(x) n7 y∈BXn4 (x)

The thirst case: If x B 4 Bn, then ∈ n \

c φn(y) φn(x)+ θy,x J(x) (x, y) sin( − ) 2 ≤ m(x) E | 2 | |yX|=|x|

c φn(y) φn(x)+ θy,x + (x, y) sin( − ) 2 m(x) E | 2 | |y|=X|x|−1 c (n8 n5 +2n4 n +1)+ n x ( x 1) + (x, y) − − | | | |− 2 m(x) E | x ( x 1)(n8 n5 +2n4 n + 1) | |y|=X|x|−1 | | | |− − − c φn(y) φn(x)+ θy,x + (x, y) sin( − ) 2 m(x) E | 2 | |y|=X|x|+1 c (n8 n5 +2n4 n +1)+ n x ( x + 1) + (x, y) − − | | | | 2 m(x) E | x ( x + 1)(n8 n5 +2n4 n + 1) | |y|=X|x|+1 | | | | − − C 1 1 (x, y) 2 + . ≤ m(x) E n d(x)d(y) y∈XBn4+1 the fourth case: If x >n4 + 1, we take into account the equation (3), we obtain J(x) = 0. | | 8 NASSIM ATHMOUNI, HATEM BALOUDI, MONDHER DAMAK, AND MARWA ENNACEUR

Remark 4.5. Let := ( , ,m,θ) be a magnetic graph with the phase G V E 1 , if d(x) = d(y), d2p(x) d2p(y) 6 θx,y =  −  0, otherwise.  By repeating the proof of Theorem 4.4, we obtain that if

(x, y) 1 sup E < and d = O(n 4 ), m(x)dp(x)dp(y) ∞ Bn(x0) x∈V Xy then is χθ complete. G − Example 4.6. Let α > 1. Let := ( , ,m,θ) be a magnetic graph with the phase given in Theorem 4.4 and G V E m(x)m(y)d(x)d(y) (x, y) := E (m(x)+ m(y))( x + y )α | | | | 1 and d(x) n 4 for all x Bn. Then, is χθ complete. ≤ ∈ G − We construct an example of graph that is χ complete and not χθ complete. − − 4.2. Case of the class of anti-tree. Let x and set 0 ∈V n(x ) := y : ρ (x ,y ) n S 0 { ∈V V 0 0 ≤ } the sphere of radius n N around x . ∈ 0 Deﬁnition 4.7. Let := ( , ,m,θ) be a magnetic graph. We say that is anti-tree if there is x0 such that G V E G ∈V

(x)= n (x ) n (x ) NG S −1 0 ∪ S +1 0 ∗ for all n N and x n(x ). ∈ ∈ S 0

S2 S0

An anti-tree with sphere , , of cardinal 1, 17, 4. S0 S1 S2 Let := ( , ,m,θ) be a magnetic anti-tree graph with m =1, 0, 1 and G V E E∈{ } x y θ := | | − | |π. x,y 2 THE DISCRETE LAPLACIAN ASSOCIATED TO EDGES WITH APPLICATION TO MAGNETIC ADJACENCY MATRIX9

If d(.) is unbounded, then is not χθ complete. Indeed, G − θ d2(x) sin2( x,y )= , 2 2 yX∼x for all x . Using Remark 4.3, we have is not χθ complete. In contrast, In [4], the authors prove that if ∈ V G − 1 =+ , ♯ n + ♯ n ∞ nX∈N S S +1 p then is χ complete. G − 4.3. Case of increasing arithmetic graph. We now strengthen the previous example and follow the ideas of [5].

Definition 4.8. A 1-dimensional decomposition of the graph is a family of finite sets (Sn)n which G ≥0 forms a partition of , that is = n Sn, and such that for all x Sn,y Sm, V V ⊔ ≥0 ∈ ∈ x y = n m 1. ∼ ⇒ | − |≤ We now introduce an example of 1 dimensional decomposition: − Definition 4.9. Let := ( , ,m) be a weighted locally finite graph and let (Sn)n be a 1 dimensional decomposition of the graphG V.E We say that is increasing arithmetic graph if − G G 1) x, y Sn = (x, y)=0 ∈ ⇒E 2) There are two integer a, k N such that d(x)= a + nk for all x Sn. ∈ ∈ Let := ( , ,m,θ) be a magnetic anti-tree graph with m =1, 0, 1 and G V E E∈{ } 1 θ := . x,y d(x) d(y) − If d(.) is unbounded then is not χθ complete. Indeed, G − θ 1 sin2( x,y )= d(x) sin2 . 2 2k yX∼x for all x . ∈V Remark 4.10. Let be a simple increasing arithmetic graph. Then G ∞ 1 ∞ 1 = = , + − √ ∞ nX=1 an + an nX=1 a + nk + p − when an = supx∈Sn ♯ G(x)) n+1 and an = supx∈Sn ♯ G(x)) n−1. Using [4, Theorem 3.20], we infer that is χ complete.N ∩ S N ∩ S G −

For f,h c( ) and g ( ), we set: ∈C V ∈C E (f+g)(x, y)= f(y)g(x, y), (h+f−)(x, y)= h(y)f(x), (f−g)(x, y)= f(x)g(x, y). and f(x)+ f(y) f(x, y)= . 2 e 10 NASSIM ATHMOUNI, HATEM BALOUDI, MONDHER DAMAK, AND MARWA ENNACEUR

Remark 4.11. 1) For g c( ) and f c( ), we have ∈C E ∈C V

f(x)δθg(x) 1 iθy,x δθ(fg)(x)= + (x, y)f(y) g(y, x) e g(x, y) . 2 4m(x) E − Xy e 2) For f,g c( ), we have ∈C V iθy,x dθ(fg)(x, y)= f(y)dθ(g)(x, y)+ g(x)dθ(f)(x, y) + (e 1)f(y)g(x). − ∗ ∗ 3) If dθ = δθ and δθ = dθ, then Dδ is essentially self-adjoint.

Theorem 4.12. Let := ( , ,m,θ) be a χθ complete graph, then Dθ is essentially self-adjoint. G V E − Proof. Let f (δ∗). By using Remark 4.3, we have ∈ D 2 2 2 2 (4) (1 χn)f 4 f , (1 χn) dθf 4 dθf , k − k ≤ k k k − + k ≤ k k

2 2 1 2 2 (5) f dθ(1 χn) m(x) f(x) (x, y) dθ(1 χn)(x, y) 2c f k + − k ≤ | | m(x) E | − | ≤ k k xX∈V Xy and

iθ.,. 2 2 θy,x 2 2 2 (6) (e 1)(1 χn) f (x, y) sin 1 χn(y) f(x) c f . k − − − +k ≤ E 2 | − | | | ≤ k k Xx,y On the other hand, we have

iθx,y (7) dθ(f χnf)= f(y)dθ(1 χn)(x, y)+(1 χn)(x)dθf(x, y) + (e 1)f(y)(1 χn)(x). − − − − − Taking into account Eqs(4-7) and the Dominated convergence theorem, we obtain

lim (f χnf) + dθ(f χnf) =0. n k − k k − k ∗ This proves that dθ = δθ . Now, let f (d∗), then ∈ D 2 (1 χn)δθf 2 δθf k − k≤ k k and

1 iθ.,y 2 1 iθy,x 2 (., y)(1 χn)(y)(f(y, .) e y(., y)) (x, y) f(y, x) e f(x, y) k4m(.) E − − k ≤ 4 E | − | Xy xX∈V Xy 2 δθf . ≤k k By the following equality

(1 χn)(x)δθf(x) 1 iθy,x δθ((1 χn)f)(x)= − + (x, y)(1 χn)(y)(f(y, x) e f(x, y)) − 2 4m(x) E − − Xy f we deduce that

lim (f χnf) + δθ(f χnf) =0. n k − k k − k ∗ So, δθ = dθ. Using Remark 4.11, we infer thatf Dθ is essentiallyf self-adjoint THE DISCRETE LAPLACIAN ASSOCIATED TO EDGES WITH APPLICATION TO MAGNETIC ADJACENCY MATRIX11

Inspired from [1, Proposition13] and [30, Theorem VIII], we obtain the following consequence:

Corollary 4.13. Let := ( , ,m,θ) be a weighted magnetic graph. If is χθ complete then ∆ ,θ is G V E G − 0 essentially self-adjoint on c( ) and ∆ ,θ is essentially self-adjoint on c( ). C V 1 C E 4.4. Some remarks on the Stieltjes and Nelson criterium. Let := ( , ,m) be a weighted graph G Vc E and o . In [4], they prove that ∆1,0 Cc (E) is essentially self-adjoint on ( ) when ∈V | skew Cskew E +∞ 1 − 2n (8) sup d (x) = . G ∞ nX=1 iY≤n x∈Bn(o) The technique of the proof is based on the quasi-analytic vectors approach. We can adapt this approach for ∆ ,θ. Indeed, let g c( ), k N, then 1 ∈C E ∈ 1 iθy,x 2 1B2(0)g, ∆1,θ1B2(0)g = (x, y) (1B2 (o)g)(y, x) e (1B2(o)g)(x, y) h k k i 4m(x) | E k − k | xX∈V Xy

1 iθy,x 2 (x, t) (x, y) (1 2 g)(y, x) e (1 2 g)(x, y) ≤ 4m(x) E E | Bk (o) − Bk(o) | xX∈V Xt Xy

1 2 2 (x, t) (x, y) (1B2(o)g)(y, x) + (1B2(o)g)(x, y) ≤ 2m(x) E E | k | | k | xX∈V Xt Xy 2 sup dG (x) 1B2(o)g . ≤ 2 k k k x∈Bk(o)

This proves that if the hypothesis (8) is satisfied, then ∆1,θ is essentially self-adjoint on c( ). We turn to the Nelson criterion, we recall [4, Theorem 5.13]: C E Theorem 4.14. Let := ( , ,m) be a locally finite graph. Set G E V (x, y) := 1 + deg (x)+deg (y). M G G Suppose that 1 sup (x, z) (x, y) (x, z) 2 < . x,y∈V,x∼y m(x) E |M −M | ∞ zX∈V c Then, ∆1,0 Cc (E) is essentially self-adjoint on ( ) and ∆1,0 Cc (E) is essentially self-adjoint on | skew Cskew E | sym c ( ). Csym E This criterion remains applicable for a magnetic graphs. Indeed, the terms f(x, y) in the proof of [4, Theorem 5.13] are replaced by f(y, x) eiθy,x g(x, y). − Remark 4.15. the Stieltjes and Nelson criterium are independent of the magnetic factor on the graph. On the contrary, the χθ depends on the choice of the phase on the edges. 4.5. Some remarks on the spectrum. Let X be a Banach space. We denote by (X) the set of all closed densely defined linear operators on X. For A (X), we write σ(A) (resp. ρ(A))C for the spectrum (resp. the resolvent set ) of A. We denote by (X)∈C the set of all compact operators on X to itself. We define the essential spectrum of the operator AKby

σess(A)= σ(A + K). K∈K\(X) 12 NASSIM ATHMOUNI, HATEM BALOUDI, MONDHER DAMAK, AND MARWA ENNACEUR

In [2], the author studies the relationship between σ(∆ , ) and σ(∆ , c ). More precisely, the author 0 0 1 0 |C (E) proves that the nonzero spectrum of σ(∆0,0) and σ(∆1,0 Cc(E)) is the same when dV (.) is bounded by using the Weyl’s criterion. This result remains true under| the inﬂuence of magnetic potential on the edges of . Indeed, ∆0,θ and ∆1,θ are non-negative operator, then ρ(∆0,θ) and ρ(∆1,θ) are non empty. By usingG [20, Theorem 1.1], we have

σ(∆ ,θ) 0 = σ(∆ ,θ) 0 . 0 \{ } 1 \{ } Going over the same techniques of the proof of [2], we obtain

σ (∆ ,θ) 0 = σ (∆ ,θ) 0 ess 0 \{ } ess 1 \{ } when dV (.) is bounded.

5. Application to the magnetic adjacency matrix The spectral theory of adjacency matrices acting on graphs is useful for the study of some gelling polymers, of some electrical networks, and number theory, e.g., [15, 14, 28]. In this section, we investigate the relationship between the magnetic adjacency matrix and the magnetic discrete laplacian acting on pseudo-cochains. Let := ( , ,m,θ) be a weighted magnetic graph. The magnetic adjacency matrix is given by G V E 1 θ (f)(x) := (x, y)f(y)eiθy,x AG m(x) E Xy with f c( ) and x . It is symmetric and thus closable. We denote it closure by the same symbol. When ∈Cis simple,V 0∈Vis unbounded if and only if it is unbounded from above and if and only if the G AG degree is unbounded see [18]. To achieve this goal, we analyze the structure of ∆1,θ. Note that skew ∆1,θ Cc (E) := (∆ ) | skew Q 1,θ θ i x,y skew c Where is the operator multiplication by the function (x, y) e 2 and ∆ : ( ) Q 7−→ 1,θ Cskew E −→ c ( ) given by Cskew E θy,x i 2 θ skew e i z,y θzy ∆ (f)(x, y) := (z,y)e 2 cos f(z,y) 1,θ m(y) E 2 Xz θx,y i θ e 2 i z,x θzx + (x, z)e 2 cos f(x, z). m(x) E 2 Xz Concerning the symmetric choice, we set: + 2 2 d : c( ) ℓ ( ) c( ) ℓ ( ) θ C V ⊂ V −→ C E ⊂ E + iθx,y dθ (f)(x, y) := f(y)+ e f(x). + The formal adjoint of dθ is given by + 2 2 δ : c( ) ℓ ( ) c( ) ℓ ( ) θ C E ⊂ E −→ C V ⊂ V 1 δ+(f)(x) := (x, y)(f(y, x)+ eiθy,x f(x, y)). θ 2m(x) E Xy We start with new and direct remarks: THE DISCRETE LAPLACIAN ASSOCIATED TO EDGES WITH APPLICATION TO MAGNETIC ADJACENCY MATRIX13

Remark 5.1. The geometric hypothesis given in Section 4.1 can be adapted in the context of d+. Indeed, we replace (ii) in the Deﬁnition 4.2 by φn converge to π and we replace Eq.(2) by

1 iθx,y 2 (x, y) χn(y)+ e χn(x) < c. m(x) E | | Xy In this case, we say that is χ+ complete. G θ − Remark 5.2. If is χ+ complete, then there is c> 0 such that G θ − 1 θ (x, y)cos2 x,y

1, if (x, y) , ∈V1 ×V2 S(x, y) :=   1, if (x, y) . − ∈V2 ×V1  Let be the following mapping from ℓ2 ( ) into ℓ2 ( ) given by: H sym E skew E

(f)(x, y) := S(x, y)f(x, y). H Then,

f,g = f, g , (g)= g and (f)= f hF i h H i FH HF for all f ℓ2 ( ) and g ℓ2 ( ). So, we have ∈ sym E ∈ skew E (f)= −1(f)= ∗(f) H F F For all f ℓ2 ( ). Therefore, ∈ sym E

θy,x i 2 θ skew −1 e i z,y θz,y ∆ (f)(x, y)= S(x, y) (z,y)S(z,y)e 2 cos f(z,y) F 1,θ F m(y) E 2 Xz θx,y i θ e 2 i z,x θz,x + S(x, y) (x, z)S(x, z)e 2 cos f(x, z) m(x) E 2 Xz

Moreover, noting that

S(x, y)S(z,y)= S(x, y)S(x, t)=1 when (z,y) > 0 and (x, t) > 0. We get ∆skew −1 = ∆sym. E E F 1,θ F 1,θ Deﬁnition 5.5. Let := ( , ,m,θ) be a weighted magnetic graph with the phase θ : ] π, π[. Set G V E E −→ − V = / , where (x, y) (y, x). The magnetic line graph of is the magnetic graph := ( , , m, θ) whereEm∼=1, ∼ G G V E b b b b b b b θs,t θx,y 1 1 ((x, y), (s,t)) := (x, y) (s,t)cos cos ( 1x s + 1y t)1 E rE E 2 2 m(x) = m(y) = (x,y)6=(s,t) b and 1 θ((x, y), (s,t)) := (θxy θst)(1x=s +1x=t) + (θts θyx)(1y=t +1y=s) 2 − − b THE DISCRETE LAPLACIAN ASSOCIATED TO EDGES WITH APPLICATION TO MAGNETIC ADJACENCY MATRIX15

A graph A line graph G G symb Theorem 5.6. Let := ( , ,m,θ) be a weighted magnetic graph. Then, ∆1,θ is unitarily equivalent to G V E θb b + ( ) AG Q R where θ 1 1 (x, y) := (x, y)cos x,y + R E 2 m(x) m(y) and ( ) is the operator of multiplication by . Q R R Proof. Set : ℓ2 ( ) ℓ2(V ) as the function given by F sym E −→ b θ (f)(x, y) := (x, y)cos x,y f(x, y). F rE 2 It is clear that 1 −1(f)(x, y)= ∗(f)(x, y)= f(x, y) F F (x, y)cos θx,y E 2 p for all f ℓ2( ). Notice now that on c( ): ∈ V C V θx,y θz,y b b (x, y) (z,y)cos cos θ −θ −1 2 2 i z,y x,y ∆ ,θ (f)(x, y)= qE E e 2 f(z,y) F 1 F m(y) Xz

θx,y θz,x (x, y) (x, z)cos cos θ −θ 2 2 i x,y x,z + qE E e 2 f(x, z) m(x) Xz This proof is complete. Corollary 5.7. Let := ( , ,m,θ) be a weighted magnetic bi-partite graph. If is bounded, then G V E R θb b sup (x, y). G A ≥− (x,y)∈E R

Corollary 5.8. Let := ( , ,m,θ) be a weighted magnetic bi-partite graph. If is χθ complete then θb G V E G − b is essentially self-adjoint. AG In [19], they introduce the deﬁnition of the ﬂux of a magnetic potential: 16 NASSIM ATHMOUNI, HATEM BALOUDI, MONDHER DAMAK, AND MARWA ENNACEUR

Deﬁnition 5.9. The space of cycles of , denoted by Z1( ), is the Z module with a basis of geometric cycles G G −

γ := (x0, x1) + (x1, x2)+ .... + (xN−1, xN ) with i=0,...,N-1, xi xi , and xN = x . A holonomy map is the map ∼ +1 0 Holθ : Z ( ) R/2πZ 1 G −→ given by

Holθ (x0, x1) + (x1, x2)+ .... + (xN−1, xN ) := θx0,x1 + ... + θxN ,x0 . b G We denote by ∆ b the magnetic physical discrete Laplacian acting on line graph. 0,θ Proposition 5.10. Let := ( , ,m,θ) be a magnetic tree, i.e., a magnetic graph with no closed path. b b G G V E G Then, ∆ b is unitarily equivalent to ∆ . 0,θ 0,0 Proof. Let γ Z ( ). Since is tree, then there is γ Z ( ) on the form ∈ 1 G G 1 ∈ 1 G

γ1 := ((x, x0), (x, x1)) + ((x, x1), (x, x2)) + ... + ((x, xn−1), (x, xn)) such that

Holθe(γ):= Holθe(γ1) n−1 = (θx,x θx,x )=0. i − i+1 Xi=0 Applying [19, Proposition 2.1], the result follows.

02,022 03,032

0,02 0,03 011 012 021 022 031 032 041 042 02,021 03,031 01,011 04,042 01 02 03 04 0,01 0,04

0 01,012 04,041 A magnetic tree A line graph G G

Corollary 5.11. Let := ( , ,m,θ) be a magnetic tree. Then, ∆skew is unitarily equivalent to ∆skew G V E 1,θ 1,0 Proof. Combine Proposition 5.10, Theorem 5.6 and Lemma 5.4. THE DISCRETE LAPLACIAN ASSOCIATED TO EDGES WITH APPLICATION TO MAGNETIC ADJACENCY MATRIX17

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Nassim Athmouni, Hatem Baloudi , Department of Mathematics, Faculty of Sciences of Gafsa, University of Gafsa, 2112 Zarroug, Tunisia

Mondher Damak, Marwa Ennaceur , Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, 3000 Sfax, Tunisia