Invertibility of graph translation and support of Laplacian Fiedler vectors
aa r X i v : . [ m a t h . F A ] M a r INVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OFLAPLACIAN FIEDLER VECTORS
MATTHEW BEGU´E AND KASSO A. OKOUDJOU
Abstract.
The graph Laplacian operator is widely studied in spectral graph theory largelydue to its importance in modern data analysis. Recently, the Fourier transform and othertime-frequency operators have been defined on graphs using Laplacian eigenvalues and eigen-vectors. We extend these results and prove that the translation operator to the i ’th nodeis invertible if and only if all eigenvectors are nonzero on the i ’th node. Because of thisdependency on the support of eigenvectors we study the characteristic set of Laplacianeigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on largeneighborhoods and then explicitly construct a family of non-planar graphs that do exhibitthis property. Introduction
Preliminaries.
Techniques and methods from spectral graph theory and applied andcomputational harmonic analysis are increasingly being used to analyze, process and makepredictions on the huge data sets being generated by the technological advances of the lastfew decades, e.g., see [20, 5, 17, 10]. At the same time these tasks on large data sets andnetworks require new mathematical technologies which are leading to the golden age ofMathematical Engineering [4, 7].As a result, theories like the vertex-frequency analysis have emerged in an effort to in-vestigate data from both a computational harmonic analysis and spectral graph theoreticalpoint of views [21]. In particular, analogues of fundamental concepts and tools such as time-frequency analysis [22], wavelets [14], sampling theory [1], have been developed in the graphcontext. Much of this is done via spectral properties of the graph Laplacian, more specificallythrough the choice of some eigenbases of the graph Laplacian. However, and to the best ofour knowledge, a qualitative analysis of the effect of this choice on the resulting theory hasnot been undertaken. In this paper we consider such qualitative analysis, focusing on theeffect of the choice of eigenbasis for the graph Laplacian, on the graph translation operatordefined in [22].Throughout this paper we shall consider finite unweighted and undirected graphs. To bespecific, a graph is defined by the pair (
V, E ) where V denotes the set of vertices and E denotes the set of edges. When the vertex and edge set ( V, E ) are clear, we will simplydenote the graph by G . We assume that the cardinality of V is N . Each element in the edgeset E is denoted as an ordered pair ( x, y ) where x, y ∈ V . If ( x, y ) ∈ E , we will often write Date : September 23, 2018.2000
Mathematics Subject Classification.
Primary 94A12, 42C15; Secondary 65F35, 90C22.
Key words and phrases.
Signal processing on graphs, time-vertex analysis, Generalized translation andmodulation, Spectral graph theory, Fiedler vectors.K.A.O was partially supported by a grant from the Simons Foundation ( x ∼ y to indicate that vertex x is connected to y . In such case, we say that y is a neighborof x .A graph is undirected if the edge set is symmetric, i.e., ( x, y ) ∈ E if and only if ( y, x ) ∈ E .In the sequel, we only consider undirected graphs. The graph is simple if there are no self-loops, that is, the edge set contains no edges of the form ( x, x ). Additionally, we assumethat graphs have at most one edge between any two pair of vertices, i.e., we do not allowmultiple-edges between vertices.The degree of vertex x ∈ V in an undirected graph equals the number of edges emminatingfrom (equivalently, to) x and is denoted d x . A graph is called regular if every vertex has thesame degree; it is called k -regular when that degree equals k ∈ N . We refer to [6] for morebackground on graphs.A path (of length m ), denoted p , is defined to be a sequence of adjacent edges, p = { ( p j − , p j ) } mj =1 . We say that the path p connects p to p m . A path is said to be simple if noedge is repeated in it. A graph is connected if for any two distinct vertices x, y ∈ V , thereexists some path connecting x and y .We make the following definition of a ball on a graph that is motivated by the definitionof a closed ball on a metric space. Definition 1.1.
Given any x ∈ V and any integer r ≥
1, we define the ball of radius r centered at x , B r ( x ) = { y ∈ V : d ( x, y ) ≤ r } , where d ( x, y ) is the shorted path length from x to y in G .1.2. The graph Laplacian.
We will consider functions on graphs that take on real (orcomplex) values on the vertices of the graph. Since V = { x i } Ni =1 is finite, it is often useful,especially when doing numerical computations, to represent f : V → R (or C ) as a vector oflength N whose i ’th component equals f ( x i ).Given a finite graph G ( V, E ) the adjacency, the adjacency matrix is the N × N matrix, A , defined by A ( i, j ) = (cid:26) , if x i ∼ x j , otherwise.The degree matrix is the N × N diagonal matrix D whose entries equal the degrees d x i ,i.e., D ( i, j ) = (cid:26) d x i , if i = j , otherwise.The main differential operator that we shall study is L , the Laplacian (Laplace’s operator,or graph Laplacian). The pointwise formulation of the Laplacian applied to a function f : V → C is given by(1.1) Lf ( x ) = X y ∼ x f ( x ) − f ( y ) . The graph Laplacian, L , can be conveniently represented as a matrix, which, by an abuseof notations, we shall also denote by L . It follows from (1.1) that the ( i, j ) th entry of L isgiven by(1.2) L ( i, j ) = d x i if i = j − x i ∼ x j NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 3 or, equivalently, L = D − A . Matrix L is called the unnormalized Laplacian to distinguishit from the normalized Laplacian, L = D − / LD − / = I − D − / AD − / , used in some ofthe literature on graphs, e.g., [6]. However, we shall work exclusively with the unnormalizedLaplacian and shall henceforth just refer to it as the Laplacian.Furthermore, it is not difficult to see that h Lf, f i = X x ∼ y | f ( x ) − f ( y ) | for any f ∈ C N . Consequently, L is a positive semidefinite matrix whose eigenvalues, { λ k } N − k =0 ⊂ [0 , ∞ ) . In addition, if the graph G ( V, E ) is connected the spectrum of the Lapla-cian L is given by 0 = λ < λ ≤ · · · ≤ λ N − . Throughout, we shall denote by Φ the set of orthonormal eigenvectors { ϕ k } N − k =0 . We abusenotations, and view Φ as a N × N orthogonal matrix whose ( k − ϕ k . Note that Φ is not unique, but for the theory that follows, we assume that one has fixedan eigenbasis and hence the matrix Φ is assumed to be fixed.In fact, the following result completely characterizes the relationship between eigenvaluesof a graph and connectedness properties of the graph. Theorem 1.2 ([6]) . If the graph G is connected then λ = 0 and λ i > for all ≤ k ≤ N − .In this case ϕ ≡ / √ N . More generally, if the graph G has m connected components, then λ = λ = · · · = λ m − = 0 and λ k > for all k = m, ..., N − . The indicator function oneach connected component (properly renormalized), forms an orthonormal eigenbasis for the m -dimensional eigenspace associated to eigenvalue 0. As seen from Theorem 1.2, the first nonzero eigenvalue of L is directly related to whetheror not the graph is connected. In fact, λ is known as the algebraic connectivity of the graph,see [12], and is widely studied. Its corresponding eigenvector, ϕ , is known as the Fiedlervector [12, 13] and will be discussed more in-depth in Section 3. If λ has multiplicity 1, thenthe corresponding Fiedler vector is unique up to a sign. The Fiedler vector is used extensivelyin dimensionality reduction techniques [4, 7, 8], data clustering [19], image segmentation [11],and graph drawing [23]. Finally, we observe that the highest λ can be is N , which happensonly for the complete graph in which case the spectrum is { , N, ..., N } .1.3. Outline of the paper.
The rest of the paper is organized as follows. In Section 2 wereview the theory of vertex-frequency analysis on graphs introduced in [22]. We primarilyfocus on the graph translation operator since it has substantial differences to the classicalEuclidean analogue of translation. In general, the translation operator is not invertible inthe graph setting. We prove when the graph translation operator acts as a semigroup andcompletely characterize conditions in which the operator is invertible and derive its inverse.In Section 3 we investigate characteristic sets (sets of zeros) of eigenvectors of the Laplacianbecause it is directly related to the theory of translation developed in Section 2. In particular,we focus on the support of the Fiedler vector of the graph. We prove in Section 3.1 that planargraphs cannot have large neighborhoods of vertices on which the Fiedler vector vanishes. Wethen introduce a family of (non-planar) graphs, called barren graphs, that have arbitrarilylarge neighborhoods on which the Fiedler vector does vanish in Section 3.2. In Section 3.3we prove results about the algebraic connectivity and Fiedler vector of a graph formed byadding multiple graphs.
M.BEGU´E AND K.A.OKOUDJOU Translation operator on graphs
The notions of graph Fourier transform, vertex-frequency analysis, convolution, transla-tion, and modulation operators were recently introduced in [22]. In this section, we focuson the translation operator and investigate certain of its properties including, semi-group(Theorem 2.2), invertibility and isometry (Theorem 2.5).Analogously to the classical Fourier transform on the real line which expressed a function f in terms of the eigenfunctions of the Laplace operator, we define the graph Fourier transform ,ˆ f , of a functions f : V → C as the expansion of f in terms of the eigenfunctions of the graphLaplacian. Definition 2.1.
Given the graph, G , and its Laplacian, L , with spectrum σ ( L ) = { λ k } N − k =0 and eigenvectors { ϕ k } N − k =0 , the graph Fourier transform of f : V → C is by(2.1) ˆ f ( λ k ) = h f, ϕ k i = N X n =1 f ( n ) ϕ ∗ k ( n ) . Notice that the graph Fourier transform is only defined on values of σ ( L ). In particular,one should interpret the notation ˆ f ( λ k ) to designate the inner product of f with the k ’theigenfunction of L . However to emphasize the interplay between the vertex and spectraldomains, we shall abuse the notation as defined here.The graph inverse Fourier transform is then given by(2.2) f ( n ) = N − X k =0 ˆ f ( λ k ) ϕ k ( n ) . It immediately follows from the above definition that Parseval’s equality holds in thissetting as well. Indeed, for any f, g : V → C , then h f, g i = h ˆ f , ˆ g i . Consequently, k f k ℓ = N X n =1 | f ( n ) | = N − X l =0 | ˆ f ( λ ℓ ) | = (cid:13)(cid:13)(cid:13) ˆ f (cid:13)(cid:13)(cid:13) ℓ . Recall that the convolution of two signals f, g ∈ L ( R ) can be defined via the Fouriertransform as \ ( f ∗ g )( ξ ) = ˆ f ( ξ )ˆ g ( ξ ). Using this approach, and by taking the inverse graphFourier transform, (2.2), we can define convolution in the graph domain. For f, g : V → C ,we define the graph convolution of f and g as(2.3) f ∗ g ( n ) = N − X l =0 ˆ f ( λ ℓ )ˆ g ( λ ℓ ) ϕ ℓ ( n ) . Many of the classical time-frequency properties of the convolution including commutativ-ity, distributivity, and associativity hold for the graph convolution, see [22, Proposition 1],and all follow directly from the definition of graph convolution (2.3).For any k = 0 , , ..., N − graph modulation operator M k : C N → C N is defined as(2.4) ( M k f )( n ) = √ N f ( n ) ϕ k ( n ) . Notice that since ϕ ≡ √ N then M is the identity operator. NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 5
An important remark is that in the classical case, modulation in the time domain representstranslation in the frequency domain, i.e., d M ξ f ( ω ) = ˆ f ( ω − ξ ). The graph modulation doesnot exhibit this property due to the discrete nature of the spectral domain. However, it isworthy to notice the special case if ˆ g ( λ ℓ ) = δ ( λ ℓ ), i.e., g is a constant function, then d M k g ( λ ℓ ) = N X n =1 ϕ ∗ ℓ ( n )( M k g )( n ) = N X n =1 ϕ ∗ ℓ ( n ) √ N ϕ k ( n ) 1 √ N = δ ℓ ( k ) . Consequently, if g is the constant unit function, M k g = ϕ k .Formally, the translation of a function defined on C is given by( T u f )( t ) = f ( t − u ) = ( f ∗ δ u )( t ) . Motivated by this example, for any f : V → C we can define the graph translation operator , T i : C N → C N via the graph convolution of the Dirac delta centered at the i ’th vertex:(2.5) ( T i f )( n ) = √ N ( f ∗ δ i )( n ) = √ N N − X k =0 ˆ f ( λ k ) ϕ ∗ k ( i ) ϕ k ( n ) . We can express T i f in matrix notation as follows:(2.6) T i f = √ N ϕ ∗ ( i ) ϕ (1) ϕ ∗ ( i ) ϕ (1) · · · ϕ ∗ N − ( i ) ϕ N − (1)... ... · · · ... ϕ ∗ ( i ) ϕ ( N ) ϕ ∗ ( i ) ϕ ( N ) · · · ϕ ∗ N − ( i ) ϕ N − ( N ) ˆ f ( λ )...ˆ f ( λ N − ) . Graph translation exhibits commutative properties, i.e., T i T j f = T j T i f , and distributiveproperties under the convolution, i.e., T i ( f ∗ g ) = ( T i f ) ∗ g = f ∗ ( T i g ), see [22, Corollary1]. Also, using the definitions of graph convolution, it is elementary to show that for any i, n ∈ { , ..., N } and for any function g : V → C we have T i g ( n ) = T n ¯ g ( i ) . Observe that if we choose real-valued eigenfunction in the definition of the graph Fouriertransform, then we simply have T i g ( n ) = T n g ( i ) . These results can be further generalized in the following theorem.
Theorem 2.2.
Assume G is a graph whose Laplacian has real-valued eigenvectors { ϕ k } N − k =0 .Let α be a multiindex, i.e. α = ( α , α , ..., α K ) where α j ∈ { , ..., N } for ≤ j ≤ K andlet α ∈ { , ..., N } . We let T α denote the composition T α K ◦ · · · T α ◦ T α . Then for any f : V → R , we have T α f ( α ) = T β f ( β ) where β = ( β , ..., β K ) and ( β , β , β , ..., β K ) is anypermutation of ( α , α , ..., α K ) .Proof. There exists a bijection between the collection of all possible T α f ( α ) for | α | = K ,1 ≤ α ≤ N , and the space of ( K + 1)-tuples with values in { , ..., N } . That is, themap that sends T α f ( α ) to ( α , α , ...., α K ) is a bijection. This enables us to define anequivalence relation on the space { , ..., N } K +1 . We write ( a , ..., a K ) ∼ = ( b , ..., b K ) if andonly if T a K ◦ · · · ◦ T a f ( a ) = T b K ◦ · · · ◦ T b f ( b ).By the commutativity of the graph translation operators, ( a , a ..., a K ) ∼ = σ ( a , a ..., a K ) =( a , a ..., a K ), i.e., σ is the permutation (1 , σ i to denote the permu-tation ( i, i + 1). Similarly, ( a , a ..., a K ) ∼ = σ i ( a , a ..., a K ) for any i = 2 , , ..., K − M.BEGU´E AND K.A.OKOUDJOU
We now have that any permutation σ i for i = 1 , ..., K − K − (cid:3) The graph translation operators are distributive with the convolution and the operatorscommute among themselves. However, the niceties end here; other properties of translationon the real line do not carry over to the graph setting. For example, we do not have thecollection of graph translation operators forming a group, i.e., T i T j = T i + j . In fact, wecannot even assert that the translation operators form a semigroup, i.e. T i T j = T i • j forsome semigroup operator • : { , ..., N } × { , ..., N } → { , ..., N } . The following theoremcharacterizes graphs which do exhibit a semigroup structure of the translation operators. Theorem 2.3.
Consider the graph, G ( V, E ) , with real-valued (resp. complex-valued) eigen-vector matrix Φ = [ ϕ · · · ϕ N − ] . Graph translation on G is a semigroup, i.e. T i T j = T i • j for some semigroup operator • : { , ..., N } × { , ..., N } → { , ..., N } , only if Φ = (1 / √ N ) H ,where H is a real-valued (resp. complex-valued) Hadamard matrix.Proof. (a) We first show that graph translation on G is a semigroup, i.e. T i T j = T i • j for some semigroup operator • : { , ..., N } × { , ..., N } → { , ..., N } , if and onlyif √ N ϕ k ( i ) ϕ k ( j ) = ϕ k ( i • j ) for all l = 0 , ..., N −
1. By the definition of graphtranslation, we have, T i T j f ( n ) = N N − X k =0 ˆ f ( λ k ) ϕ ∗ k ( j ) ϕ ∗ k ( i ) ϕ k ( n )and T ℓ f ( n ) = √ N N − X k =0 ˆ f ( λ k ) ϕ ∗ k ( ℓ ) ϕ k ( n ) . Therefore, T i T j f = T i • j f will hold for any function f : V → R if and only if √ N ϕ k ( i ) ϕ k ( j ) = ϕ k ( i • j ) for every k ∈ { , ..., N − } .(b) We show next that √ N ϕ k ( i ) ϕ k ( j ) = ϕ k ( i • j ) for all k = 0 , ..., N − / √ N by the orthonormality ofthe eigenvectors. Assume √ N ϕ k ( i ) ϕ k ( j ) = ϕ k ( i • j ), which, in particular, implies √ N ϕ k ( i ) ϕ k ( i ) = √ N ϕ k ( i ) = ϕ k ( i • i ).Suppose that there exists k ∈ { , ..., N − } such that | ϕ k ( a ) | > / √ N for some a ∈ { , ..., N } . Then √ N | ϕ k ( a ) | > | ϕ k ( a ) | and so a • a = a for some a ∈{ , ..., N } . Note that a = a , if not | ϕ k ( a ) | = √ N | ϕ k ( a ) | which is impossible.Thus, a ∈ { , ..., N } \ { a } . Then since | ϕ k ( a ) | = √ N | ϕ k ( a ) | > | ϕ k ( a ) | > / √ N we can repeat the same argument to assert a • a = a for some a ∈ { , ..., N } \{ a , a } . Repeat this procedure N times give | ϕ k ( n ) | > / √ N for all vertices n ∈{ , ..., N } , which contradicts the notion that k ϕ k k = 1.Therefore we have shown that the graph translation operator forms a semigrouponly if | ϕ k ( n ) | ≤ / √ N for all n = 1 , ..., N and k = 0 , ..., N = 1. But again,since each eigenvector must satisfy k ϕ k k = 1, we can strengthen this condition to | ϕ k ( n ) | = 1 / √ N for all n = 1 , ..., N and k = 0 , ..., N = 1. Since Φ is an orthogonalmatrix, i.e. ΦΦ ∗ = Φ ∗ Φ = I , then Φ = (1 / √ N ) H , where H is a Hadamard matrix. (cid:3) NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 7
Remark 2.4. (a) If we relax the constraint that Φ must be real-valued, we can obtaingraphs with constant-amplitude eigenfunctions that allow the translation operatorsto form a (semi)group. For the cycle graph on N nodes, C N , one can choose Φ equalto the discrete Fourier transform (DFT) matrix, where Φ nm = e − πi ( n − m − /N .Under this construction, we have T i T j = T i + j (mod N ) .(b) It is shown in [3, Theorem 5] that if Φ = (1 / √ N ) H for Hadamard H , then thespectrum of the Laplacian, σ ( L ), must consist entirely of even integers. The authorsof [9] explore graphs with integer spectrum but do not address the case of a spectrumof only even integers.(c) The converse to Theorem 2.3 does not necessarily hold. That is, if the eigenvectormatrix Φ = 1 / √ N H , for a renormalized Hadamard matrix H , then the translationoperators on G need not form a semigroup. For example, consider the real Hadamardmatrix, H , of order 12 given by H = − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − . Then the second and third columns multiplied componentwise equals the vector[1 , − , , − , − , , , − , , , − , − ⊤ , which does not equal any of the columns of H .(d) What kinds of graphs have a Hadamard eigenvector matrix? The authors of [3] provethat if N is a multiple of 4 for which a Hadamard matrix exists, then the completegraph on N vertices, K N , is one such graph.Unlike in the classical case in R d , graph translation is not an isometric operation, i.e., k T i f k = k f k . However, [22, Lemma 1] provides the following estimates:(2.7) | ˆ f (0) | ≤ k T i f k ≤ √ N max k ∈{ , ,...,N − } | ϕ ℓ ( i ) | k f k ≤ √ N max k ∈{ , ,...,N − } k ϕ ℓ k ∞ k f k Furthermore, unlike the Euclidean notion of translation, graph translation need not beinvertible. Theorem 2.5 characterizes all graphs for which the operator T i is not invertible.Additionally, Hadamard matrices appear again in characterizing when graph translation doesact as a unitary operator. Theorem 2.5.
The graph translation operator T i fails to be invertible if and only if thereexists some k = 1 , ..., N − for which ϕ k ( i ) = 0 . In particular, the nullspace of T i has abasis equal to those eigenvectors that vanish on the i ’th vertex. M.BEGU´E AND K.A.OKOUDJOU
Additionally, T i is unitary if and only if | ϕ k ( i ) | = 1 / √ N for all k = 0 , , ..., N − and allgraph translation operators are unitary if and only if √ N Φ is a Hadamard matrix.Proof. By (2.6), the operator T i can be written as the matrix T i = √ N ϕ ∗ ( i ) ϕ (1) ϕ ∗ ( i ) ϕ (1) · · · ϕ ∗ N − ( i ) ϕ N − (1)... ... · · · ... ϕ ∗ ( i ) ϕ ( N ) ϕ ∗ ( i ) ϕ ( N ) · · · ϕ ∗ N − ( i ) ϕ N − ( N ) Φ ∗ =: √ N A i Φ ∗ (2.8)We can compute the rank of T ∗ i T i = N Φ A ∗ i A i Φ ∗ . Since Φ is an N × N matrix of full rank,we can express the rank of T i solely in terms of the matrix A i , i.e.,rank( T i ) = rank( T ∗ i T i ) = rank(Φ A ∗ i A i Φ ∗ ) = rank( A ∗ i A i ) . We can explicitly compute for any indices n, m ∈ { , ..., N } ,( A ∗ i A i )( n, m ) = N X k =1 A i ( k, n ) A i ( k, m ) = N − X k =0 ϕ ∗ n ( k ) ϕ n ( i ) ϕ m ( k ) ϕ ∗ m ( i )= ϕ n ( i ) ϕ ∗ m ( i ) N − X k =0 ϕ ∗ n ( k ) ϕ m ( k )= ϕ n ( i ) ϕ ∗ m ( i ) δ n ( m ) . Hence, A ∗ i A i is a diagonal matrix with diagonal entries ( A ∗ i A i )( n, n ) = | ϕ n ( i ) | . Therefore,(2.9) T ∗ i T i = N N − X k =0 | ϕ k ( i ) | ϕ k ⊗ ϕ ∗ k . Consequently, this proves that rank( T i ) = |{ k : ϕ k ( i ) = 0 }| and hence T i is invertible ifand only if ϕ k ( i ) = 0 for all k .Furthermore, T i T ∗ i = NN P N − k =0 ϕ k ⊗ ϕ ∗ n = I if and only if | ϕ n ( i ) | = 1 / √ N for all n =0 , , ..., N − ϕ k j ( i ) = 0 for { k j } Kj =1 ⊆ { , ..., N − } . Hence, rank( T i ) = N − K .Then for each j ∈ { , ..., K } and any n ∈ { , ..., N } we have T i ϕ k j ( n ) = √ N N − X k =0 ˆ ϕ k j ( λ k ) ϕ ∗ k ( i ) ϕ k ( n ) = √ N ϕ ∗ k j ( i ) ϕ k j ( n ) = 0 . Therefore, ϕ k j is in the null space of T i for every j = 1 , ..., K . Thus { ϕ k j } Kj =1 is a collectionof K orthogonal unit-norm vectors in the null space which has dimension N − rank( T i ) = K ,hence they form an orthonormal basis for the null space of T i which proves the claim aboutthe null space of T i .Finally, if √ N | ϕ n ( i ) | = 1 for all n, i = 1 , ..., N then √ N Φ is Hadamard which concludesproof. (cid:3)
NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 9
Corollary 2.6. If ϕ k ( i ) = 0 for all k = 1 , ..., N − , then the graph translation operator T i is invertible and its inverse is given by T − i = 1 √ N Φ ϕ ∗ (1) ϕ ∗ ( i ) − ϕ ∗ (2) ϕ ∗ ( i ) − · · · ϕ ∗ ( N ) ϕ ∗ ( i ) − ... ... · · · ... ϕ ∗ N − (1) ϕ ∗ N − ( i ) − ϕ ∗ N − (2) ϕ ∗ N − ( i ) − · · · ϕ ∗ N − ( N ) ϕ N − ( i ) − . Proof.
We shall first prove that the inverse to the matrix A i given in (2.8) is given by A − i = ϕ ∗ (1) ϕ ∗ ( i ) − ϕ ∗ (2) ϕ ∗ ( i ) − · · · ϕ ∗ ( N ) ϕ ∗ ( i ) − ... ... · · · ... ϕ ∗ N − (1) ϕ ∗ N − ( i ) − ϕ ∗ N − (2) ϕ ∗ N − ( i ) − · · · ϕ ∗ N − ( N ) ϕ N − ( i ) − . We can then compute A i A − i ( n, m ) = N X k =1 A i ( n, k ) A − i ( k, m ) = N − X k =0 ϕ ∗ k ( i ) ϕ k ( n ) ϕ ∗ k ( m ) ϕ ∗ k ( i ) − = N − X k =0 ϕ k ( n ) ϕ ∗ k ( m ) = δ n ( m ) , and similarly A − i A i ( n, m ) = N X k =1 A − i ( n, k ) A i ( k, m ) = N X k =1 ϕ ∗ n − ( k ) ϕ ∗ n − ( i ) − ϕ m − ( k ) ϕ ∗ m − ( i )= ϕ ∗ n − ( i ) − ϕ ∗ m − ( i ) N X k =1 ϕ ∗ n − ( k ) ϕ m − ( k ) = ϕ ∗ n − ( i ) − ϕ ∗ m − ( i ) δ n ( m ) , which proves A − i A i = A i A − i = I N .Thus we can verify by the orthonormality of Φ that T i T − i = A i Φ ∗ Φ A − i = I N = Φ A − i A i Φ ∗ = T − i T i . (cid:3) Since the invertibility of the graph translation operators depends entirely on when andwhere eigenvectors vanish, Section 3 is devoted to studying the support of graph eigenvectors.
Remark 2.7.
The results of Theorem 2.5 and its corollary are not applicable solely to thegraph translation operators. They can be generalized to a broader class of operators ongraphs, in particular, operators that act as Fourier multipliers. An operator A is a Fouriermultiplier with symbol a if c Af ( ξ ) = ˆ a ( ξ ) ˆ f ( ξ )for some function a defined in the spectral domain.Indeed graph translation is defined as a Fourier multiplier since it is defined as d T i f ( λ k ) = ϕ k ( i ) ˆ f ( λ k ) . Hence, Theorem 2.5 and Corollary 2.6 can be generalized to Fourier multipliers in the fol-lowing way
Corollary 2.8.
Let A be a Fourier multiplier whose action on f : V → C is defined in thespectral domain c Af ( λ k ) = ˆ a ( λ k ) ˆ f ( λ k ) . Then A is invertible if and only if ˆ a ( λ k ) = 0 for all λ = 0 , , ..., N − . Furthermore, its inverse A − will be given by the Fourier multiplier [ A − f ( λ k ) = ˆ a ( λ k ) − ˆ f ( λ k ) . Support of Laplacian Fiedler vectors on graphs
This section proves results about the support of Laplacian eigenvectors on graphs. Inparticular, we characterize and describe the set on which eigenvectors vanish. The Fiedlervector, ϕ , has unique properties that enable us to prove our main result, Theorem 3.9, thatplanar graphs cannot have large regions on which ϕ vanishes. We then construct a family of(non-planar) graphs, called the barren graphs, and prove in Theorem 3.12 that their Fiedlervectors do vanish on large regions. As seen in Theorem 2.5, the support of eigenvectors willinfluence the behavior of the graph translation operators defined in the last section.3.1. The characteristic set of the Fiedler vector.
Let ϕ denote a Fiedler vector for L on G . We can decompose the vertex set, V , into three disjoint subsets, V = V + ∪ V − ∪ V ,where V + = { x ∈ V : ϕ ( x ) > } , V − = { x ∈ V : ϕ ( x ) < } , and V = { x ∈ V : ϕ ( x ) = 0 } .The set V , the set of vertices on which the Fiedler vector vanishes, is referred to in literatureas the characteristic set of the graph [2]. This vertex decomposition is not a unique propertyto the graph G ; any graph can allow multiple such decompositions of the vertex set V . Inthe case that the algebraic connectivity has higher multiplicities, i.e., λ = λ = · · · = λ m for some 2 ≤ m ≤ N −
1, then each ϕ s is a Fiedler vector for 1 ≤ s ≤ m . Futhermore,any linear combination of { ϕ s } ms =1 will also be a Fiedler vector and yield a different vertexdecomposition. Even in the case when the algebraic connectivity of G is simple, then − ϕ is also a Fiedler vector for G . In this case, V + and V − can be interchanged but the set V isunique to G .We wish to describe and characterize the sets V + , V − , and V for graphs. Fiedler provedin [13] that the subgraph induced on the vertices { v ∈ V : ϕ ( v ) ≥ } = V + ∪ V forms aconnected subgraph of G . Similarly, V − ∪ V form a connect subgraph of G . Recently, it wasproved in [24] that we can relax the statement and show that the subgraphs on V + and V − are connected subgraphs of G .The following result guarantees that V + and V − are always close in terms of the shortestpath graph distance. Lemma 3.1.
Let G ( V, E, ω ) with Fiedler vector ϕ inducing the partition of vertices V = V + ∪ V − ∪ V . Then d ( V + , V − ) ≤ .Proof. First consider the case in which V = ∅ . In this case, there necessarily exists an edge e = ( x, y ) with x ∈ V + and y ∈ V − and hence d ( V + , V − ) = 1.Now consider the case in which V = ∅ . Since G is connected we are guaranteed theexistence of some x ∈ V and some y ∼ x with either y ∈ V + or y ∈ V − . Since x ∈ V , wehave(3.1) 0 = λ ϕ ( x ) = Lϕ ( x ) = X z ∼ x ϕ ( z ) − ϕ ( x ) = X z ∼ x ϕ ( z ) . Therefore, (3.1) implies the existence of at least one other neighbor of x , call it y ′ , suchthat ϕ ( y ′ ) has the opposite sign of ϕ ( y ). Hence we have now constructed a path, namely( y, x, y ′ ) connecting V + and V − and the lemma is proved. (cid:3) NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 11
Figure 3.1.
A graph with arbitrarily large set V ( ϕ )Many graphs exhibit the property that eigenvectors ϕ k for large values of k are highlylocalized and vanish on large regions of the graph; see [15] for an experimental excursion onthis phenomenon. It is perhaps a misconception that eigenvectors corresponding to smalleigenvalues, or in particular, the Fiedler vector of graphs have full support. Indeed theFiedler vector of the Minnesota graph never achieves value zero. On the other hand theFiedler vector of the the graph approximations to the Sierpeinski gasket SG n can vanish butonly along the small number of vertices symmetrically in the center of the graph.It was shown in [2], that the cardinality of V can be arbitrarily large. Figure 3.1 showsa family of graphs that yield sets V with arbitrarily large cardinality. The family is a pathgraph P N on an odd number of vertices, except the middle vertex and its edges are duplicatedan arbitrarily large number of times. As evident from Figure 3.1, the set V is not connected;in fact, no vertex in V is connected to any other vertex of V .For the sake of thoroughness we introduce a family of graphs also with arbitrarily large V but that is also connected. We call the family of graphs the generalized ladder graphs , denotedLadder( n, m ). The standard ladder graphs, Ladder( n, n , P n , and the path graph of length 1, P .The graph Ladder( n,
2) resembles a ladder with n rungs. The generalized ladder graphs,Ladder( n, m ), are ladders with n rungs and each rung contains m vertices. Provided thatthe number of rungs, n , is odd, then V will be the middle rung and will clearly be connected.This gives | V | = m . Figure 3.2 shows a generalized ladder graph and its Fiedler vector.The generalized ladder graph provides an example of a graph with a connected charac-teristic set. Observe in Figure 3.2 however, that each vertex in V is connected to at mosttwo vertices. We then pose the question as to whether or not there exist graphs for which avertex V has three or more neighbors all contained in V . It is simpler to state this propertyusing the definition of a graph ball, given in Definition 1.1.The following proposition shows that the Fiedler vector cannot be constant-valued on anyballs within V + and V − . Figure 3.2.
Left: The generalized ladder graph, Ladder(3 , ϕ on Ladder(3 , V consistsof the three vertices making the middle rung of the ladder and contains oneball of three vertices. Proposition 3.2.
Let ϕ be the Fiedler vector for the Laplacian on graph G and suppose B r ( x ) ⊆ V + or B r ( x ) ⊆ V − for r ≥ . Then ϕ cannot be constant-valued on B r ( x ) .Proof. It suffices to prove the claim for r = 1. Without loss of generality, assume B ( x ) ⊆ V + and suppose that ϕ is constant on B ( x ). Then Lϕ ( x ) = X y ∼ x ω x,y ( ϕ ( x ) − ϕ ( y )) = 0 , since y ∼ x implies y ∈ B ( x ) and ϕ is constant on that ball. However, Lϕ ( x ) = λ ϕ ( x ) > λ > ϕ ( x ) > V + . This is a contradiction and the proof is complete. (cid:3) The result of Proposition 3.2 can be formulated in terms on any non-constant eigenvectorof the Laplacian, not just a Fiedler vector.
Corollary 3.3.
Any non-constant eigenvector of the Laplacian, ϕ k , associated with eigen-value λ k > cannot be constant on any ball contained in the positive vertices { i ∈ V : ϕ k ( i ) > } or negative vertices { i ∈ V : ϕ k ( i ) < } associated to that eigenvector.Proof. Suppose there existed a ball B ( x ) ⊆ { i ∈ V : ϕ k ( i ) > } on which ϕ k was constant.Then just as in the previous proof we could calculate Lϕ k ( x ) = X y ∼ x ω x,y ( ϕ k ( x ) − ϕ k ( y )) = 0 , which contradicts Lϕ k ( x ) = λ k ϕ k ( x ) > (cid:3) We wish to extend Proposition 3.2 to the set V . However, as seen in generalized laddergraphs, Ladder( n, m ) for n odd and m >
2, for which V contains a ball of radius 1. Thisball, however, contains 3 vertices (the center vertex and its two neighbors). The next goalis to characterize graphs whose characteristic set V contains a ball of radius 1 containing atleast four vertices. We prove that this is impossible for planar graphs. NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 13
Definition 3.4. A planar graph is a graph whose vertices and edges can be embedded in R with edges intersecting only at vertices.In 1930, Kazimierz Kuratowski characterized all planar graphs in terms of subdivisions. Definition 3.5. A subdivision of a graph G ( V, E ), also referred to as an expansion , is thegraph H ( ˜ V , ˜ E ) where the vertex set is the original vertex set with an added vertex, w , andthe edge set replaces an edge ( u, v ) with the two edges ( u, w ) and ( w, v ). That is, ˜ V = V ∪{ w } and ˜ E = E \ { ( u, v ) } ∪ { ( u, w ) , ( w, v ) } . Theorem 3.6 (Kuratowski’s Theorem, [18]) . A finite graph, G , is planar if and only if itdoes not contain a subgraph that is a subdivision of K or K , , where K is the completegraph on 5 vertices and K , is the complete bipartite graph on six vertices (also known asthe utility graph), see Figure 3.3. A weaker formulation of Kuratowski’s Theorem can be stated in terms of graph minors.
Definition 3.7.
Given an undirected graph G ( V, E ), consider edge e = ( u, v ) ∈ E . Con-tracting the edge e entails deleting edge e and identifying u and v as the same vertex. Theresulting graph H ( ˜ V , ˜ E ) has one fewer edge and vertex as G .An undirected graph is called a minor of G if it can be formed by contracting edges of G . Theorem 3.8 (Wagner’s Theorem, [25]) . A finite graph is planar if and only if it does nothave K or K , as a minor. Because of the importance of K and K , in identifying non-planar graphs, there arereferred to as forbidden minors . Figure 3.3.
The forbidden minors. Left: The complete graph on five vertices.Right: The complete bipartite graph on six vertices.One of the main results in this section shows that planar graphs cannot have large ballscontained in the characteristic set V . Theorem 3.9.
Let G ( V, E ) be a planar graph with Fiedler vector ϕ . Then the zero set of ϕ contains no balls of radius r = 1 with more than three vertices. Proof.
Suppose that V contains a ball, B ( x ), centered at vertex x ∈ V and comprised of atleast four vertices. Without loss of generality, we can assume that the connected componentof V containing x equals B ( x ). If not, then contract edges so that the connected componentof V containing x equals a ball of radius 1. Since | B ( x ) | ≥
4, then we have d x ≥ { y i } d x i =1 denote the neighbors of x . Then as constructed, B ( x ) = { x, y , y , ..., y d x } .By Lemma 3.1, for i = 1 , ,
3, each vertex y i has at least one neighbor in V + and at leastone in V − ; pick one neighbor from each set and denote them p i and n i , respectively. It isproved in [24] that V + and V − are connected subgraphs of G . Therefore, there is a path ofedges that connect p , p , and p (if p = p = p , then this path is empty). We create aminor of G by contracting the path connecting p , p , and p to create one vertex p ∈ V + .Similarly, since V − is connected, we can contract the path connecting n , n , and n , tocreate one vertex n ∈ V − .Consider the subgraph of the now minorized version of G consisting of vertices { x, p, n, y , y , y } .This subgraph is K , , the complete bipartite graph on six vertices since the vertices { x, p, n } are all connected to { y , y , y } . Thus by Wagner’s Theorem, G is not a planar graph, whichis a contradiction. This completes the proof. (cid:3) The result of Theorem 3.9 does not hold for general graphs. We construct a family of(nonplanar) graphs for which V contains a ball with a large number of vertices. Since theset of vertices for which the Fiedler vector vanishes is large, we call this family of graphs the barren graphs . The barren graph with | V | = N + 7 and | V | = N + 1 is denoted Barr( N ).3.2. Construction of the barren graph, Barr ( N ) . The barren graph will be constructedas a sum of smaller graphs.
Definition 3.10.
Let G ( V, E ) and G ( V, E ) be two graphs. The sum of graphs G and G is the graph G ( V, E ) where E = E ∪ E .The barren graph Barr( N ) is defined as follows Definition 3.11.
Let K ( V i , V j ) denote the bipartite complete graph between vertex sets V i and V j , that is, the graph with vertex set V = V i ∪ V j and edge set E = { ( x, y ) : x ∈ V i , y ∈ V j } . For N ≥ N ), is a graph with N + 7 vertices. Let { V i } i =1 denote distinct vertex sets with given cardinalities {| V i |} i =1 = { N, , , , , } . Then thebarren graph is the following graph sum of the 5 complete bipartite graphsBarr( N ) = K ( V , V ) + K ( V , V ) + K ( V , V ) + K ( V , V ) + K ( V , V ) . As constructed, Barr( N ) itself is bipartite; all edges connect the sets V ∪ V ∪ V to V ∪ V ∪ V . Figure 3.4 shows two examples of barren graphs.We shall show that for any N , the Fiedler vector for Barr( N ) vanishes on V ∪ V whichhas cardinality N + 1. Hence, the Fiedler vector for Barr( N ) has support on exactly sixvertices for any N ≥
3. In order to prove this, we explicitly derive the entire spectrum andall eigenvectors of the Laplacian.
Theorem 3.12.
The barren graph, Barr ( N ) , has the spectrum given in Table 1. In partic-ular, the Fiedler vector of Barr ( N ) vanishes on vertices V ∪ V and hence | supp( ϕ ) | = 6 for any N .Proof. Firstly, the graph Barr( N ) is connected and so we have λ = 0 with ϕ ≡ ( N + 7) − / .All other eigenvalues must be positive. NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 15
Figure 3.4.
The barren graph Barr(4) (top) and Barr(6) (bottom). The set V is denoted with N blue dots; the vertex set V is the black vertex in thecenter; the sets V and V are denoted with red dots; the sets V and V aredenoted with green dots.We will next show that the structure and support of the function shown in Figure 3.5 isan eigenvector for two eigenvalues of Barr( N ).One can check upon inspection that the shown function ϕ is orthogonal to the constantfunction. Then, if the function shown in Figure 3.5, call it ϕ , is an eigenvector, then theeigenvalue equation, Lx = λx is satisfied at each vertex. It is easy to verify that Lϕ ( x ) = 0 foreach x ∈ V ∪ V . For x ∈ V or x ∈ V the eigenvalue equation becomes Lϕ ( x ) = 2( b − a ) = λb .For any x ∈ V or x ∈ V , the eigenvalue equation gives Lϕ ( x ) = N a + ( a − b ) = λa . Finally,we also impose that the condition that the eigenvectors are normalized so that k ϕ k = 1.Therefore, the function ϕ shown in Figure 3.5 is an eigenvector of L if and only if the λ k value eigenvector λ λ (cid:0) N + 3 − √ N − N + 9 (cid:1) Figure 3.5 λ y Figure 3.6 λ = · · · = λ N +1 V λ N +2 y Figure 3.6 λ N +3 = λ N +4 N + 1 Figure 3.7 λ N +5 12 (cid:0) N + 3 + √ N − N + 9 (cid:1) Figure 3.5 λ N +6 y Figure 3.6
Table 1.
The spectrum of the barren graph, Barr( N ). The values y , y , y are the roots to the cubic polynomial (3.2). ba a − a − a − b Figure 3.5.
Support and function values for the eigenvectors associated witheigenvalues λ and λ N +5 .following system of equations has a nontrivial solution: a + 2 b = 12( b − a ) = λbN a + ( a − b ) = λa. The first equation is not linear, but we can still solve this system by hand with substitutionto obtain the following two solutions: a = q N − N +9 ∓ ( N − √ N − N +92( N − N +9) b = q N − N +9 ± ( N − √ N − N +9 N − N +9 λ = (cid:0) N + 3 ± √ N − N + 9 (cid:1) . This gives two orthogonal eigenvectors and their eigenvalues.Consider now the vector shown in Figure 3.6 with full support, yet only taking on fourdistinct values.
NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 17 dc cc cdb aaaa
Figure 3.6.
Support and function values for the eigenvectors associated witheigenvalues λ , λ N +2 , and λ N +6 .Similar to the previous example, we obtain a system of equations by imposing the condi-tions k ϕ k = 1, h ϕ, i = 0, and from writing out the eigenvalue equations at each vertex classfrom V , V , V and V which gives: N a + b + 4 c + 2 d = 1 ( k ϕ k = 1) N a + b + 4 c + 2 d = 0 ( ϕ ⊥ a − c ) + ( a − b ) = λa ( Lϕ ( x ) = λϕ ( x ) : x ∈ V ) N ( b −
1) = λb ( Lϕ ( x ) = λϕ ( x ) : x ∈ V )( c − d ) + N ( c − a ) = λc ( Lϕ ( x ) = λϕ ( x ) : x ∈ V ∪ V )2( d − c ) = λd ( Lϕ ( x ) = λϕ ( x ) : x ∈ V ∪ V ) . Again, this system cannot be solved with linear methods. However, by tedious substitutionswe can reduce the system (assuming each of the variables a, b, c, d, λ are nonzero) to solvingfor the roots of the following cubic polynomial in λ :(3.2) λ + ( − N − λ + ( N + 10 N + 15) λ + ( − N − N ) = 0The cubic polynomial x + c x + c x + c = 0 has three distinct real roots if its discriminant,∆ = 18 c c c − c c + c c − c − c , is positive. The discriminant of (3.2) is positivefor all N > y < y < y denote the three positive roots which make up λ , λ N +2 , and λ N +6 , respectively. By substituting back into the system of equations, one canobtain values for a, b, c, d for each of the λ = y , y , y .The roots y , y , y monotonically increase in N . A simple calculation shows that y = 2for N = 3 and y > N >
3. Hence λ < λ = y for all N . Also observe that y < N <
5, so the ordering of the eigenvalues in Table 1 can vary but their values are accurate.One can verify that the three eigenvectors obtained from Figure 3.6 are linearly indepen-dent and orthogonal to each eigenvector derived so far.Consider now the two functions shown in Figure 3.7. The eigenvalue equation gives Lϕ ( x ) = 0 for every except for those x ∈ V ∪ V in which case we have Lϕ ( x ) = ( N + 1) ϕ ( x ).The two functions shown in Figure 3.7 are orthogonal and linearly independent to each otherand every eigenvector derived thus far and hence N + 1 is an eigenvalue of Barr( N ) withmultiplicity two. a − ab − b a − a − b b Figure 3.7.
Support and function values for the eigenvectors associated witheigenvalues λ N +3 and λ N +4 .Finally, we will construct eigenfunctions that are supported only on the N vertices in V .Observe that if a function, f , is supported on V then for any x ∈ V , the eigenvalue equationgives Lf ( x ) = 5 f ( x ) since x neighbors five vertices on which f vanishes. Therefore, Barr( N )has eigenvalue 5. To construct the corresponding eigenbasis, one can choose any orthonormalbasis for the subspace of the N -dimensional vector space that is orthogonal to the constantvector. Any basis for this ( N − V . Finally one can verify by inspection that these N − N ) corresponding to the eigenvalues given in Table 1 (cid:3) As a remark, observe the behavior of the spectrum of Barr( N ) as N → ∞ . For everynatural number N , λ < N →∞ λ = 2. Using a symbolic solver, one can prove thatlim N →∞ λ = lim N →∞ y = 2 as well. Furthermore, the other two roots of the polynomial(3.2) tend to infinity as N → ∞ . Therefore, as N → ∞ , Barr( N ) has spectrum approaching0 (with multiplicity 1), 2 (with multiplicity 2), 5 (with multiplicity N − ∞ .3.3. Characteristic vertices and graph adding.
In this subsection, we prove resultsabout eigenvectors and their characteristic vertices for graph sums as defined in Definition3.10. We borrow the following notation from [24] for clarity.
Definition 3.13.
For any function f , Let i ( f ) = { i ∈ V : f ( i ) = 0 } ,i + ( f ) = { i ∈ V : f ( i ) > } ,i − ( f ) = { i ∈ V : f ( i ) < } . Observe that the set V (resp. V + and V − ) from Section 3.1 is equal to i ( ϕ ) (resp. i + ( ϕ )and i − ( ϕ )). NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 19
Theorem 3.14.
Consider n ≥ connected graphs, { G j ( V j , E j ) } nj =1 and suppose that all n graph Laplacians, L j , share a common eigenvalue λ > with corresponding eigenvectors ϕ ( j ) .Each graph’s vertex set, V j , assumes a decomposition V j = i + ( ϕ ( j ) ) ∪ i − ( ϕ ( j ) ) ∪ i ( ϕ ( j ) ) andsuppose that i ( ϕ ( j ) ) = ∅ for all j . Consider the graph G ( V, E ) = G ( ∪ nj =1 V j , ∪ nj =1 E j ∪ E ) where the edge set E = { ( x i , y i ) } Ki =1 for x i ∈ i ( ϕ j ) , y i ∈ i ( ϕ ℓ ) , and j = ℓ is nonempty.Define ϕ on G by ϕ ( x ) = ϕ ( j ) ( x ) for x ∈ V j . Then, λ is an eigenvalue of G and ϕ is acorresponding eigenvector.Furthermore, if we add the assumption that the common eigenvalue λ > is the algebraicconnectivity, i.e., the lowest nonzero eigenvalue of the graphs G j , then λ is an eigenvalue of G ( V, E ) = G ( ∪ nj =1 V j , ∪ nj =1 E j ∪ E ) but not the smallest positive eigenvalue. Hence, ϕ ( x ) isan eigenvector of G but not its Fiedler vector.Proof. We will verify that Lϕ ( x ) = λϕ ( x ) for every x ∈ V . Every x ∈ V lies in exactly one V j and every edge connecting to x must be in either E j or E . Suppose x contains no edgesfrom E . Then Lϕ ( x ) = L j ϕ ( j ) ( x ) = λϕ ( j ) ( x ) = λϕ ( x ).Suppose instead that x does contain at least one edge from E . Then by construction ofthe set E , we have ϕ ( x ) = 0 and ϕ ( y ) = 0 for all ( x, y ) ∈ E . This allows us to compute Lϕ ( x ) = X y ∼ x ( ϕ ( x ) − ϕ ( y )) = X y ∼ x ( x,y ) ∈ E j ( ϕ ( x ) − ϕ ( y )) + X y ∼ x ( x,y ) ∈ E ( ϕ ( x ) − ϕ ( y ))= L j ϕ ( j ) ( x ) + 0 = λϕ ( j ) ( x ) = λϕ ( x ) . Hence for any vertex in V, the vector ϕ satisfies the eigenvalue equation and the first partof the proof is complete.For the second claim of the theorem, let G ( V , E ) and G ( V , E ) have equal algebraicconnectivities and Fiedler vectors ϕ (1) and ϕ (2) , respectively. We can decompose each ver-tex set into its positive, negative, and zero sets, i.e., V j = i + ( ϕ ( j ) ) ∪ i − ( ϕ ( j ) ) ∪ i ( ϕ ( j ) ).Furthermore, i + ( ϕ ( j ) ) and i − ( ϕ ( j ) ) are connected subgraphs of G j .Now consider the larger graph G ( V, E ). The function ϕ ( x ) := ϕ ( j ) ( x ) for x ∈ V j is aneigenfunction of G by the first part of the theorem. However, now, the sets i + ( ϕ ) and i − ( ϕ )are disconnected. Indeed, let x ∈ i + ( ϕ (1) ) and y ∈ i + ( ϕ (2) ). Then any path connecting x and y must contain an edge in E since all E contains all edges connecting G to G . Andhence any path connect x to y will contain at least two vertices in i ( ϕ ).Then by [24] since i + ( ϕ ) and i − ( ϕ ) are both disconnected, then ϕ cannot be the Fiedlervector of G and λ is not the smallest nonzero eigenvalue. (cid:3) We can prove a stronger statement in the specific case where the graphs share algebraicconnectivity, λ .We can state a generalization of Theorem 3.14 for eigenvectors supported on subgraphsof G . Theorem 3.15.
Consider the graph G ( V, E ) . Let S ⊆ V and let H ( S, E S ) be the resultingsubgraph defined by just the vertices of S . Suppose that ϕ ( S ) is an eigenvector of L S , theLaplacian of subgraph H , with corresponding eigenvalue λ . If E ( S, V \ S ) = E ( i ( ϕ ( S ) ) , V \ S ) ,that is, if all edges connecting graph H to its complement have a vertex in the zero-set of ϕ ( S ) , then λ is an eigenvalue of G with eigenvector ϕ ( x ) = (cid:26) ϕ ( S ) ( x ) x ∈ S x / ∈ S. Proof.
The proof is similar to the proof of Theorem 3.14 in that we will simply verify that Lϕ ( x ) = λϕ ( x ) at every point x ∈ V . For any x in the interior of S , then Lϕ ( x ) = L S ϕ ( S ) ( x ) = λϕ ( S ) ( x ) = λϕ ( x ). For any x in the interior of V \ S , then Lϕ ( x ) = 0 since ϕ vanishes at x and all of its neighbors. For x ∈ δ ( S ) (recall δ ( S ) = { x ∈ S : ( x, y ) ∈ E and y ∈ V \ S } ), we have Lϕ ( x ) = X y ∼ x ( ϕ ( x ) − ϕ ( y )) = X y ∼ xy ∈ S ( ϕ ( x ) − ϕ ( y )) + X y ∼ xy / ∈ S ( ϕ ( x ) − ϕ ( y ))= L S ϕ ( S ) ( x ) + (0 −
0) = λϕ ( S ) ( x ) = λϕ ( x ) , where the term (0 −
0) arises from the fact that ϕ ( y ) = 0 since y / ∈ S and since ( x, y ) ∈ E then by assumption x ∈ i ( ϕ ( S ) ) and hence ϕ ( x ) = 0. The same logic shows that Lϕ ( x ) = 0for x ∈ δ ( V \ S ). Hence, we have shown that Lϕ ( x ) = λϕ ( x ) for every possible vertex x ∈ V and the proof is complete. (cid:3) Theorem 3.15 is interesting because it allows us to obtain eigenvalues and eigenvectors ofgraphs by inspecting for certain subgraphs. Furthermore since the eigenvector is supportedon the subgraph, it is sparse and has a large nodal set.
Example 3.16.
Consider the star graph S N ( V, E ) which is complete bipartite graph between N vertices in one class ( V A ) and 1 vertex in the other ( V B ). Let S be the subgraph formedby any two vertices in V A and the one vertex in V B . Then the resulting subgraph on S isthe path graph on 3 vertices, P . It is known that P has Fiedler vector ϕ ( S ) = ( √ , , −√ λ = 1. Then by Theorem 3.15, the star graph S N has eigenvalue λ = 1 witheigenvector supported on two vertices. In fact, S N contains exactly (cid:0) N (cid:1) path subgraphs allof which contain the center vertex and have ϕ ( S ) as an eigenvector. However, only N − S N has eigenvalue 1 with multiplicity N − References
1. Aamir Anis, Akshay Gadde, and Antonio Ortega,
Towards a sampling theorem for signals on arbitrarygraphs , IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2014.2. RB Bapat and Sukanta Pati,
Algebraic connectivity and the characteristic set of a graph , Linear andMultilinear Algebra (1998), no. 2-3, 247–273.3. S. Barik, S. Fallat, and S. Kirkland, On Hadamard diagonalizable graphs , Linear Algebra and its Appli-cations (2011), no. 8, 1885–1902.4. Mikhail Belkin and Partha Niyogi,
Laplacian eigenmaps for dimensionality reduction and data represen-tation , Neural computation (2003), no. 6, 1373–1396.5. Maria Cameron and Eric Vanden-Eijnden, Flows in complex networks: theory, algorithms, and applica-tion to lennard–jones cluster rearrangement , Journal of Statistical Physics (2014), no. 3, 427–454.6. Fan RK Chung,
Spectral Graph Theory , vol. 92, American Mathematical Soc., 1997.7. Ronald R Coifman and St´ephane Lafon,
Diffusion maps , Applied and Computational Harmonic Analysis (2006), no. 1, 5–30.8. Wojciech Czaja and Martin Ehler, Schroedinger eigenmaps for the analysis of bio-medical data , IEEETrans Pattern Anal Mach Intell. (2013), no. 5, 1274–80.9. Shaun M Fallat, Stephen J Kirkland, Jason J Molitierno, and M Neumann, On graphs whose laplacianmatrices have distinct integer eigenvalues , Journal of Graph Theory (2005), no. 2, 162–174.10. Pedro F Felzenszwalb and Daniel P Huttenlocher, Efficient graph-based image segmentation , Interna-tional journal of computer vision (2004), no. 2, 167–181. NVERTIBILITY OF GRAPH TRANSLATION AND SUPPORT OF FIEDLER VECTORS 21
11. ,
Efficient graph-based image segmentation , International Journal of Computer Vision (2004),no. 2, 167–181.12. Miroslav Fiedler, Algebraic connectivity of graphs , Czechoslovak Mathematical Journal (1973), no. 2,298–305.13. , A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory ,Czechoslovak Mathematical Journal (1975), no. 4, 619–633.14. David K Hammond, Pierre Vandergheynst, and R´emi Gribonval, Wavelets on graphs via spectral graphtheory , Applied and Computational Harmonic Analysis (2011), no. 2, 129–150.15. Steven M Heilman and Robert S Strichartz, Localized eigenfunctions: Here you see them, there you dont ,Notices of the AMS (2010), no. 5, 624–629.16. Wilfried Imrich, Sandi Klavzar, and Douglas F Rall, Topics in graph theory: Graphs and their cartesianproduct , CRC Press, 2008.17. David Kempe, Jon Kleinberg, and ´Eva Tardos,
Maximizing the spread of influence through a socialnetwork , Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery anddata mining, ACM, 2003, pp. 137–146.18. Casimir Kuratowski,
Sur le probl`eme des courbes gauches en topologie , Fundamenta mathematicae (1930), no. 15, 271–283.19. Mark EJ Newman, The structure and function of complex networks , SIAM review (2003), no. 2,167–256.20. Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry Winograd, The pagerank citation ranking:Bringing order to the web. , Tech. report, Stanford InfoLab, 1999.21. David I Shuman, Sunil K Narang, Pascal Frossard, Antonio Ortega, and Pierre Vandergheynst,
Theemerging field of signal processing on graphs: Extending high-dimensional data analysis to networks andother irregular domains , IEEE Signal Processing Magazine (2013), no. 3, 83–98.22. David I Shuman, Benjamin Ricaud, and Pierre Vandergheynst, Vertex-frequency analysis on graphs ,Applied and Computational Harmonic Analysis (2016), no. 2, 260–291.23. Daniel A Spielman, Spectral graph theory , Lecture Notes, Yale University (2009), 740–776.24. John C Urschel and Ludmil T Zikatanov,
Spectral bisection of graphs and connectedness , Linear Algebraand its Applications (2014), 1–16.25. Klaus Wagner, ¨Uber eine eigenschaft der ebenen komplexe , Mathematische Annalen (1937), no. 1,570–590.
Matthew Begu´e, Department of Mathematics, University of Maryland, College Park,MD, 20742 USA
E-mail address : [email protected] Kasso A. Okoudjou, Department of Mathematics, University of Maryland, College Park,MD, 20742 USA
E-mail address ::