Spectrum of signless 1-Laplacian on simplicial complexes

We first develop a general framework for signless 1-Laplacian defined in terms of the combinatorial structure of a simplicial complex. The structure of the eigenvectors and the complex feature of eigenvalues are studied. The Courant nodal domain theorem for partial differential equation is extended to the signless 1-Laplacian on complex. We also study the effects of a wedge sum and a duplication of a motif on the spectrum of the signless 1-Laplacian, and identify some of the combinatorial features of a simplicial complex that are encoded in its spectrum. A special result is that the independent number and clique covering number on a complex provide lower and upper bounds of the multiplicity of the largest eigenvalue of signless 1-Laplacian, respectively, which has no counterpart of p-Laplacian for any p > 1.


Introduction
Laplacian on graphs was studied by many authors and it has been extended to simplicial complex by B. Eckmann [8] which encode its basic topology. Many recent results [9,10] involve lots of algebraic and geometry properties for complex. In graph theory, (sign/signless) 1-Laplacian has been established [1][2][3][4]11] which performs better than Laplacian in the discrete and combinatorial properties, such as the Cheeger constant and dual Cheeger constant of a graph. In detail, the second eigenvalue of 1-Laplacian equals to the Cheeger constant and the corresponding eigenvector provides the Cheeger cut, while the first eigenvalue of signless 1-Laplacian equals to the dual Cheeger constant. In a word, to a graph, the 1-Laplacian performs well on discrete and combinatorial properties, while the Laplacian closes to algebraic and geometry properties. This paper devotes to the study of signless 1-Laplacian on complex.
An abstract simplicial complex K = (V, S) on a finite vertices set V is a collection of subsets of V which is closed under inclusion, i.e. if F ′ ⊂ F ∈ S, then F ′ ∈ S. An n-face is an element of cardinality n + 1. The collection of all n-faces of a simplicial complex is denoted by S n (K) and the number of all n-faces is denoted by #S n (K). For two n-faces F i and F j sharing a (n − 1)-face, we say they are (n − 1)-down neighbours and is denoted by F i down ∼ F j . For two n-faces F i and F j which are boundaries of a (n + 1)-simplex, we use the term of (n + 1)-up neighbours and write as F i up ∼ F j . LetF = {v 0 , v 1 , . . . , v n+1 } be a (n+1)-face of a complex K = (V, S), then F i = {v 0 , . . . ,v i , . . . , v n+1 } is an n-face ofF and is denoted by F i ≺F . For every (n + 1)-dim simplicial face, there are n + 2 boundaries and the boundary set, denoted by ∂F , is . According to the combinatorial structure of simplicial complex related to up or down adjacency, we define signless up 1-Laplacian and signless down 1-Laplacian respectively. For simplicity, we will omit the word 'signless' from now on.
Let p = #S n+1 (K), q = #S n (K) and l = #S n−1 (K). The incidence matrix corresponding to S n+1 (K) and S n (K) is defined by B up n = (b up Similarly, the incidence matrix corresponding to S n (K) and S n−1 (K) is B down Let Sgn(x) = (Sgn(x 1 ), Sgn(x 2 ), · · · , Sgn(x q )) T , where Define respectively up and down 1-Laplacian as follows: where x = (x 1 , x 2 , ..., x q ) is a real q-dimensional vector with components corresponding to the function values on every n-face, i.e., x i = x(F i ). Hereafter, for convenience, with some abuse of notation, x i also represents the n-dim simplicial face with subscript i. where D up = diag(d up 1 , . . . , d up q ) and d up i is the number of (n + 1)-faces that contain x i in boundary. A pair (µ, x) ∈ R × (R q \ {0}) is called an unnormalized eigenpair of the up 1-Laplacian ∆ up 1,n if 0 ∈ ∆ up 1,n x − µ Sgn(x) (or µ Sgn(x) ∆ up 1,n x = ∅).
(1.2) Definition 2 (eigenvalue problem for down 1-Laplacian). A pair (µ, ) and d down i = (n + 1), i.e., the number of (n − 1)-faces of Direct calculation shows where the summation j 1 ,...,j n+1 z ij 1 ···j n+1 is taken over all n-faces that x j 1 , · · · , x j n+1 and x i are in a same (n + 1)-face. Moreover, z ij 1 ···j n+1 is symmetric on its indices. In coordinate form, the eigenvalue problem (1.1) for up 1-Laplacian is to solve µ ∈ R and x ∈ R n \ {0} such that there exists z ij 1 ···j n+1 satisfying Similarly, the coordinate form of down 1-Laplacian reads as where the summation i 1 ,··· ,im z ii 1 ···im is taken over all n-faces that x i 1 , . . . , x im and x i are all the n-faces that share a same (n − 1)-face. Moreover, z ii 1 ···im is symmetric on its indices. The coordinate form of eigenvalue problem for down 1-Laplacian (1.3) is: From the variational point of view, ∆ up 1,n and ∆ down 1,n are respectively the subdifferential of the following convex functions i.e., ∆ up 1,n x = ∂I up (x) and ∆ down 1,n x = ∂I down (x). Indeed, the eigenvalue problem (1.1) for ∆ up 1,n (resp. (1.3) for ∆ down 1,n ) could be derived from the variational problem of I up (·) (resp. I down (·)) on the piecewise linear manifold determined by Proposition 1. Some sufficient conditions for 0 being an eigenvalue of ∆ up 1,d are listed below.
(1) #S d (K) > #S d+1 (K); (2) the up-degree of each d-face is less than or equal to d + 2 and there is at least one d-face with the degree less than d + 2; (3) S d (K) is d + 2 colorable, i.e., the colors of faces of each (d + 1)-dim simplex are pairwise different. Proof.
(1) It is easy to see that 0 is an eigenvalue if and only if I up (x) = 0 has a solution, i.e., . Since x has #S d (K) coordinate components and #S d+1 (K) absolute terms, and by #S d (K) > #S d+1 (K), the related linear system of equations has nonzero solution x. So an eigenvector corresponding to 0 exists.
The organization of the present paper is as follows. After preparing some useful lemmas in Section 2, the Courant nodal domain theorem will be investigated in Section 3. Some nontrivial relations to some parameters such as chromatic number, independent number and clique covering number on complexes are given in Section 4. Finally, we study the spectrum for 1-Laplacian on complexes constructed via wedges and duplication of motifs in Section 5.

Preliminary Lemmas
Let X be a real normed linear space and let F : X → R be a convex functional. The sub-gradient (or sub-derivative) ∇F (x) is defined as the collection of u ∈ X * such that F (y) − F (x) ≥ (u, y − x), ∀y ∈ X. The concept of sub-gradient has been extended to Lipschitz functionals in the name of Clarke derivative. And it was generalized to the class of lower semi-continuous functions. In this section, we use ∇ to denote the Clarke derivative.

Eigenvalue problem for one-homogeneous functions
Let X and Y be two real linear spaces. The functional F : X → Y is said to be one-homogeneous if and only if F (tx) = tF (x) for any x ∈ X and t ≥ 0. Then we have the Euler equality for one-homogeneous function.
Lemma 1 (Euler identity). Let F (x) be a one-homogeneous function. If u ∈ ∇F (x), then (u, x) = F (x). Now we establish the eigenvalue problem of two one-homogeneous functions.
Definition 3. Let F (x) and G(x) be two one-homogeneous functions. We call (µ, x) an eigenpair with respect to the pair In detail, x is called an eigenvector and µ is said to be an eigenvalue.
is an eigenvector with respect to (F, G).
Proof. Since F and G are Lipschitz, with the aid of Proposition 2.3.14 in [7], we have . This means that any critical point of G(x) in the sense of Clarke derivative is an eigenvector with respect to (F, G). Proof.

On functions with the form
Here (e k ) m k=1 is the standard orthogonal basis. Proof. The proof follows from the basic results ∇|x j | = Sgn(x j )e j and ∇ n i=1 Let x be an 1-norm with the form x = m i=1 d i |x i |. The eigenvalue problem for the energy function F tries to find the eigenpair (µ, x) with respect to (F, · ).

Remark 1. Lemma 2 implies that any critical point of F (x)
x in the sense of Clarke derivative is an eigenvector.
Lemma 5. Given a function with the form F (x) = n i=1 | m j=1 a ij x j |, and a constraint X = {x : x = 1}, there exists a finite set W ⊂ X such that if (µ, x) is an eigenpair, then there exists y ∈ W such that (µ, y) is also an eigenpair.
Proof. Let the map F : X → {−1, 0, 1} n+m define as It is easy to see that #W ≤ 3 m+n . So W is a finite set. Of course, the choose of W is not unique. Next we prove that if (µ, x) is an eigenpair, then there exists y ∈ W such that (µ, y) is also an eigenpair.
In fact, for such eigenvector x, let a = F(x). Then x ∈ F −1 (a) and thus F −1 (a) = ∅. Hence, there exists a unique y ∈ W such that F(y) = a = F(x). That is to say, According to Lemma 4, we obtain ∇F (x) = ∇F (y) and ∇ x = ∇ y , then Lemma 3 implies that F (y)/ y = µ = F (x)/ x and (µ, y) is also an eigenpair.
Proof. According to Lemma 4, we obtain ∇F (x) ⊂ ∇F (y) and ∇ x ⊂ ∇ y , then Lemma 3 implies that F (y)/ y = µ = F (x)/ x and (µ, y) is also an eigenpair. and respectively. From the view of geometry, the hyperplanes partition the constraint X into finite polyhedrons. So, there are only finite vertices of these polyhedrons.
The set W should guarantee that for any eigenvector x ∈ X, there exists y ∈ W such that (2.3) and (2.4) hold. We can let W be the collection of the vertices of these polyhedrons. Furthermore, we may let W be a subset of the collection of the vertices of these polyhedrons as long as each polyhedron intersect with W .

Analysis of I up (x) and I down (x)
Let I(x) be I up (x) or I down (x), and let x be the corresponding norm x up or x down . Some basic properties are collected below. Proof. This is a special case of Remark 1.
Proof. This is a direct consequence of Lamma 3.
Applying the results in Section 2.2 to complex, we have: Then there exists a positive integer C, which is only based on N and d, such that the vertices of the polyhedron contained in For example, such C can be choosen as (N − 1)!.
Proof. According to Cramer's rule and the fact that the determine of an N order (0, 1)-matrix must less than or equal to N !, we may easily obtain our desired result by taking C = (N − 1)!. Indeed, by the knowledge of Hadamard matrix, the bound could reduce to 2( √ N + 1/2) N +1 , and then the number C can be choosen as 2( Furthermore, we obtain that the eigenvalues of signless 1-Laplacian contained in Especially, we have Example 1. We show the detailed computing of spectrum of signless 1-Laplacian on the tetrahedron.
Remark 4. In fact, the above example could be generalized to ∆ up 1,n−1 for n-dim simplex. That is, the spectrum of ∆ up 1,n−1 for n-dim simplex is {0, 1}.

Courant nodal domain theorem
In this section, we develop Courant nodal domain theorem for 1-Laplacian on complexes. Similar to the reason (see [11] Page 8), we should modify the definition of nodal domain as follows.
Proof. It can be directly verified that i up ∼ j (resp. i down ∼ j) if and only if x i and x j appear in a same term | · | in I up (·) (resp. I down (·)). Hence, we derive that Sgn(x i j ) ⊃ Sgn(x j ) and Sgn(x i j + x i j 1 + · · · + x i jm ) ⊃ Sgn(x j + x j 1 + · · · + x jm ), j ′ ∼ j, j = 1, 2, · · · , n, i = 1, 2, · · · , k. Then, by Lemma 6, we complete the proof. Now, fixed the dimension d, I(·) = I up (·) (resp. I down (·)) and · = · up (resp. · down ). We apply the Liusternik-Schnirelmann theory to ∆ 1 . Note that I(x) is even, and X = {x : x = 1} is symmetric. For a symmetric set T ⊂ X, i.e., −T = T , the Krasnoselski genus of T [5,6] is defined to be Obviously, the genus is a topological invariant. Let us define By the same way as already used in [2], it can be proved that these c k are critical values of I(x).
One has c 1 ≤ c 2 ≤ · · · ≤ c n , and if 0 ≤ · · · ≤ c k−1 < c k = · · · = c k+r−1 < c k+r ≤ · · · ≤ 1, the multiplicity of c k is defined to be r. The Courant nodal domain theorem for the signless 1-Laplacian reads Theorem 1. Let x k be an eigenvector with eigenvalue c k and multiplicity r, and let S(x k ) be the number of nodal domains of x k . Then we have Proof. Assume there are n d-faces in the complex. Suppose the contrary, that there exists an eigenvector x k = (x 1 , x 2 , · · · , x n ) corresponding to the variational eigenvalue c k with multiplicity r such that S(x k ) ≥ k + r. Let D 1 (x k ), · · · , D k+r (x k ) be the nodal domains of x k . Let y i = (y i 1 , y i 2 , · · · , y i n ), where for i = 1, 2, · · · , k + r, j = 1, 2, · · · , n. By the construction of y i , i = 1, 2, · · · , k + r, we have: (1) The nodal domain of y i is the i-th nodal domain of x k , i.e., D(y i ) = D i (x k ); (2) D(y i ) ∩ D(y j ) = ∅, i = j; (3) By Proposition 3, y 1 , · · · , y k+r are all eigenvectors with the same eigenvalue c k . Now, for any x ∈ span(y 1 , · · · , y k+r ) ∩ X, there exist a 1 , . . . , a k+r such that And for any l ∈ {1, 2, . . . , n}, there exists a unique j such that x l = a j y j l . Hence, |x l | = k+r j=1 |a j ||y j l |. Since x ∈ X, y j ∈ X, j = 1, · · · , k + r, we have 1 = Note that I(·) is convex and even. Therefore, we have Note that y 1 , · · · , y k+r are non-zero orthogonal vectors, so span(y 1 , · · · , y k+r ) is a k + r dimensional linear space. It follows that span(y 1 , · · · , y k+r ) ∩ X is a symmetric set which is homeomorphous to S k+r−1 . Obviously, γ(span(y 1 , · · · , y k+r ) ∩ X) = k + r. Therefore, we derive that It contradicts with c k < c k+r . So the proof is completed.

Relationship with some parameters on complex
In this section, we will concentrate on the relationships between the eigenvalues of signless 1-Laplacian and other attractive parameters, such as chromatic number, independent number and clique covering number.

Independent number and chromatic number for vertices
Firstly, we recall the concepts of independent number and chromatic number of a hypergraph. The definition of chromatic number of hypergraphs generalize chromatic number of graphs in various ways, see for example [13]. In [13],the chromatic number is defined for r-unifrorm hypergraphs as while the independent number of a hypergraph is defined by Note that a simplicial complex can be considered as a hypergraph, the definition of independent number and chromatic number for simplicial complexes can be defined as follows: In [14], the independent number and chromatic number of simplicial complex are respectively defined by α = max{|A| : A ⊃ F, ∀ maximal face F } and However, in the proof of main theorems in [14], the author essentially deals with α d and χ d , where d is the dimension of complex. So, the results still hold if we replace 'α' and 'χ' by 'α d ' and 'χ d ' in those theorems. In this subsection, we will concentrate on χ s and α s and study relation between eigenvalues of 1-Laplacians and them. An elementary result for the relations of these concepts are: (1) χ s α s ≥ |V |, χ s ≤ ⌈χ 1 /s⌉, for all s. Here ⌈x⌉ is the ceiling function on x, i.e., the minimum integer that does not less than x. Moreover, χ s ≤ ⌈χ t /⌊s/t⌋⌉, where ⌊x⌋ is the Gauss function on x, i.e., the maximum integer that does not exceed x.
(3) The above definitions for independent number and chromatic number are respectively equivalent to α ′ s = max{|A| : |A ∩ F | ≤ s, ∀s-face F } and Proof.
(2) χα ≥ |V | and χ ≤ ⌈χ 1 /d⌉ are proved in [14]. Now we prove α ≤ α d . Since the dimension of K is d, the maximal face F has at most d + 1 elements. So F ⊂ A implies |F ∩ A| ≤ |F | − 1 ≤ d, for any maximal face F . Thus, |F ∩ A| ≤ d holds for any face F . This deduces that A maximal face (i.e., facet) is not need to be a d-face. But if the complex is homogeneous (or pure), then facets must coincide with d-faces. So, it is easy to check that α = α d .
(3) It is easy to check that α ′ s ≥ α s . Next we will show the reverse inequality. Indeed, it is enough to show that for any set A with |A ∩ F | ≤ s for all s-faces F , we have |A ∩ F | ≤ s for any face whose dimension is larger than s. For the contrary, there is some (s + k)-face F ′ satisfying |A ∩ F ′ | ≥ s + 1. Then there is one s-face whose vertices are in A, which is a contradiction. The proof for χ ′ s = χ s is similar.
where M i is the largest degree of vertices, µ d−1 is the minimum unnormalized eigenvalue of ∆ up Proof. Let A ⊂ V be the largest independent set with |A| = α d , and for some fixed a, b ∈ R, let Taking a = |A c | and b = |A|, we get . (4.1) Let M i and m i be respectively maximum and minimum number of (i + 1)-faces that contains an i-face, i ∈ {0, 1, . . . , d − 1}.
Since A is independent, for each (d − 1)-face F ⊂ A, the number of d-faces that contains F is larger than or equals to m d−1 . And it can be proved that m 0 > m 1 > · · · > m d−1 .
Claim: for any i-face F ⊂ A with i ∈ {0, . . . , d − 1}, the number of (i + 1)-faces that contains F and meets A c is larger than or equals to m d−1 .
We prove the above claim by mathematical induction on i from (d − 1) to 0. For i = d − 1, the claim holds accroding to definition of m d−1 and the independence of A. Suppose the claim holds for i. Then for the case of (i − 1), for any (i − 1)-face F , there are two subcases: * The eigenvalues which we concern are the unnormalized eigenvalues of ∆ up 1,d−1 , i.e., the corresponding constraint relates to x 1 := (d−1)-face F |xF |, not the normalized version x := (d−1)-face F deg F |xF |

F contains in an i-face F ′ with all its vertices in A.
In this case, using the inductive hypothesis for F ′ , there exist v 1 , . . . , v m d−1 ∈ A c such that F ′ ∪ {v 1 }, · · · , F ′ ∪ {v m d−1 } are (i + 1)-faces. In consequence, F ∪ {v 1 }, · · · , F ∪ {v m d−1 } are i-faces, which means that the claim holds for such F .
2. there is no i-face with all its vertices in A that contains F .
In this case, all i-faces that contains F must meet A c , and the number of i-faces containing F is at least m i−1 ≥ m d−1 . Thus the claim holds.
In particular, for i = 0, it means that |E({i}, A c )| ≥ m d−1 for any i ∈ A. Thus, Taking Combining (4.1) and (4.2), we have which derives our desired inequality.
This implies Combining (4.3) and (4.2), we have Since A is the maximal independent set, for any j ∈ A c , A ∪ {j} is not independent, i.e., there is a d-face in A ∪ {j} containing j as its vertex, which implies that |E(A, {j})| ≥ d. Thus, Combining (4.1) with (4.4), we have So far, we focused on vertices of simplicial complexes, i.e. 0-faces. In fact, we can also define independent number and chromatic number for any i-face of simplicial complexes. In the following subsection, we will give corresponding definitions and study the relationships between eigenvalues of 1-Laplacian and them.

Chromatic number for i-faces
Definition 8 (chromatic number). A d-face coloring of a simplicial complex assigns a color to each d-face so that no two faces that contain in the same (d + 1)-face have the same color. The smallest number of colors needed is called its chromatic (or coloring) number.
One can easily see that the chromatic number of all d-faces is at least d + 2.
Theorem 3. Let χ be the chromatic number of S d (the set of d-faces) with respect to the up-adjacent relation. Assume vol : S d → (0, +∞) is a given degree function on S d and vol(S) := i∈S vol(i) is the volume of S, for any S ⊂ S d . Let x = i∈S d vol(i)|x i | for any x ∈ R S d . Then where e d+1 (S d ) is the number of (d + 1)-simplexes with d-faces in S d , c up 1 = min x . Furthermore, this upper bound is sharp.
Proof of Theorem 3. Let S 1 d , · · · , S χ d be the color classes of S d . Given an integer k ∈ {1, 2, . . . , χ}, we define the vector x by and where e 0,d (S k d , S d \ S k d ) counts the number of (d + 1)-simplexes with one d-face in S k d and others in S d \ S k d , and e d+1 (S d \ S k d ) (resp. e d+1 (S d )) is the number of (d + 1)-simplexes with d-faces in S d \ S k d (resp. S d ).
In summary, for every k = 1, · · · , χ, we have Summing these inequalities for k = 1, 2, · · · , χ, we obtain Elementary computation gives Now we get It is easy to see that max (a,b) =(0,0) where the maximum arrives at t = −d − 1, and this implies the desired inequality Next we prove that the bound is sharp. In fact, if K is a (d + 1)-simplex, then the equality holds.

Independent number and clique covering number for i-faces
Similar to the results in [12], we give the counterpart of connections between the independent number and clique covering number of K as well as the multiplicity of the eigenvalue 1 of ∆ 1 . The definitions and notions are listed below.
• t: times of 1 appearing in the sequence of variational eigenvalues (c k ) n k=1 .
• α: independent number of S d , i.e., the cardinality of the largest subset of d-faces that does not adjacent. It is defined by α = max{p : there exist p faces in S d which are pairwise non-adjacent}.
• κ: the clique covering number (the smallest number of cliques of S d whose union covers S d ).
Here a clique is a subset of S d such that any two d-faces in such clique are adjacent.
Our purpose is to show γ(I 1 S d ) ≤ κ. We first prove that for any disjoint subsets W 1 and W 2 , Here * is the topological join. Given a ∈ I 1 W 1 ∪W 2 , let Since W 1 and W 2 are disjoint, one can easily verify that a = a 1 + a 2 and 1 = I(a) ≤ I(a 1 ) + I(a 2 ) ≤ a 1 + a 2 = a = 1. Hence, I(a 1 ) = a 1 and I(a 2 ) = a 2 hold. Taking x = a 1 / a 1 and y = a 2 / a 2 , one has I(x) = x = I(y) = y = 1, which implies that x ∈ I 1 W 1 and y ∈ I 1 W 2 . Therefore, we have a = a 1 x + a 2 y = tx + (1 − t)y, where t = a 1 . Then we obtain Combining the subadditivity of Krasnoselski genus with respect to topological join, we have According to the mathematical induction, we can easily deduce that which provides a recurrence method to estimate a large complex by smaller one. Note that for a clique W of S d , We can easily construct an odd continuous function f : This means that γ(I 1 W ) = 1. Then (4.8) implies that 1 : cliques W 1 , · · · , W l form a partition of S d = κ.

Constructions and their effect on the spectrum
This section follows the lines of Horak and Jost [9]. We denote by spec(K) the set of all eigenvalues of ∆ 1 . Consider the topological multiplicity, we use Spec(K) to denote the multiset of ∆ 1 -eigenvalues. Further, A • ∪ B is the multiset sum of two multisets A and B. Our results are similar to [9], but most of the proofs are different.

Wedges
Definition 9. The combinatorial k-wedge sum of simplicial complexes K 1 and K 2 , K 1 ∨ k K 2 , is defined as the quotient of their disjoint union by the identification F 1 ∼ F 2 , that is where F 1 and F 2 are the k-dim simplicial faces in K 1 and K 2 respectively. This definition could be generalized to the k-wedge sum of arbitrary many simplicial complexes. For example, the 1-wedge of some tetrahedrons is shown as below.
Proof. Since K 1 and K 2 are identified by a k-face, then by noticing that k < i, any i-face of K 1 is non-adjacent to i-faces of K 2 in K 1 ∨ k K 2 . Consequently, if (µ, x 1 ) is an eigenpair of ∆ up 1,i (K 1 ), then letting it is easy to check that (µ, x) is an eigenpair of ∆ up 1,i (K 1 ∨ k K 2 ). The same property holds for ∆ up 1,i (K 2 ). Moreover, if (µ, x 1 ) and (µ, x 2 ) are eigenpairs of ∆ up 1,i (K 1 ) and ∆ up 1,i (K 2 ), respectively, then is can be easily verified (µ, x) is an eigenpair of ∆ up 1,i (K 1 ∨ k K 2 ), where x is defined by So, we have proved that For the converse, let (µ, x) be an eigenpair of ∆ up 1,i (K 1 ∨ k K 2 ), and let x 1 (resp. x 2 ) be the restriction of x on S i (K 1 ) (resp. S i (K 2 )). Since x = 0, at least one of x 1 and x 2 is not 0. Suppose x 1 = 0. Then, there exist z j 1 ···j i+1 j ∈ Sgn(x j 1 + · · · + x j i+1 + x j ) such that . Therefore, for j ∈ S i (K 1 ), the above equation holds and thus (µ, x 1 ) is an eigenpair of ∆ up 1,i (K 1 ). If x 2 = 0, then the same process deduces that (µ, x 2 ) is an eigenpair of ∆ up 1,i (K 2 ). Hence, Remark 9. This is a signless 1-Laplacian counterpart of Theorem 6.1 [9].
Similar proof, we have the following Theorem 6.
for all i, k with i > k + 1.
Theorem 7. Let K 1 and K 2 be simplicial complexes, for which the spectrum of ∆ up 1,i (K 1 ) and ∆ up 1,i (K 2 ) both contain the eigenvalue µ, and let x 1 , x 2 be their corresponding eigenvectors. If an i-wedge K = K 1 ∨ i K 2 is obtained by identifying i-faces i 1 and i 2 , for which x 1 i 1 = x 2 i 2 , then the spectrum of ∆ up 1,i (K) contains the eigenvalue µ, too.
Proof. Note that we have identified i 1 with i 2 in K. So, we can assume . It is easy to see that Now we are going to prove that is an eigenvector of ∆ up 1,i (K) corresponding to the eigenvalue µ. In fact, since (µ, x 1 ) is an eigenpair of ∆ up 1,i (K 1 ), there exist z 1 j 1 ···j i+1 j ∈ Sgn(x 1 j 1 + · · · + x 1 j+1 + x 1 j ) and z 1 j ∈ Sgn(x 1 j ) such that for any j ∈ S i (K 1 ) \ {i 1 }, and j 1 ,···j i+1 z 1 j 1 j 2 ···i 1 = µd up i 1 (K 1 )z 1 i 1 .

Duplication of motifs
Given a simplicial complex K = (V, S) and a collection of simplicial faces M . The closure ClM of M is the smallest subcomplex of K that contains each simplex in M and is obtained by repeatedly adding to M each face of every simplex in M . The star StM of M is the set of all simplices in K that have a face in M . The link lkM of M is ClStM − StClM .
From the definition of link, the vertices in motif M are different from that in lkM. Let l 0 , ..., l m be vertices of lkM and p 0 , ..., p k be the vertices of M. Duplication of the i-motif M is defined as follows.
Definition 11 (duplication of the i-motif M ). Let M ′ be a simplicial complex on the vertices p ′ 0 , ..., p ′ k and the map h : p ′ i → p i be a simplicial isomorphism between M ′ and M . Let K M := K ∪ {{p ′ i 0 , · · · , p ′ i k , l j 1 , · · · , l j l }|{p i 0 , · · · , p i k , l j 1 , · · · , l j l } ∈ K}. We call K M the duplication of i-motif of M .
The following proposition is proved by the similar methods in [9]. For completeness, we give the proof. which confirms the claim.
The same to [9], we have the following corollary.
Corollary 1. If the spectrum of the simplicial complex ClStM contains the eigenvalue µ, with an eigenvector h that is identically equal to zero on lkM, then µ is also the eigenvalue of K M .
Using the same methods in the proof of Theorem 8, we have the following Theorem 9. Let c j be the eigenvalue of ∆ up 1,i (ClStM )| StM and c ′ j be the eigenvalue of ∆ up 1,i (ClStM ). Then c ′ j ≤ c j .