Eigenvalues of Laplace operators on non-bipartite graphs
aa r X i v : . [ m a t h . SP ] F e b EIGENVALUES OF LAPLACE OPERATORS ONNON-BIPARTITE GRAPHS
HONGJUN W ANG , HONGMEI SONG , AND JIA ZHAO , ∗ Abstract.
This paper considers the comparison between the eigenvalues ofLaplace operators with the standard conditions and the anti-standard condi-tions on non-bipartite graphs which are equilateral or inequilateral. First of all,we show the calculation of the eigenvalues of Laplace operators on equilateralmetric graphs with arbitrary edge length. Based on this method, we use theproperties of the cosine function and the arccosine function to find the compar-ison between the eigenvalues of Laplace operators with the standard conditionsand the anti-standard conditions on equilateral non-bipartite graphs. In addi-tion, we give the inequalities between standard and anti-standard eigenvalueson a special inequilateral non-bipartite graph. Introduction
Differential operators on metric graphs are a class of unbounded linear opera-tors which are widely used in physics, chemistry, and engineering, see [1–3]. Thespectral theory of differential operators on metric graphs has become an importantpart of the spectral theory of differential operators in recent decades, for exam-ple [4]. In this paper, we focus on the Laplace operators on metric graphs, i.e., thesecond derivative operators on each edge of the graphs. The most common vertexconditions for the Laplace operators on metric graphs are standard conditions (orKirchhoff conditions) and anti-standard conditions (or anti-Kirchhoff conditions),see Section 2 below for more details. We denote Laplace operators subjected tostandard conditions and anti-standard conditions on metric graph Γ by L st (Γ) and L a/st (Γ).Our main aim is to provide the comparison between the eigenvalues of L st (Γ)and L a/st (Γ) on non-bipartite graphs with arbitrary edge length. There have beensome results about the comparison between the eigenvalues of L st (Γ) and L a/st (Γ)on bipartite graphs which can be found in [5–7]. Inspired by [6] which treats theeigenvalues of L st (Γ) and L a/st (Γ) on equilateral metric graphs with edge length 1,we give the calculation of the eigenvalues of L st (Γ) and L a/st (Γ) on an equilateralmetric graph Γ with arbitrary edge length. In [7], the authors show that the pos-itive eigenvalues of Laplace operators with standard conditions and anti-standardconditions on bipartite graphs are equal, and the equation related to the number of Mathematics Subject Classification.
Key words and phrases.
The metric graphs; Laplace operators; Vertex conditions; Eigenvalues.Hongjun Wang: Department of Mathematics, Hebei University of technology, Tianjin 300401,People’s Republic of China. email: [email protected] Song: Department of Mathematics, Hebei University of technology, Tianjin 300401,People’s Republic of China. email: [email protected].*Corresponding author: Jia Zhao, Department of Mathematics, Hebei University of technology,Tianjin 300401, People’s Republic of China. email: [email protected]. HONGJUN W ANG , HONGMEI SONG , AND JIA ZHAO , ∗ even cycles was given. Next, we assume that graph Γ is an equilateral non-bipartitegraph, then provide necessary and sufficient conditions for the eigenvalues of L st (Γ)and L a/st (Γ) to satisfy the following inequality:(1.1) λ k +1 ( L st (Γ)) ≥ λ N − n + k ( L a/st (Γ)) , k ∈ N , where N denotes the number of edges, n denotes the number of vertices.The paper is organized as follows. We introduce two main vertex conditionsfor Laplace operators in Section 2 and give the calculation of the eigenvalues of L st (Γ) and L a/st (Γ) on an equilateral metric graph Γ with arbitrary edge lengthin Section 3. Based on the calculation of the eigenvalues of L st (Γ) and L a/st (Γ),we give the comparison between the eigenvalues of L st (Γ) and L a/st (Γ) on a non-bipartite graph Γ with arbitrary edge length by using the properties of the cosinefunction and the arccosine function in Section 4. We show how the eigenvalues of L a/st (Γ) change when vertices of an equilateral odd cycle are increased in corollary4.2. The comparison between the eigenvalues of L st (Γ) and L a/st (Γ) on equilateralnon-bipartite graphs is described in the previous part. As a supplement, we givethe inequalities between standard and anti-standard eigenvalues on a special in-equilateral non-bipartite graph in the final Section. We find that there are similarresults on a special inequilateral non-bipartite graph as on equilateral non-bipartitegraphs. 2. Preliminaries
A metric graph Γ in the paper consists of a finite set of vertices V = { v , · · · , v n } and a finite set E = { e , · · · , e N } of edges connecting the vertices, where n, N denotethe number of vertices and edges respectively. Moreover, the graph Γ is connected,and each edge e i is assigned a positive length l e i ∈ (0 , ∞ ). Finally, the graph Γ isalso assumed to be simple, i.e., the graph Γ contains no loops, and at most one edgecan join two vertices in Γ. We denote the total length of a graph by L (Γ) = P l e i and say that the graph Γ is equilateral if l e i = l e j for any e i , e j ∈ E . Any edge e i ∈ E will be identified with interval [0 , l e i ], then we write that o ( e i ) and t ( e i )denote the starting vertex and ending vertex of edge e i respectively.Let V , V ⊂ V be two disjoint sets and satisfy V ∪ V = V . If e i ∩ V = ∅ , e i ∩ V = ∅ for any edge e i ∈ E , we say that Γ is bipartite.Two vertices v i and v j will be called adjacent (denoted v i ∼ v j ) if there is anedge e k connecting v i and v j . We give the adjacency matrix A := ( a ij ) n × n of Γ by(2.1) a ij = (cid:26) , if v i ∼ v j , , if v i ≁ v j , then A is symmetric. We define the transition matrix Z := ( z ij ) n × n by Z := Diag( A e ) − A . where e := (1) n × . The signed incidence matrix D := ( d ij ) n × N of Γ is given by(2.2) d ij = − , if v i = o ( e j ) , , if v i = t ( e j ) , , else. IGENVALUES OF LAPLACE OPERATORS ON NON-BIPARTITE GRAPHS 3
We give the index mapping s ( i, j ): I V × I V → I E as follows(2.3) s ( i, j ) = ( k, if e k = { v i , v j } , , else. where I V := { , ..., n } and I E := { , ..., N } . Let L (Γ) denote the Hilbert space ⊕ i L [0 , l e i ] with the inner product( f, g ) L (Γ) = X i Z l ei f i ( x ) g i ( x )d x, where f i ( x ) denotes the restriction f ( x ) | e i . We denote the maximal Laplace oper-ator L max as follows(2.4) L max f = − f ′′ ( x ) , f ∈ Dom( L max ) , Dom( L max ) = { f ∈ L (Γ) : f i , f ′ i ∈ AC [0 , l e i ] ,f or each i ∈ I E , L max f ∈ L (Γ) } , where AC [0 , l e i ] represents the set of all absolutely real-valued continuous functionson interval [0 , l e i ]. The standard Laplace operator L st (Γ) (also called KirchhoffLaplace operator) is defined by(2.5) L st (Γ) f = − f ′′ ( x ) , f ∈ Dom( L st (Γ)) , Dom( L st (Γ)) = { f ∈ Dom( L max ) : f ( v ) = f ( v ) = · · · = f d v ( v ) ,f ′ ( v ) + f ′ ( v ) + · · · + f ′ d v ( v ) = 0 } , where d v represents the degree of the vertex v , and f ′ i ( v ) is the derivative at the ver-tex v taken along the edge e i in the outgoing direction. The anti-standard Laplaceoperator L a/st (Γ) (also called anti-Kirchhoff Laplace operator) is defined by(2.6) L a/st (Γ) f = − f ′′ ( x ) , f ∈ Dom( L a/st (Γ)) , Dom( L a/st (Γ)) = { f ∈ Dom( L max ) : f ′ ( v ) = f ′ ( v ) = · · · = f ′ d v ( v ) ,f ( v ) + f ( v ) + · · · + f d v ( v ) = 0 } . The spectra of L st (Γ) and L a/st (Γ) consist of the isolated, nonnegative eigenvalueswith finite multiplicities.3. Standard and anti-standard eigenvalues on equilateral metricgraphs
We assume that the graph Γ is an equilateral metric graph. For each directed edge e i of Γ, let [0 , l e ] be a real interval of length l e . We now find the real-valued function u ∈ Dom( L max ) that satisfies the standard conditions or satisfies the anti-standardconditions. We define the boundary value function matrix U ( x ) as follows U ( x ) := ( u ij ( x )) n × n , u ij ( x ) = a ij u s ( i,j ) ( l e + l e d i,s ( i,j ) − xd i,s ( i,j ) ) , where u i ( x ) denotes the restriction u ( x ) | e i . Hence the equation − u ′′ i ( x ) = λu i ( x ) , ∀ i ∈ { , · · · , N } can be translated into(3.1) − U ′′ ( x ) = λU ( x ) . HONGJUN W ANG , HONGMEI SONG , AND JIA ZHAO , ∗ Then we have U ∗ ( x ) = U ( l e − x ) , U ′ ( x ) ∗ = − U ′ ( l e − x ) . We define the two matrices Φ := U (0) , Ψ := U ′ (0) . We consider the standard eigenvalue problem(3.2) − u ′′ j ( x ) = λu j ( x ) , ∀ j ∈ { , · · · , N } ,u j ( v i ) = u k ( v i ) , ∀ v i ∈ e j ∩ e k , N X j =1 d ij u ′ j ( v i ) = 0 , ∀ i ∈ { , · · · , n } , then they could be represented by U ( x ) as follows(3.3) − U ′′ ( x ) = λU ( x ) ,U (0) = φe ∗ · A ,U ′ (0) e = 0 , where φ := ( u ( v i )) n × . We consider the anti-standard eigenvalue problem(3.4) − u ′′ j ( x ) = λu j ( x ) , ∀ j ∈ { , · · · , N } ,d ij u ′ j ( v i ) = d ik u ′ k ( v i ) , ∀ v i ∈ e j ∩ e k , N X j =1 d ij u j ( v i ) = 0 , ∀ i ∈ { , · · · , n } , then they could be represented by U ( x ) as follows(3.5) − U ′′ ( x ) = λU ( x ) ,U ′ (0) = ψe ∗ · A ,U (0) e = 0 , where ψ := ( u ′ ( v i )) n × . Let U ( x ) be a nontrivial solution of (3.1) corresponding tothe eigenvalue λ . Then λ ∈ [0 , ∞ ) and the form of U ( x ) is(3.6) U ( x ) = Φ + Φ ∗ − Φ l e x, λ = 0 , cos( √ λx )Φ + sin( √ λx ) √ λ Ψ , λ > . Let P := ( p ij ) n × m , Q := ( q ij ) n × m and R := ( r ij ) n × m be three real matrices.We define the Hadamard product as follows P · Q = ( p ij q ij ) n × m , then P · Q = Q · P, ( P · Q ) ∗ = P ∗ · Q ∗ , ( P + Q ) · R = P · R + Q · R. Moreover, if x := ( x i ) n × we have(( P · Q ) x ) i = ( P Diag( x i ) Q ∗ ) ii . IGENVALUES OF LAPLACE OPERATORS ON NON-BIPARTITE GRAPHS 5
Let Γ be a metric graph with adjacency matrix A := ( a ij ) n × n . We define the threematrix spaces M (Γ) := { M = ( m ij ) n × n : a ij = 0 ⇒ m ij = 0 } , M − (Γ) := { M ∈ M (Γ) : M ∗ = − M and M e = 0 } , M + (Γ) := { M ∈ M (Γ) : M ∗ = M and M e = 0 } . We need to know the multiplicities of the eigenvalues of L st (Γ) and L a/st (Γ) bythe dimensions of the space M − (Γ) and M + (Γ)which can be found in [8]. Thedimensions of M − (Γ) and M + (Γ) are(3.7) dim( M − ) = N − n + 1 , (3.8) dim( M + ) = ( N − n + 1 if Γ is bipartite,N − n if Γ is non − bipartite. Let D be a n × n real matrix. We call that D is reducible if there exists a n × n permutation matrix P such that P DP ⊤ = (cid:20) D D D (cid:21) , where D is r × r matrix and D is ( n − r ) × r matrix and D is ( n − r ) × ( n − r )matrix. We say that D is irreducible if there is no such a matrix P .We want to obtain the eigenvalues of L st (Γ) and L a/st (Γ) on equilateral metricgraphs by the eigenvalues of the transition matrix Z . By using the theory in [6,Lemma 2.36], we can give the following lemma. Lemma 3.1.
Let graph Γ be a metric graph with the transition matrix Z . Then thetransition matrix Z is irreducible and P nj =1 z ij = 1 for all i ∈ { , · · · n } . Moreover Z has the real eigenvalues µ , µ , · · · , µ n which can be sorted. According to theirmultiplicities, we have µ > µ ≥ · · · ≥ µ n ≥ − . Furthermore, if µ n = − then Γ is bipartite. Theorem 3.2.
Let Γ be an equilateral metric graph with the transition matrix Z .Then the spectrum of L st (Γ) is given by σ ( L st (Γ)) = { } ∪ { λ > √ λl e ) ∈ σ ( Z ) } ∪ { λ > √ λl e ) = − } . The multiplicities of the eigenvalues are given by m ( λ ) = , if λ = 0 , dimker( Z − cos( √ λl e ) I ) , if sin( √ λl e ) = 0 ,N − n + 2 , if cos( √ λl e ) = 1 and λ > ,N − n + 2 , if cos( √ λl e ) = − and Γ is bipartite,N − n, if cos( √ λl e ) = − and Γ is non − bipartite, where I is an identity matrix. HONGJUN W ANG , HONGMEI SONG , AND JIA ZHAO , ∗ Proof.
We consider the first case λ = 0. From (3.6) we know that all solutions of(3.1) corresponding to λ = 0, and the respective derivatives are obtained by U ( x ) = Φ + Φ ∗ − Φ l e x, U ′ ( x ) = Φ ∗ − Φ l e . By using Φ = U (0) = φe ∗ · A , we get a new expression U ′ (0) = Φ ∗ − Φ l e = ( eφ ∗ − φe ∗ ) · A l e . Since U ′ (0) e = 0, then we have( A · eφ ∗ ) e = ( A · φe ∗ ) e ⇒ ( A φe ∗ ) ii = ( A eφ ∗ ) ii ⇔A φ = Diag( A e ) φ ⇔ Diag( A e ) − A φ = φ ⇔Z φ = φ. Thus φ is an eigenvector of Z belonging to the eigenvalue 1 with the multiplicity1, then the multiplicity of the eigenvalue λ = 0 is 1.We consider the second case λ >
0. From (3.6) we see that all solutions of (3.1),and the respective derivatives are obtained by U ( x ) = cos( √ λx )Φ + sin( √ λx ) √ λ Ψ , U ′ ( x ) = −√ λ sin( √ λx )Φ + cos( √ λx )Ψ . We assume that sin( √ λl e ) = 0. Since U ( l e ) = U ∗ (0) = Φ ∗ and Φ = U (0) = φe ∗ · A , we obtain thatΨ = √ λ sin( √ λl e ) (Φ ∗ − Φ cos( √ λl e ))= √ λ sin( √ λl e ) ( eφ ∗ − cos( √ λl e ) φe ∗ ) · A . Due to U ′ (0) e = 0 and Ψ = U ′ (0), we have( eφ ∗ · A ) e = (cos( √ λl e ) φe ∗ · A ) e ⇒ ( A φe ∗ ) ii = cos( √ λl e )( A eφ ∗ ) ii ⇔A φ = cos( √ λl e )Diag( A e ) φ ⇔Z φ = cos( √ λl e ) φ. Then multiplicity of the eigenvalue λ is dimker( Z− cos( √ λl e ) I ) when sin( √ λl e ) = 0.For the remaining case of sin( √ λl e ) = 0, we will distinguish the cases cos( √ λl e ) =1 and cos( √ λl e ) = − √ λl e ) = 1. Since every solution Φ ∗ =Φ = φe ∗ · A , the solution space with vanishing Ψ is one dimension. On the otherhand, we have Ψ ∗ = − Ψ by using U ′ ( l e ) = − U ′ (0) ∗ = − Ψ ∗ . Then according toΨ e = U ′ (0) e = 0, the solution space with vanishing Φ of (3.3) is isomorphic to M − with dim( M − ) = N − n + 1. Hence the multiplicity of λ is N − n + 2.Case 2: When cos( √ λl e ) = −
1, we have Ψ = Ψ ∗ by using U ′ ( l e ) = − U ′ (0) ∗ = − Ψ ∗ . Then according to Ψ e = U ′ (0) e = 0, the solution space with vanishing Φ IGENVALUES OF LAPLACE OPERATORS ON NON-BIPARTITE GRAPHS 7 of (3.3) is isomorphic to M + with dim( M + ) which is either N − n + 1 or N − n depending on whether Γ is bipartite or not. On the other hand, since Φ = φe ∗ ·A , thesolution space with vanishing Ψ is one dimension. But since U ( l e ) = U ∗ (0) = Φ ∗ ,we have Φ = − Φ ∗ which is possible for non-trivial Φ if Γ is bipartite, we cannotobtain Φ = − Φ ∗ if Γ is non-bipartite. Hence the multiplicity of λ is N − n + 2 ifΓ is bipartite, the multiplicity of λ is N − n if Γ is non-bipartite. We have proventhe theorem. (cid:3) Theorem 3.3.
Let graph Γ be an equilateral metric graph with the transition matrix Z . Then spectrum of L a/st (Γ) is given by σ ( L a/st (Γ)) = { } ∪ { λ > − cos( l e √ λ ) ∈ σ ( Z ) } ∪ { λ > l e √ λ ) = 1 } . The multiplicities of the eigenvalues are given by m ( λ ) = N − n + 1 , if λ = 0 and Γ is bipartite,N − n, if λ = 0 and Γ is non − bipartite, dimker( Z + cos( √ λl e ) I ) , if sin( √ λl e ) = 0 ,N − n + 2 , if cos( √ λl e ) = − ,N − n + 2 , if cos( √ λl e ) = 1 and Γ is bipartite,N − n, if cos( √ λl e ) = 1 and Γ is non − bipartite. Proof.
We consider the first case λ = 0. From (3.6) we know that all solutions of(3.1) with λ = 0, and the respective derivatives are obtained by U ( x ) = Φ + Φ ∗ − Φ l e x, U ′ ( x ) = Φ ∗ − Φ l e . Using integration by parts, we obtain(3.9) 0 = N X j =1 Z l e u ′′ j u j d x j = N X j =1 [ u ′ j u j ] l e − N X j =1 Z l e u ′ j d x j . Simplify the first term on the right side of the equation as follows N X j =1 [ u ′ j u j ] l e = N X j =1 u ′ j ( l e ) u j ( l e ) − N X j =1 u ′ j (0) u j (0)= N X j =1 X v i ∈ e j d ij u ′ j ( v i ) u j ( v i )= n X i =1 d vi X j =1 d ij u ′ j ( v i ) u j ( v i ) , where d v i denotes the degree of the vertex v i . Considering d ij u ′ j ( v i ) = d ik u ′ k ( v i ) atfixed vertex v i , we can write u ′ ( v i ) := d ij u ′ j ( v i ). Therefore we have(3.10) N X j =1 [ u ′ j u j ] l e = n X i =1 d vi X j =1 d ij u ′ j ( v i ) u j ( v i ) = n X i =1 u ′ ( v i ) d vi X j =1 u j ( v i ) = 0 . HONGJUN W ANG , HONGMEI SONG , AND JIA ZHAO , ∗ Then (3.9) can be written as(3.11) 0 = N X j =1 Z l e u ′′ j u j d x j = − N X j =1 Z l e u ′ j d x j ≤ . Thus u ′ j ≡
0. We have Φ = Φ ∗ , and according to Φ e = 0, this solution space isisomorphic to M + . Hence the multiplicity of λ = 0 is N − n + 1 if Γ is bipartite,the multiplicity of λ = 0 is N − n if Γ is non-bipartite.We consider the second case λ >
0. From (3.6) we know that all solutions of(3.1) and the respective derivatives are obtained by U ( x ) = cos( √ λx )Φ + sin( √ λx ) √ λ Ψ , U ′ ( x ) = −√ λ sin( √ λx )Φ + cos( √ λx )Ψ . We assume that sin( √ λl e ) = 0. Using U ′ ( l e ) = − U ′ (0) ∗ = − Ψ ∗ , we obtain − Ψ ∗ = −√ λ sin( √ λl e )Φ + cos( √ λl e )Ψ ⇔ Φ = 1 √ λ sin( √ λl e ) (cos( √ λl e )Ψ + Ψ ∗ ) . Due to Ψ = U ′ (0) = ψe ∗ · A and Φ e = U (0) e = 0, we have( A · eψ ∗ ) e = − cos( √ λl e )( A · ψe ∗ ) e Z ψ = − cos( √ λl e ) ψ. Then multiplicity of the eigenvalue λ is dimker( Z +cos( √ λl e ) I ) when sin( √ λl e ) = 0.For sin( √ λl e ) = 0, we will distinguish the cases cos( √ λl e ) = 1 and cos( √ λl e ) = − √ λl e ) = −
1, since Ψ ∗ = Ψ = ψe ∗ · A , thesolution space with vanishing Φ of (3.5) is one dimension. On the other hand, wehave Φ ∗ = − Φ by using U ( l e ) = U ∗ (0) = Φ ∗ . Then according to Φ e = U (0) e = 0,the solution space with vanishing Ψ of (3.5) is isomorphic to M − with dim( M − ) = N − n + 1. Hence the multiplicity of λ is N − n + 2.Case 2: When cos( √ λl e ) = 1, we have Φ ∗ = Φ by using U ( l e ) = U (0) ∗ = Φ ∗ .Then according to Φ e = U (0) e = 0, the solution space with vanishing Ψ of (3.5) isisomorphic to M + with dim( M + ) which is either N − n + 1 or N − n depending onwhether Γ is bipartite or not. On the other hand, since Ψ = ψe ∗ · A , the solutionspace with vanishing Φ is one dimension. But since U ′ ( l e ) = − U ′ (0) ∗ = − Ψ ∗ , wesee Ψ = − Ψ ∗ which is possible for non-trivial Ψ if Γ is bipartite, we cannot obtainΨ = − Ψ ∗ if Γ is non-bipartite. Hence the multiplicity of λ is N − n + 2 if Γ isbipartite, the multiplicity of λ is N − n if Γ is non-bipartite. We have proven thetheorem. (cid:3) The relation between standard and anti-standard eigenvalues onequilateral non-bipartite graphs
The relation between the eigenvalues of L st (Γ) and L a/st (Γ) on equilateral bi-partite graphs has achieved some good results, please refer to [5] and [7] for details.However, it has remained unnoticed that there is a relation between the eigenval-ues of L st (Γ) and L a/st (Γ) on equilateral non-bipartite graphs. The comparisonbetween the eigenvalues of Laplace operators with the standard conditions and theanti-standard conditions on equilateral non-bipartite graphs is as follows. IGENVALUES OF LAPLACE OPERATORS ON NON-BIPARTITE GRAPHS 9
We assume that the graph Γ is an equilateral non-bipartite graph with edgelength l e . The number 1 must be the eigenvalue of the corresponding transitionmatrix Z , and let the eigenvalues of Z be µ , µ , · · · , µ n − , µ n , then we have1 = µ > µ ≥ · · · µ n − ≥ µ n > − . Define the N ( m )0 and the N ( m ) as follows N ( m )0 = { , · · · , | {z } m , , · · · , | {z } m , · · · } , N ( m ) = { , · · · , | {z } m , , · · · , | {z } m , · · · } . The l e times eigenvalues of L st (Γ) and L a/st (Γ) are listed in Table 1,standard anti-standard λ = 0 { } { } or ∅ cos( l e √ λ ) = 1 { (2 kπ ) : k ∈ N ( N − n +2) } { (2 kπ ) : k ∈ N ( N − n ) } sin( l e √ λ ) = 0 { (2 kπ ± arccos( µ m )) : { ((2 k + 1) π ± arccos( µ m )) : k ∈ N , m ∈ { , · · · , n }} k ∈ N , m ∈ { , · · · , n }} cos( l e √ λ ) = − { (2 k + 1) π : k ∈ N ( N − n )0 } { (2 k + 1) π : k ∈ N ( N − n +2)0 } . Table 1Theorem 4.1.
Let graph Γ be an equilateral non-bipartite graph with the transitionmatrix Z , then we have (4.1) λ k +1 ( L st (Γ)) ≥ λ k + N − n ( L a/st (Γ)) , k ∈ N hold if and only if (4.2) µ ≤ − µ n , µ ≤ − µ n − , · · · , µ n − ≤ − µ , µ n ≤ − µ , (4.3) µ ≥ − µ n − , µ ≥ − µ n − , · · · , µ n − ≥ − µ , µ n − ≥ − µ , where µ , µ , · · · , µ n − , µ n are the eigenvalues of the Z .Proof. We can assume that each edge has length l e . Because the arccosine func-tion y = arccos( x ): [-1,1] → [0 , π ], and y is monotonically decreasing in interval[-1,1], the spectra of L st (Γ) and L a/st (Γ) are σ ( L st (Γ)) = ( , (cid:18) arccos( µ ) l e (cid:19) , (cid:18) arccos( µ ) l e (cid:19) , · · · , (cid:18) arccos( µ n ) l e (cid:19) , (cid:18) πl e (cid:19) , · · · , (cid:18) πl e (cid:19) | {z } N − n , (cid:18) π − arccos( µ n ) l e (cid:19) , · · · , (cid:18) π − arccos( µ ) l e (cid:19) , (cid:18) π − arccos( µ ) l e (cid:19) , (cid:18) πl e (cid:19) , · · · , (cid:18) πl e (cid:19) | {z } N − n +2 , · · · ) ,λ k (cid:0) L st (Γ) (cid:1) ≤ λ k +1 (cid:0) L st (Γ) (cid:1) , k ∈ N . HONGJUN W ANG , HONGMEI SONG , AND JIA ZHAO , ∗ σ ( L a/st (Γ)) = ( , · · · , | {z } N − n , (cid:18) π − arccos( µ n ) l e (cid:19) , · · · , (cid:18) π − arccos( µ ) l e (cid:19) , (cid:18) π − arccos( µ ) l e (cid:19) , (cid:18) πl e (cid:19) , · · · , (cid:18) πl e (cid:19) | {z } N − n +2 , (cid:18) π + arccos( µ ) l e (cid:19) , (cid:18) π + arccos( µ n ) l e (cid:19) , (cid:18) πl e (cid:19) , · · · , (cid:18) πl e (cid:19) | {z } N − n , · · · ) ,λ k ( L a/st (Γ)) ≤ λ k +1 ( L a/st (Γ)) , k ∈ N . We now consider the comparison between the eigenvalues of L st (Γ) and L a/st (Γ)in interval (cid:16) , ( πl e ) i . The number of the eigenvalues of L st (Γ) and L a/st (Γ) is samein interval (cid:16) , ( πl e ) i . We consider the following inequality:(4.4) arccos( µ ) ≥ ( π − arccos( µ n )) . The function cos( x ) acts on both sides of inequality (4.4), and since cosine function y = cos( x ) is monotonically decreasing in interval (0 , π ], we have µ ≤ − µ n . For the following inequalities:arccos( µ ) ≥ ( π − arccos( µ n − )) , · · · , arccos( µ n − ) ≥ ( π − arccos( µ )) , arccos( µ n ) ≥ ( π − arccos( µ )) , repeating our analysis, we have µ ≤ − µ n − , · · · , µ n − ≤ − µ , µ n ≤ − µ . Since the arccosine function y = arccos( x ): [-1,1] → [0 , π ], then we have(2 π − arccos( µ n )) ≥ π, (2 π − arccos( µ n − )) ≥ π. We analyze following inequality:(4.5) (2 π − arccos( µ n − )) ≥ ( π + arccos( µ )) , The function cos( x ) acts on both sides of inequality (4.5), and since cosine function y = cos( x ) is monotonically increasing in interval ( π, π ], we have µ n − ≥ − µ . For the following inequalities:(2 π − arccos( µ n − )) ≥ ( π + arccos( µ )) , · · · , (2 π − arccos( µ )) ≥ ( π + arccos( µ n − )) , (2 π − arccos( µ )) ≥ ( π + arccos( µ n − )) , then we have µ n − ≥ − µ , · · · , µ ≥ − µ n − , µ ≥ − µ n − . IGENVALUES OF LAPLACE OPERATORS ON NON-BIPARTITE GRAPHS 11
For the comparison between the eigenvalues of L st (Γ) and L a/st (Γ) in interval (cid:16) ( kπl e ) , ( k +1) πl e ) i , k ∈ N . Because the number of eigenvalues of L st (Γ) and L a/st (Γ) is same in interval (cid:16) ( kπl e ) , ( k +1) πl e ) i , k ∈ N , and according to cos(2 kπ + √ λ ) = cos( √ λ ) , √ λ ∈ (0 , πl e ], the comparison between the eigenvalues of L st (Γ)and L a/st (Γ) in interval (cid:16) ( kπl e ) , ( k +1) πl e ) i , k ∈ N can get the same results as ininterval (cid:16) , ( πl e ) i . Conversely, based on (4.2) and (4.3), we get (4.1) by using theproperties of the arccosine function. We have proven the theorem. (cid:3) For a given finite non-bipartite graph, the eigenvalues of the corresponding transi-tion matrix can be easily calculated. Thus we can easily judge whether the eigenval-ues of the Laplace operators satisfy (4.1) by judging whether the eigenvalues of tran-sition matrix satisfy (4.2) and (4.3). To provide one of the simplest examples, con-sider that Γ is a regular pentagon with edge length 1 which is a cycle with 5 edges.In this case, all eigenvalues of Z are µ = 1 , µ = µ = √ − , µ = µ = − √ − .The values of arccosine function of µ , µ , µ and µ are not clear. It is not easyfor us to directly compare the size between the eigenvalues of L st (Γ) and L a/st (Γ),but the eigenvalues of Z satisfy µ ≤ − µ , µ ≤ − µ , µ ≥ − µ , µ ≥ − µ , we have λ k +1 ( L st (Γ)) ≥ λ k ( L a/st (Γ)) , k ∈ N . We next consider the changing rules of the anti-standard eigenvalues when ver-tices of the metric graphs are increased. Let e Γ be a metric graph obtained by addingsome vertices to each edge of the metric graph Γ, then we have λ k ( L st (Γ)) = λ k ( L st ( e Γ)) , k ∈ N . Corollary 4.2.
Assume that graph Γ is an odd equilateral cycle, and let e Γ be anequilateral bipartite graph obtained by adding some vertices in the middle of eachedge of the graph Γ . If the eigenvalues of the transition matrix Z satisfy (4.2) and (4.3) , then we have λ k +1 ( L a/st ( e Γ)) ≥ λ k ( L a/st (Γ)) , k ∈ N . Proof.
For the graph Γ which is an equilateral non-bipartite graph, if the eigenvaluesof the transition matrix Z satisfy (4.2) and (4.3), then we have λ k +1 ( L st (Γ)) ≥ λ k ( L a/st (Γ)) , k ∈ N . After adding vertices, the graph e Γ is an equilateral bipartite graph, we have λ k ( L st ( e Γ)) = λ k ( L a/st ( e Γ)) , k ∈ N . The following relation between the eigenvalues of L st (Γ) and L st ( e Γ) λ k ( L st (Γ)) = λ k ( L st ( e Γ)) , k ∈ N . Hence λ k +1 ( L a/st ( e Γ)) ≥ λ k ( L a/st (Γ)) , k ∈ N . This completes the proof. (cid:3) HONGJUN W ANG , HONGMEI SONG , AND JIA ZHAO , ∗ Inequalities between standard and anti-standard eigenvalues ona special inequilateral non-bipartite graph
The previous method is only valid for the calculation of the eigenvalues of Laplaceoperators on equilateral graphs. Thus in this section, we will present a differentapproach which solves the eigenvalues of Laplace operators on inequilateral metricgraphs.Since the eigenvalues of L st (Γ) and L a/st (Γ) are always non-negative real num-bers, we have λ = k . We denote 2 N × N global scattering matrix by S ( k ), andby L denote 2 N × N diagonal matrix Diag( l b ), see [6] for details. Lemma 5.1.
Let L (Γ) be a Laplace operator on graph Γ with the global scatteringmatrix, then k ∈ C \{ } is an eigenvalue of L (Γ) with the multiplicity m k if andonly if k is a root of the secular equation (5.1) with the same multiplicity, (5.1) det( I − S ( k ) e ikL ) = 0 . This lemma was proven in [9, Theorem 3.7.1] and [6, Theorem 3.34]. It playsthe crucial role in finding the eigenvalues of L st (Γ) and L a/st (Γ) on inequilateralmetric graphs. The following calculation of the eigenvalues of L st (Γ) and L a/st (Γ)on the special inequilateral non-bipartite graph is based on Lemma 5.1.Let graph Γ be a right triangle with edge lengths 3, 4, 5. We denote the globalscattering matrices corresponding to the standard conditions and the anti-standardconditions by S st and S a/st , then S st = − S a/st .The secular equation corresponding to the standard conditions isdet( I − S st e ikL ) = ( e ik − = 0 , then we have k = π , π , π , π , π , π , π , π , π , · · · . Since 0 is an eigenvalue of L st (Γ) with multiplicity 1, then according to the Lemma5.1, the spectrum of L st (Γ) is σ ( L st (Γ)) = { , ( π , ( π , ( π , ( π , ( π , ( π , ( 2 π ( 2 π , ( 5 π , · · · } . The secular equation corresponding to the anti-standard conditions isdet( I − S a/st e ikL ) = ( e ik + 1) = 0 . Through a simple calculation, we can obtain that k = π , π , π , π , π , π , π , π , π , · · · , and the spectrum of L a/st (Γ) is σ ( L a/st (Γ)) = { ( π
12 ) , ( π
12 ) , ( π , ( π , ( 5 π
12 ) , ( 5 π
12 ) , ( 7 π
12 ) , ( 7 π
12 ) , ( 3 π , · · · } . Let e Γ be an equilateral bipartite graph with edge length 1 obtained by addingsome vertices on each edge of the metric graph Γ, and let e A and e Z denote theadjacency matrix and corresponding transition matrix respectively. The eigenvaluesof e Z are given by 0 , , − , , − , − , , , − √ , − √ , √ , √ , and the eigenvalues of L st ( e Γ) and L a/st ( e Γ) are listed in Table 2,
IGENVALUES OF LAPLACE OPERATORS ON NON-BIPARTITE GRAPHS 13 standard anti-standard λ = 0 { } { } cos √ λ > { (2 kπ ) : k ∈ N } { (2 kπ ) : k ∈ N } sin √ λ > = 0 { (2 kπ ± π ) : k ∈ N }∪ { (2 k ± π ± π ) : k ∈ N }∪{ (2 kπ ± π ) : k ∈ N }∪ { ((2 k ± π ± π ) : k ∈ N }∪{ (2 kπ ± π ) : k ∈ N }∪ { (2 k ± π ± π ) : k ∈ N }∪{ (2 kπ ± π ) : k ∈ N }∪ { (2 k ± π ± π ) : k ∈ N }∪{ (2 kπ ± π ) : k ∈ N } { (2 k ± π ± π ) : k ∈ N } cos √ λ = − { (2 k + 1) π : k ∈ N } { (2 k + 1) π : k ∈ N } . Table 2
Hence the spectra of L st ( e Γ) and L a/st ( e Γ) are σ ( L st ( e Γ)) = σ ( L a/st ( e Γ)) = { , ( π , ( π , ( π , ( π , ( π , ( π , ( 2 π , ( 2 π , ( 5 π , ( 5 π , · · · } . We have the following results: λ k ( L st ( e Γ)) = λ k ( L a/st ( e Γ)) , λ k ( L a/st (Γ)) ≤ λ k +1 ( L a/st ( e Γ)) , k ∈ N . We can verify that eigenvalues of L st (Γ) and L a/st (Γ) satisfy λ k +1 ( L st (Γ)) ≥ λ k ( L a/st (Γ)) , k ∈ N . This research was supported by the Natural Science Foundation of Hebei Provinceunder Grant No. A2019202205. We thank the reviewers for useful comments andsuggestions.
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