The Gel'fand's inverse problem for the graph Laplacian
TThe Gel’fand’s inverse problem for the graphLaplacian
Emilia Bl˚asten, Hiroshi Isozaki, Matti Lassas and Jinpeng Lu
Abstract
We study the discrete Gel’fand’s inverse boundary spectral problem ofdetermining a finite weighted graph. Suppose that the set of vertices of thegraph is a union of two disjoint sets: X = B ∪ G , where B is called the setof the boundary vertices and G is called the set of the interior vertices. Weconsider the case where the vertices in the set G and the edges connecting themare unknown. Assume that we are given the set B and the pairs ( λ j , φ j | B ),where λ j are the eigenvalues of the graph Laplacian and φ j | B are the valuesof the corresponding eigenfunctions at the vertices in B . We show that thegraph structure, namely the unknown vertices in G and the edges connectingthem, along with the weights, can be uniquely determined from the givendata, if every boundary vertex is connected to only one interior vertex andthe graph satisfies the following property: any subset S ⊆ G of cardinality | S | (cid:62) x ∈ S is called an extreme pointof S if there exists a point z ∈ B such that x is the unique nearest point in S from z with respect to the graph distance. This property is valid for severalstandard types of lattices and their perturbations.
1. Introduction
In this paper, we consider the discrete version of the Gel’fand’s inverse boundaryspectral problem, defined for a finite weighted graph and the graph Laplacian on it.We assume that we are given the Neumann eigenvalues of the graph Laplacian andthe values of the corresponding Neumann eigenfunctions at a pre-designated subsetof vertices, called the boundary vertices.The Gel’fand’s inverse boundary spectral problem was originally formulated in[41] for partial differential equations. For partial differential operators, one considersan n -dimensional Riemannian manifold ( M, g ) with boundary and the Neumanneigenvalue problem − ∆ g Φ j ( x ) = ω j Φ j ( x ) , for x ∈ M, (1.1) ∂ ν Φ j | ∂M = 0 , (1.2)where ∆ g is the Laplace–Beltrami operator with respect to the Riemannian metric g on M , and Φ j : M → R are the eigenfunctions corresponding to the eigenvalues ω j ∈ R . In local coordinates ( x i ) ni =1 , the Laplacian has the representation∆ g u = det( g ) − n X i,j =1 ∂∂x i (cid:18) det( g ) g ij ∂∂x j u (cid:19) , (1.3)where g ( x ) = [ g ij ( x )] ni,j =1 , det( g ) = det( g ij ( x )) and [ g ij ] ni,j =1 = g ( x ) − .The Gel’fand’s inverse problem is to find the topology, differential structureand Riemannian metric of ( M, g ) when one is given the boundary ∂M and thepairs ( ω j , Φ j | ∂M ) , j = 1 , , . . . , where ω j are the Neumann eigenvalues and Φ j | ∂M are the Dirichlet boundary values of the corresponding eigenfunctions. Here, the1 a r X i v : . [ m a t h . SP ] F e b igenfunctions Φ j are assumed to form a complete orthonormal family in L ( M ).We review earlier results on this problem and the related problems in Section 1.2.To formulate the discrete Gel’fand’s inverse problem, we consider a finite weightedgraph. We use the following terminology. When X is the set of vertices of a finitegraph, we can declare any subset B ⊆ X to be the set of the boundary vertices,denoted by B = ∂G , and call the set G = X − B the set of the interior vertices of X . This terminology is motivated by inverse problems where one typically aims toreconstruct objects in a set Ω ⊆ R n using observations on the boundary ∂ Ω. In ourcase, we aim to reconstruct objects in a vertex set G ⊆ X from observations on theboundary ∂G .For x, y ∈ X = G ∪ ∂G , we denote x ∼ y if there is an edge in the edge set E connecting x to y , that is, { x, y } ∈ E . Every edge { x, y } ∈ E has a weight g xy = g yx > x ∈ G has a measure µ x >
0. For a function u : G ∪ ∂G → R defined on the whole vertex set, the graph Laplacian ∆ G on G isdefined by (cid:0) ∆ G u (cid:1) ( x ) = 1 µ x X y ∼ xy ∈ G ∪ ∂G g xy (cid:0) u ( y ) − u ( x ) (cid:1) , x ∈ G, (1.4)and the Neumann boundary value ∂ ν u of u is defined by (cid:0) ∂ ν u (cid:1) ( z ) = 1 µ z X x ∼ zx ∈ G g xz (cid:0) u ( x ) − u ( z ) (cid:1) , z ∈ ∂G. (1.5)We consider the Neumann eigenvalue problem − ∆ G φ j ( x ) = λ j φ j ( x ) , for x ∈ G, (1.6) ∂ ν φ j | ∂G = 0 . (1.7)The discrete Gel’fand’s inverse problem is to find the set of interior vertices G , theedge structure of ( G ∪ ∂G, E ) and the weights g, µ , when one is given the boundary ∂G and the pairs ( λ j , φ j | ∂G ) , j = 1 , , . . . , N , N = | G | , where | G | is the number ofelements in G . Here, { φ j } Nj =1 is a complete orthonormal family of eigenfunctionsand their Dirichlet boundary values, φ j | ∂G = (cid:0) φ j ( z ) (cid:1) z ∈ ∂G , are vectors in R | ∂G | .We mention that with suitable choices of g, µ , our definition of the graph Lapla-cian (1.4) includes widely used Laplacians in graph theory, in particular, the com-binatorial Laplacian when g, µ ≡
1, and the transitive Laplacian when g ≡ , µ x =deg( x ). The spectra of these two particular operators are mostly unrelated forgeneral graphs and were usually studied separately.Solving the discrete Gel’fand’s inverse problem is not possible without furtherassumptions due to the existence of isospectral graphs, see [35, 39, 66]. One of themain difficulties we encounter in solving the problem is that the graph Laplaciancan have nonzero eigenfunctions which vanish identically on a part of the graph.This phenomenon, intuitively caused by the symmetry of the graph, can make onepart of the graph invisible to the spectral data measured at another part. Thereforeone needs to impose appropriate assumptions. On one hand, the assumptions haveto break some symmetry of the graph to make the inverse problem solvable, and alsodesignate sufficiently many boundary vertices to measure data on. On the other,the assumptions need to include a large class of interesting graphs besides trees,since trees are already well-understood. In this paper, we introduce the Two-PointsCondition (Assumption 1), and prove that the inverse boundary spectral problemon finite graphs is solvable with this assumption. Our result can be applied to detectlocal perturbations and recover potential functions on periodic lattices ([3, 4]), inparticular, to probe graphene defects from the scattering matrix. We will addresspotential applications in another work. 2e start by defining the notations for undirected simple graphs, where weightson vertices and edges are considered. These weights are related to physical situationswhere graph models are applicable. A graph is generally denoted by a pair (
X, E ) with X being the set of vertices and E being the set of edges between vertices. A graph ( X, E ) is finite if both X and E are finite. A graph is said to be simple if there is at most one edge between anypair of vertices and no edge between the same vertex. For undirected simple graphs,edges are two-element subsets of X . We endow a general graph with the followingadditional structures that affect wave propagation on the graph. Definition 1.1 (Weighted graph with boundary) . We say that G = ( G, ∂G, E, µ, g )is a weighted graph with boundary if the following conditions are satisfied.• G ∪ ∂G is the set of vertices (points), G ∩ ∂G = ∅ ; E is the set of edges. Ele-ments of G are called interior vertices and elements of ∂G are called boundaryvertices. We require ( G ∪ ∂G, E ) to be an undirected simple graph.• µ : G ∪ ∂G → R + is the weight function on the set of vertices.• g : E → R + is the weight function on the set of edges.We use the following terminology. A graph with boundary G is connected (resp. finite ) if ( G ∪ ∂G, E ) is connected (resp. finite). We say that G is strongly connected ,if it is still connected after one removes all edges connecting boundary vertices toboundary vertices (see Definition 2.4). Vertices x and y are connected, denoted by x ∼ y or more precisely { x, y } ∈ E , if there is an edge between them. When x ∼ y ,we denote by g xy , or equivalently g yx , the weight of the edge connecting x to y . Wewrite µ x short for µ ( x ).The degree of a vertex x of G is defined as the number of vertices connected to x by edges in E , denoted by deg E ( x ) or deg G ( x ). The neighbourhood N ( ∂G ) of ∂G is defined by N ( ∂G ) = { x ∈ G : x ∼ z for some z ∈ ∂G } ∪ ∂G. When the weights are not relevant in a specific context, we make use of the notation(
G, ∂G, E ) for an unweighted graph with boundary.
Definition 1.2 (Paths and metric) . Let x, y ∈ G ∪ ∂G . A path of ( G ∪ ∂G, E ) from x to y is a sequence of vertices ( v j ) Jj =0 satisfying v = x , v J = y and v j ∼ v j +1 for j = 0 , . . . , J −
1. The length of the path is J . The distance between x and y ,denoted by d ( x, y ), is the minimal length among all paths from x to y . In otherwords, the distance d ( x, y ) is the minimal number of edges in paths that connect x to y . The distance is defined to be infinite if there is no path from x to y . Anundirected graph ( G ∪ ∂G, E ) can be considered as a discrete metric space equippedwith the distance function d .In our setting, every pair of connected vertices has distance 1, while differentchoices of distances appear in other settings. If the graph sits in a manifold, it ismore natural to use the intrinsic distance of the underlying manifold. For this typeof graphs, additionally with geometric choices of weights, the graph Laplacian (1.4)can be used to approximate the standard Laplacians on the manifold, as long as thegraphs are sufficiently dense ([20, 21, 22, 58]). Definition 1.3.
Given a subset S ⊆ G , we say a point x ∈ S is an extreme pointof S with respect to ∂G , if there exists a point z ∈ ∂G such that x is the uniquenearest point in S from z , with respect to the distance d on ( G ∪ ∂G, E ).3 ssumption 1. We impose the following assumptions on the finite graph ( G, ∂G, E ) .(1) For any subset S ⊆ G with cardinality at least , there exist at least twoextreme points of S with respect to ∂G . We refer to this condition as theTwo-Points Condition.(2) For any z ∈ ∂G and any pair of points x, y ∈ G , if x ∼ z, y ∼ z , then x ∼ y . Observe that Item 2 of Assumption 1 is satisfied if every boundary point isconnected to only one interior point. Hence any graph can be adjusted to satisfyItem 2 by attaching an additional edge to every boundary point and declaring theadded vertices as the new boundary points. We remark that Item 2 is essential forproper wavefront behaviour (see Lemma 3.4).One can view the Two-Points Condition (Item 1 of Assumption 1) as a criterionof choosing appropriate boundary points for solving the inverse boundary spectralproblem. As an intuitive example in the continuous setting, any compact subset ofa square in R has at least two extreme points unless it is a single point set. In thiscase, two extreme points can be chosen by taking a point achieving the maximalheight and a point achieving the minimal height with respect to one edge of thesquare. The boundary points realizing the extreme point condition are the verticalprojections of those two chosen points to the proper edges (see Figure 1). Severaltypes of graphs satisfying the Two-Points Condition are discussed in Section 1.3.Figure 1: Any non-singleton compact subset of the unit square in R has at leasttwo extreme points with respect to the boundary of the square.From now on, let G be a finite weighted graph with boundary. For a function u : G ∪ ∂G → R , its graph Laplacian on G is defined by the formula (1.4). Recallthat the Neumann boundary value of u is defined by the formula (1.5), see e.g.[26, 30]. For u , u : G ∪ ∂G → R , we define the L ( G )-inner product by h u , u i L ( G ) = X x ∈ G µ x u ( x ) u ( x ) . (1.8)For a finite graph with boundary, the function space L ( G ) is exactly the space ofreal-valued functions on G ∪ ∂G equipped with the inner product (1.8). Note thatthe inner product is calculated only on the interior G and not on the boundary ∂G . The main reason for such consideration is that we mostly deal with functions u satisfying ∂ ν u | ∂G = 0, in which case the values of u on ∂G are uniquely and linearlydetermined by the values on G , see (3.2).Let q : G → R be a potential function, and we consider the following Neumanneigenvalue problem for the discrete Schr¨odinger operator − ∆ G + q . ( ( − ∆ G + q ) u ( x ) = λu ( x ) , x ∈ G, λ ∈ R ,∂ ν u | ∂G = 0 . (1.9)Note that all Neumann eigenvalues are real, since the Neumann graph Laplacianis a self-adjoint operator on real-valued functions on G with respect to the innerproduct (1.8) due to Lemma 2.1. 4 efinition 1.4. Let G be a finite weighted graph with boundary, and q : G → R be a potential function. A collection of data ( λ j , φ j | ∂G ) Nj =1 is called the Neumannboundary spectral data of ( G , q ), if • λ j ∈ R , φ j : G ∪ ∂G → R , N = | G | is the number of interior vertices of G ; • the functions φ j are Neumann eigenfunctions with respect to Neumann eigen-values λ j for the equation (1.9), namely( − ∆ G + q ) φ j = λ j φ j , ∂ ν φ j | ∂G = 0; (1.10) • the functions φ j form an orthonormal basis of L ( G ). Remark . There are multiple choices of Neumann boundary spectral data for agiven graph. More precisely, given two choices of Neumann boundary spectral data( λ j , φ j | ∂G ) Nj =1 and (˜ λ j , ˜ φ j | ∂G ) Nj =1 of ( G , q ), they are equivalent if(i) there exists a permutation σ of { , . . . , N } such that ˜ λ σ ( j ) = λ j for all j ;(ii) for any fixed k , there exists an orthogonal matrix O such that˜ φ σ ( i ) | ∂G = X j ∈ L k O ij φ j | ∂G , for all i ∈ L k , where L k = { j | λ j = λ k } and the matrix O is of dimension | L k | .In fact, this is the only non-uniqueness in the choice of Neumann boundaryspectral data. In other words, there is exactly one equivalence class of Neumannboundary spectral data on any given finite weighted graph with boundary, and anyrepresentative of that class is a choice of Neumann boundary spectral data. Wemention that the Neumann boundary spectral data is related to other types of dataon graphs, such as the Neumann-to-Dirichlet map (see [47, 48] for the manifoldcase).Next, we define our a priori data. In order to uniquely determine the graphstructure, not only do we need to know the Neumann boundary spectral data, butsome structures related to the boundary also need to be known. In essence, thisextra knowledge is the number of interior points connected to the boundary, andthe edge structure between the boundary and its neighbourhood. Definition 1.6.
Let G , G be two finite graphs with boundary. We say that G , G are boundary-isomorphic , if there exists a bijection Φ : N ( ∂G ) → N ( ∂G ) with thefollowing properties:(i) Φ | ∂G : ∂G → ∂G is bijective;(ii) for any z ∈ ∂G , y ∈ N ( ∂G ), we have y ∼ z if and only if Φ ( y ) ∼ Φ ( z ),where ∼ denotes the edge relation of G .We call Φ a boundary-isomorphism . Definition 1.7.
Let G , G be two finite weighted graphs with boundary, and q, q be real-valued potential functions on G, G . We say ( G , q ) is spectrally isomorphic to ( G , q ) (with a boundary-isomorphism Φ ), if(i) there exists a boundary-isomorphism Φ : N ( ∂G ) → N ( ∂G );(ii) the Neumann boundary spectral data of ( G , q ) and ( G , q ) have the samenumber of eigenvalues counting multiplicities;(iii) there exists a choice of Neumann boundary spectral data of ( G , q ) and ( G , q ),such that λ j = λ j and φ j | ∂G = φ j ◦ Φ | ∂G for all j .Now we state our main results, Theorems 1 and 2.5 heorem 1. Let G = ( G, ∂G, E, µ, g ) , G = ( G , ∂G , E , µ , g ) be two finite, stronglyconnected, weighted graphs with boundary satisfying Assumption 1. Let q, q bereal-valued potential functions on G, G . Suppose ( G , q ) is spectrally isomorphic to ( G , q ) with a boundary-isomorphism Φ . Then there exists a bijection Φ : G ∪ ∂G → G ∪ ∂G such that(1) Φ | ∂G = Φ | ∂G ,(2) for any pair of vertices x , x of G , we have x ∼ x if and only if Φ( x ) ∼ Φ( x ) .Remark. It may happen that Φ and Φ differ on G ∩ N ( ∂G ), for example if thereexist points y , y ∈ G ∩ N ( ∂G ) which are connected to the same set of boundaryvertices but connected to different parts in the interior. Theorem 2.
Take the assumptions of Theorem 1, and identify vertices of G withvertices of G via the bijection Φ . Assume furthermore that µ z = µ z , g xz = g xz forall z ∈ ∂G , x ∈ G , where µ , g denote the weights of G . Then the following twoconclusions hold.(1) If µ = µ , then g = g and q = q .(2) If q = q = 0 , then µ = µ and g = g .In particular, if µ = deg G and µ = deg G , then g = g and q = q . The Gel’fand’s inverse problem [41] for partial differential equations has been aparadigm problem in the study of the mathematical inverse problems and imagingproblems arising from applied sciences. The combination of the boundary controlmethod, pioneered by Belishev on domains of R n and by Belishev and Kurylev onmanifolds [15], and the Tataru’s unique continuation theorem [67] gave a solutionto the inverse problem of determining the isometry type of a Riemannian manifoldfrom given boundary spectral data. Generalizations and alternative methods tosolve this problem have been studied e.g. in [1, 10, 14, 24, 44, 50, 53, 55], seeadditional references in [12, 47, 54]. The inverse problems for the heat, wave andSchr¨odinger equations can be reduced to the Gel’fand’s inverse problem, see [10, 47].In fact, all these problems are equivalent, see [48]. Also, for the inverse problemfor the wave equation with the measurement data on a sufficiently large finite timeinterval, it is possible to continue the data to an infinite time interval, which makesit possible to reduce the inverse problem to the Gel’fand’s inverse problem, see[47, 52]. The stability of the solutions of these inverse problems have been analyzedin [1, 18, 23, 38, 63]. Numerical methods to solve the Gel’fand’s inverse problemshave been studied in [13, 36, 37]. The inverse boundary spectral problems havebeen extensively studied also for elliptic equations on bounded domains of R n . Inthis setting, the Gel’fand’s problems can be solved by reducing it, see [59, 60], toCalder´on’s inverse problems for elliptic equations that were solved using complexgeometrical optics, see [65].An intermediate model between discrete and continuous models is the quantumgraphs, namely graphs equipped with differential operators defined on the edges.In this model, a graph is viewed as glued intervals, and the spectral data that aremeasured are usually the spectra of differential operators on edges subject to theKirchhoff condition at vertices. For such graphs, two problems have attracted muchattention. In the case where one uses only the spectra of differential operatorsas data, Yurko ([69, 70, 71]) and other researchers ([5, 19, 51]) have developed socalled spectral methods to solve inverse problems. Due to the existence of isospectral6igure 2: Left:
Finite hexagonal lattice. The white vertices are considered to bethe boundary vertices for the set of the blue (interior) vertices.
Right:
A finitehexagonal lattice with one blue edge removed. Theorems 1 and 2 show that the exactstructure of such graphs can be uniquely recovered from the boundary spectral data.trees, one spectrum is not enough to determine the operator and therefore multiplemeasurements are necessary. It is known in [71] that the potential can be recoveredfrom appropriate spectral measurements of the Sturm-Liouville operator on anyfinite graph. An alternative setting is to consider inverse problems for quantumgraphs when one is given the eigenvalues of the differential operator and the values ofthe eigenfunctions at some of the nodes. Avdonin and Belishev and other researchers([5, 6, 7, 8, 11, 16]) have shown that it is possible to solve a type of inverse spectralproblem for trees (graphs without cycles). With this method, one can recover boththe tree structures and differential operators.In this paper, we consider inverse problems in the purely discrete setting, thatis, for the discrete graph Laplacian. In this model, a graph is a discrete metric spacewith no differential structure on edges. The graph can be additionally assigned withweights on vertices and edges. The spectrum of the graph Laplacian on discretegraphs is an object of major interest in discrete mathematics ([26, 35, 64]). It iswell-known that the spectrum is closely related to geometric properties of graphs,such as the diameter (e.g. [28, 29, 30]) and the Cheeger constant (e.g. [25, 27, 40]).There were inverse problems, especially the inverse scattering problem, consideredon periodic graphs (e.g. [2, 45, 49]). However, due to the existence of isospectralgraphs ([35, 39, 66]), few results are known regarding the determination of the exactstructure of a discrete graph from spectral data.There have been several studies with the goal of determining the structure orweights of a discrete weighted graph from indirect measurements in the field of in-verse problems. These studies mainly focused on the electrical impedance tomogra-phy on resistor networks ([17, 34, 57]), where electrical measurements are performedat a subset of vertices called the boundary. However, there are graph transforma-tions which do not change the electrical data measured at the boundary, such aschanging a triangle into a Y-junction, which makes it impossible to determine theexact structure of the inaccessible part of the network in this way. Instead, thefocus was to determine the resistor values of given networks, or to find equivalenceclasses of networks (with unknown topology) that produce a given set of boundarydata ([31, 32, 33]).
As primary examples, we consider several standard types of graphs satisfying theTwo-Points Condition (Item 1 of Assumption 1).
Example 1.
All finite trees satisfy the Two-Points Condition, with the boundaryvertices being all vertices of degree 1.This fact can be shown as follows. Recall that a tree is a connected graphcontaining no cycles. For any subset S ⊆ G with | S | (cid:62)
2, pick a point O ∈ G such7igure 3: Left:
Finite triangular lattice. The white vertices are the boundaryvertices; the blue vertices are the interior vertices.
Right:
Finite two-level squareladder, made out of two layers of square lattices.that S does not lie on the same branch from O . Take any point x ∈ S on any branchfrom O , and consider the subtree starting from x in that branch. If the subtree doesnot intersect with S , then any boundary point on this subtree realizes the extremepoint condition for x . Otherwise if the subtree intersects with S at another point x ∈ S , then we consider the subtree starting from x . Repeat this procedure untilwe find a subtree which does not intersect with S , and the procedure stops in finitesteps since the graph is finite. This procedure on two different branches from O gives two extreme points, which yields the Two-Points Condition.For general cyclic graphs, it is often not easy to see if the Two-Points Condition issatisfied. The following proposition shows a concrete way to test for the Two-PointsCondition. Proposition 1.8.
For a finite graph with boundary ( G, ∂G, E ) , the Two-PointsCondition follows from the existence of a function h : G ∪ ∂G → R satisfying thefollowing conditions:(1) the Lipschitz constant of h is bounded by , i.e. if x ∼ y then | h ( x ) − h ( y ) | (cid:54) ;(2) | N ± ( x ) | = 1 for all x ∈ G , and | N ± ( z ) | (cid:54) for all z ∈ ∂G , where N + ( x ) = { y ∈ G ∪ ∂G : y ∼ x , h ( y ) = h ( x ) + 1 } ,N − ( x ) = { y ∈ G ∪ ∂G : y ∼ x , h ( y ) = h ( x ) − } . We call N + ( x ) the discrete gradient of h at x , and N − ( x ) the discrete gradient of − h at x .Proof. For any S ⊆ G with | S | (cid:62)
2, take the points where the function h achievesmaximum and minimum in S . Let x ∈ S be any maximal point. By condition(2), we can take the unique path, denoted by γ x , starting from x such that eachstep increases the function h by 1. This path γ x can only pass each point of thegraph at most once, and therefore the path must end somewhere since the wholegraph is finite. Let z be the point where the path γ x ends. By constructionwe know | N + ( z ) | = 0, which indicates z ∈ ∂G by condition (2). Observe that h ( z ) − h ( x ) (cid:62) d ( x , z ) as γ x may not be distance-minimizing.We claim that x is the unique nearest point in S from z (i.e. x is an extremepoint of S ). Suppose not, and there exists another x ∈ S, x = x such that d ( x , z ) (cid:54) d ( x , z ). Then condition (1) implies that | h ( x ) − h ( z ) | (cid:54) d ( x , z ) (cid:54) d ( x , z ) . (1.11)We claim that the two equalities in (1.11) cannot hold at the same time. Supposeboth equalities are achieved. We take the shortest path from z to x , and then thelength of this path is equal to d ( x , z ) by the second equality. The function h can8nly change d ( x , z ) times along the shortest path from z to x , and hence everychange must be decreasing by 1 in order to make both equalities hold. However,there can only exist one such path starting from z due to condition (2), which isexactly the backward direction from z to x . Along this path, x would be reachedin exactly d ( x , z ) steps and hence x = x . Hence the equalities in (1.11) cannothold at the same time, and we have h ( x ) > h ( z ) − d ( x , z ). On the other hand,we already know h ( x ) (cid:54) h ( z ) − d ( x , z ) indicating h ( x ) > h ( x ), which is acontradiction to the maximality of x . This shows x is an extreme point of S .The same argument shows that any minimal point is also an extreme point of S . Therefore the Two-Points Condition follows from the condition | S | (cid:62) Example 2.
Finite square, hexagonal (Figure 2, Left), triangular (Figure 3, Left),graphite and square ladder (Figure 3, Right) lattices satisfy the Two-Points Condi-tion with the set of boundary vertices being the domain boundary.We can apply Proposition 1.8 to verify this. For the square, hexagonal, graphiteand square ladder lattices, the function h can be chosen as the standard heightfunction with respect to a proper floor. Note that for these lattices, it suffices tochoose only the floor and the ceiling as the boundary. For triangular lattices, thefunction h can be constructed as a group action, such that h changes by 1 / Example 3.
In the finite square, hexagonal, triangular, graphite and square ladderlattices, any horizontal edges can be removed and the obtained graphs still satisfythe Two-Points Condition. Here, the horizontal edges refer to the edges in the non-gradient directions with respect to the function h . See Figure 2 (Right). This isbecause removing such edges does not affect the conditions for the function h inProposition 1.8.A finite square lattice with an interior vertex and all its edges removed also satis-fies the Two-Points Condition. Essentially, one can repeat the proof of Proposition1.8 to show this particular situation. However, it is necessary to choose all foursides as the boundary and use two different choices of the function h . (Intuitivelyspeaking, removing one small square does not affect the ability to find maximal andminimal points in at least two directions.) More generally, removing one square ofany size from a finite square lattice does not affect the Two-Points Condition.However, the Two-Points Condition does not hold (without declaring more bound-ary vertices) if one adds an additional vertex and connects it to any interior vertexwith an additional edge. This is because the set of the two endpoints of that addi-tional edge violates the Two-Points Condition. Example 4.
Assume that a function h : G ∪ ∂G → R satisfies the conditions(1, 2) in Proposition 1.8 for a finite graph with boundary ( G, ∂G, E ). Then one canadd to the graph (
G, ∂G, E ) additional edges { x, y } that connect any two vertices x, y ∈ G ∪ ∂G satisfying | h ( x ) − h ( y ) | <
1. Similarly, one can remove from the graphany edges { x, y } that connect vertices x, y ∈ G ∪ ∂G satisfying | h ( x ) − h ( y ) | < h satisfy the conditions (1, 2) in Proposition1.8, and hence the Two-Points Condition. This procedure can be used, for example,to add additional horizontal edges in the finite hexagonal lattice in Figure 2. Example 5.
Graphs that satisfy the conditions given in Proposition 1.8 can beconnected together so that these conditions stay valid. To do this, assume thatreal-valued functions h and h , defined on disjoint finite graphs with boundary( G , ∂G , E ) and ( G , ∂G , E ), satisfy the conditions (1, 2) in Proposition 1.8,respectively. Moreover, assume that there are c ∈ R and ordered sets X = { x , x , . . . , x k } ⊆ G ∪ ∂G and X = { x , x , . . . , x k } ⊆ G ∪ ∂G , such that | h ( x j ) + c − h ( x j ) | < j = 1 , , . . . , k . (In particular, such sets and c alwaysexist for k = 1.) Then we consider ( G, ∂G, E ) for G = G ∪ G , ∂G = ∂G ∪ ∂G E = E ∪ E ∪ E , where E = (cid:8) { x j , x j } : j = 1 , , . . . , k (cid:9) . We define afunction h on G ∪ ∂G by h ( x ) = ( h ( x ) , for x ∈ G ∪ ∂G ,h ( x ) + c, for x ∈ G ∪ ∂G . (1.12)Then this function h satisfies the conditions (1, 2) in Proposition 1.8 for ( G, ∂G, E ),and therefore the graph (
G, ∂G, E ) satisfies the Two-Points Condition. Figure 3 isa special case of this example.This paper is organized as follows. We introduce relevant definitions and basicfacts in Section 2. In Section 3, we define the discrete wave equation and study thewavefront propagation. Section 4 is devoted to proving our main results.
2. Preliminaries
In this section, let G = ( G, ∂G, E, µ, g ) be a finite weighted graph with boundary.First, we derive an elementary but important Green’s formula.
Lemma 2.1 (Green’s formula) . For two functions u , u : G ∪ ∂G → R , we have X x ∈ G µ x (cid:0) u ( x )∆ G u ( x ) − u ( x )∆ G u ( x ) (cid:1) = X z ∈ ∂G µ z (cid:0) u ( z ) ∂ ν u ( z ) − u ( z ) ∂ ν u ( z ) (cid:1) . Proof.
By definition (1.4), X x ∈ G µ x u ( x )∆ G u ( x ) = X x ∈ G X y ∼ xy ∈ G ∪ ∂G g xy u ( x ) (cid:0) u ( y ) − u ( x ) (cid:1) = X x ∈ G (cid:0) X y ∼ x,y ∈ G + X y ∼ x,y ∈ ∂G (cid:1) g xy u ( x ) (cid:0) u ( y ) − u ( x ) (cid:1) . Observe that the indices and summations can be switched in the following way: X x ∈ G X y ∼ x,y ∈ G g xy u ( x ) u ( y ) = X y ∈ G X x ∼ y,x ∈ G g yx u ( y ) u ( x ) = X x ∼ y,x ∈ G X y ∈ G g yx u ( y ) u ( x ) . Hence the summation over x, y ∈ G cancels out, and we get X x ∈ G µ x (cid:0) u ( x )∆ G u ( x ) − u ( x )∆ G u ( x ) (cid:1) = X x ∈ G X y ∼ x,y ∈ ∂G g xy (cid:0) u ( x ) u ( y ) − u ( x ) u ( y ) (cid:1) , where we have used the fact that the weights are symmetric: g xy = g yx . Then thelemma follows from (1.5) and the following identity: u ( x ) u ( y ) − u ( x ) u ( y ) = u ( y ) (cid:0) u ( x ) − u ( y ) (cid:1) + u ( y ) (cid:0) u ( y ) − u ( x ) (cid:1) . Next we consider the boundary distance functions and the closely related resolv-ing sets of a graph, see [43, 62] and their generalizations in [42].
Definition 2.2.
Let (
G, ∂G, E ) be a finite connected graph with boundary. We say ∂G = { z i } m − i =0 is a resolving set for ( G, ∂G, E ), if the boundary distance coordinate (cid:0) d ( · , z ) , d ( · , z ) , · · · , d ( · , z m − ) (cid:1) : G → R m is injective, where d denotes the distance on ( G ∪ ∂G, E ).For any point x ∈ G , we denote by r x the boundary distance function r x : ∂G → R , r x ( z ) = d ( x, z ) . (2.1)The set of boundary distance functions of a graph G is denoted by R ( G ). If ∂G isa resolving set, the map x r x from G to R ( G ) is a bijection.10he minimal cardinality of resolving sets for a graph is called the metric dimen-sion of the graph [43]. The boundary distance functions are extensively used in thestudy of inverse problems on manifolds, see e.g. [46, 47, 56].The concept of resolving sets gives a rough idea on how to choose boundarypoints such that the inverse problem may be solvable. If the chosen boundarypoints do not form a resolving set, then there is little hope to solve the inverseproblem from spectral data measured at those points. Lemma 2.3.
The Two-Points Condition (Item 1 of Assumption 1) implies that ∂G is a resolving set for ( G, ∂G, E ) .Proof. Suppose that ∂G is not a resolving set for ( G, ∂G, E ). Then by Definition 2.2,there exist two points x , x ∈ G such that d ( x , z ) = d ( x , z ) for all z ∈ ∂G .However, the set S = { x , x } is a contradiction to the Two-Points Condition for( G, ∂G, E ).We point out that the Neumann spectral data for the equation (1.9) are notaffected at all by edges between boundary points, since the Neumann boundaryvalue (1.5) only counts edges to interior points. In other words, the edges betweenboundary points are invisible to our Neumann spectral data. However, this limi-tation does not matter to us since the structure of the boundary is a priori given.What we will reconstruct in the next few sections is actually the reduced graph of G , which is defined as follows. Definition 2.4 (Reduced graph) . Let G = ( G, ∂G, E, µ, g ) be a weighted graphwith boundary. The reduced graph of G is defined as G re = ( G, ∂G, E re , µ, g | E re ) , where E re = E − (cid:8) { x, y } ∈ E (cid:12)(cid:12) x ∈ ∂G and y ∈ ∂G (cid:9) . A graph with boundary being strongly connected is equivalent to its reduced graphbeing connected. Note that G and G re have identical Neumann spectral data dueto the definition of the Neumann boundary value (1.5).Reducing a graph affects distances as paths along edges between boundary pointsbecome forbidden. In the same way as Definition 1.2, the distance d re ( x, y ) onthe reduced graph ( G ∪ ∂G, E re ) is defined through paths of ( G ∪ ∂G, E re ) from x to y , instead of along paths of the original graph ( G ∪ ∂G, E ). Then clearly d re ( x, y ) (cid:62) d ( x, y ) for any x, y ∈ G ∪ ∂G . The change of distances also affects the r -neighbourhood N re ( x, r ) of x ∈ G ∪ ∂G , which is defined by N re ( x, r ) = (cid:8) y ∈ G ∪ ∂G (cid:12)(cid:12) d re ( y, x ) (cid:54) r (cid:9) . However, reducing a graph does not affect the Two-Points Condition.
Lemma 2.5. If x ∈ S is an extreme point of a subset S ⊆ G realized by some z ∈ ∂G , then there exists z ∈ ∂G also realizing the extreme point condition of x such that none of the shortest paths from x to z pass through any other boundarypoint.As a consequence, if ( G, ∂G, E ) satisfies the Two-Points Condition (Item 1 ofAssumption 1), then so does ( G, ∂G, E re ) with respect to its distance function d re .Proof. If any shortest path from x to z passes through another boundary point z ∈ ∂G , then x is an extreme point also realized at z . Then we consider the setof all the boundary points with respect to which x is an extreme point, and take apoint z (not necessarily unique) in this set with the minimal distance from x . Itfollows that z is the desired boundary point; otherwise there is another boundarypoint in the set with a smaller distance from x .Let x ∈ S be an extreme point of S with respect to ∂G realized by z ∈ ∂G .By the argument above, we may assume that none of the shortest paths from x z pass through any boundary point except for z . Reducing the graph will notaffect this path or its length. On the other hand, no distances between points maydecrease in the reduction. So this path is still the shortest path between S and z in the reduced graph. Hence x is also an extreme point of S with respect to ∂G inthe reduced graph.
3. Wave Equation
Definition 3.1 (Time derivatives) . For a function u : G × N → R , we define thediscrete first and second time derivatives at ( x, t ) by D t u ( x, t ) = u ( x, t + 1) − u ( x, t ) , t (cid:62) ,D tt ( x, t ) = u ( x, t + 1) − u ( x, t ) + u ( x, t − , t (cid:62) . These are sometimes called the forward difference and the second-order centraldifference in time.We consider the following initial value problem for the discrete wave equationwith the Neumann boundary condition: D tt u ( x, t ) − ∆ G u ( x, t ) + q ( x ) u ( x, t ) = 0 , x ∈ G, t (cid:62) ,∂ ν u ( x, t ) = 0 , x ∈ ∂G, t (cid:62) ,D t u ( x,
0) = 0 , x ∈ G,u ( x,
0) = W ( x ) , x ∈ G ∪ ∂G, (3.1)where the values of u on ∂G are uniquely determined by the values on G via theNeumann boundary condition at each time step. More precisely, the definition ofthe Neumann boundary value (1.5) yields that u ( z ) = X x ∼ zx ∈ G g xz u ( x ) . X x ∼ zx ∈ G g xz , z ∈ ∂G. (3.2)We require ∂ ν W | ∂G = 0 for the compatibility of the initial value and Neumannboundary condition. The initial conditions and the boundary condition imply that u ( x,
1) = u ( x,
0) = W ( x ) for all x ∈ G ∪ ∂G . Definition 3.2 (Waves) . Given W : G ∪ ∂G → R satisfying ∂ ν W | ∂G = 0, denoteby u W : ( G ∪ ∂G ) × N → R the solution of the discrete wave equation (3.1) withthe initial condition u ( · ,
0) = W ( · ) on G ∪ ∂G . The function W is called the initialvalue . In this paper, a wave refers to a solution of the wave equation (3.1).The discrete wave equation has a unique solution and it is solved in the followingway: the solution on G ∪ ∂G at times t = 0 and t = 1 are determined by the initialconditions. Afterwards, the value on G at time t (cid:62) G ∪ ∂G ) × { t − } and on G × { t − } by the equation D tt u − ∆ G u + qu = 0.Then the formula (3.2) gives the value on ∂G at time t .The main purpose of this section is to prove a wavefront lemma which will be usedfrequently in the next section. Item 2 of Assumption 1 is essential for the wavefrontlemma, as the wave propagation may “speed up” due to the instantaneous effect ofthe boundary condition if a shortest path goes through the boundary. Under Item 2of Assumption 1, distances of the reduced graph are realized by avoiding boundarypoints, which is essential to guarantee proper wave behaviour. Lemma 3.3.
Let G be a finite connected graph with boundary satisfying Item 2 ofAssumption 1. Suppose the reduced graph of G is connected. Let x ∈ G and z ∈ ∂G .If x ∼ z , then d re ( x, p ) (cid:54) d re ( z, p ) for any p ∈ G ∪ ∂G − { z } . roof. Let x ∈ G be a point such that x ∼ z and d re ( x , p ) = d re ( z, p ) −
1. Thispoint exists since distances are realised by paths in a connected graph. Such apoint x cannot be in ∂G , because there are no edges between boundary pointsin the reduced graph. We have x, x ∈ G and x ∼ z , x ∼ z . Then by Item 2of Assumption 1, we have x ∼ x if x = x . Hence the triangle inequality yieldsthat d re ( x, p ) (cid:54) d re ( x, x ) + d re ( x , p ) = d re ( z, p ). If x = x , then d re ( x, p ) = d re ( z, p ) − < d re ( z, p ).Recall that the Neumann boundary value (1.5) does not take into account theedges between boundary points. This means that waves cannot propagate from oneboundary point to another without going through the interior. Hence the wavefrontpropagates by the distance function d re of the reduced graph, instead of the distancefunction of the original graph. Lemma 3.4 (Wavefront) . Let G be a finite connected weighted graph with boundarysatisfying Item 2 of Assumption 1. Suppose the reduced graph of G is connected.Let x ∈ G, z ∈ ∂G and t = d re ( x, z ) . Suppose W : G ∪ ∂G → R is an initial valuesatisfying ∂ ν W | ∂G = 0 , W ( z ) = 0 and (cid:8) y ∈ N re ( z, t ) ∩ G (cid:12)(cid:12) W ( y ) = 0 (cid:9) ⊆ { x } . (3.3) Then the following properties hold for the wave u W generated by W :(1) If W ( x ) > , then t (cid:62) , u W ( z, t ) > and u W ( z, t ) = 0 for all t < t ;(2) If W ( x ) = 0 , then u W ( z, t ) = 0 for all t (cid:54) t .Proof. Let us prove the first claim of the lemma with W ( x ) >
0. To start with,we show that t = d re ( x, z ) (cid:62)
2. Suppose d re ( x, z ) = 1. Let x, y , . . . , y J ∈ G bethe interior points connected to z ∈ ∂G . The boundary conditions W ( z ) = 0 and ∂ ν W ( z ) = 0 imply that X y ∼ zy ∈ G g yz W ( y ) = 0 . On the other hand, (3.3) implies that W ( y j ) = 0 for j = 1 , . . . , J . Hence theequation above reduces to g xz W ( x ) = 0, which is a contradiction as g is defined tobe positive. Hence d re ( x, z ) (cid:62) t (cid:62) W ( z ) = 0 due to the Neumannboundary condition. Let z ∈ ∂G , z = z . If d re ( z , z ) (cid:54) t −
1, then for any x ∈ G with x ∼ z , Lemma 3.3 implies that d re ( x , z ) (cid:54) d re ( z , z ) (cid:54) t −
1, andhence W ( x ) = 0 by (3.3). Since this holds for all the interior points connectedto z , the Neumann boundary condition yields that W ( z ) = 0. On the otherhand, if d re ( z , z ) = t , then for any x ∈ G with x ∼ z , Lemma 3.3 implies that d re ( x , z ) (cid:54) t , and hence W ( x ) (cid:62)
0. Then the Neumann boundary condition gives W ( z ) (cid:62)
0. Combining these observations, for any p ∈ G ∪ ∂G , we have W ( p ) = 0 , p ∈ G ∪ ∂G, d re ( p, z ) (cid:54) t − , (cid:62) , p ∈ G ∪ ∂G, d re ( p, z ) = t ,> , p = x. (3.4)The main task is to prove the following statement, by induction on τ = 1 , , . . . , t −
1, that for any p ∈ G ∪ ∂G , u W ( p, τ ) ( = 0 , p ∈ G ∪ ∂G, d re ( p, z ) (cid:54) t − τ, (cid:62) , p ∈ G ∪ ∂G, d re ( p, z ) = t − τ + 1 , (3.5)and that u W ( x τ , τ ) > , (3.6)13or some x τ ∈ G satisfying d re ( x τ , z ) = t − τ + 1. By the initial conditions of thewave u W , we have u W ( p,
1) = u W ( p,
0) = W ( p ) for all p ∈ G ∪ ∂G . Thus (3.4)implies (3.5) when τ = 1, and also implies (3.6) by choosing x τ = x . This verifiesthe initial conditions for the induction. Assume that (3.5) and (3.6) hold for some τ ∈ { , , . . . , t − } , we need to prove that (3.5) and (3.6) hold for τ + 1. We willspend most of the proof to argue this.By the wave equation (3.1), we have u W ( p, τ + 1) = 2 u W ( p, τ ) − u W ( p, τ −
1) + ∆ G u W ( p, τ ) − q ( p ) u W ( p, τ ) , (3.7)when p ∈ G and τ (cid:62)
1. This formula and the Neumann boundary condition arewhat the induction is based on. First, we prove that (3.5) holds for τ + 1.Let p ∈ G satisfying d re ( p, z ) (cid:54) t − ( τ + 1). Then we see that the terms2 u W ( p, τ ) , u W ( p, τ −
1) and q ( p ) u W ( p, τ ) in (3.7) are all equal to zero by the induc-tion assumption. Moreover, since u W ( p, τ ) = 0, we have∆ G u W ( p, τ ) = 1 µ p X y ∼ py ∈ G ∪ ∂G g py u W ( y, τ ) . Let y ∈ G ∪ ∂G be any point connected to p . Then d re ( y, z ) (cid:54) d re ( y, p ) + d re ( p, z ) (cid:54) t − ( τ + 1) = t − τ , and hence u W ( y, τ ) = 0 by the induction assumption.Thus (3.7) shows that u W ( p, τ + 1) = 0 for all p ∈ G , d re ( p, z ) (cid:54) t − ( τ + 1).On the other hand, if p ∈ G satisfying d re ( p, z ) = t − ( τ + 1) + 1 = t − τ , thenfor the same reason as above, we see that u W ( p, τ + 1) = ∆ G u W ( p, τ ) = 1 µ p X y ∼ py ∈ G ∪ ∂G g py u W ( y, τ ) . If y ∈ G ∪ ∂G satisfies y ∼ p , then d re ( y, z ) (cid:54) d re ( y, p ) + d re ( p, z ) = 1 + t − τ ,and hence u W ( y, τ ) (cid:62) u W ( p, τ + 1) (cid:62)
0. Itremains to consider the case of p ∈ ∂G , and find x τ +1 for (3.6).Let p ∈ ∂G . Instead of using (3.7) which is valid only in the interior, wecan determine the sign of u W ( p, τ + 1) by using the Neumann boundary condition ∂ ν u W ( p, τ + 1) = 0. Namely, u W ( p, τ + 1) = X x ∼ px ∈ G g xp u W ( x, τ + 1) .X x ∼ px ∈ G µ p g xp . (3.8)Suppose p = z and d re ( p, z ) (cid:54) t − ( τ + 1). Any interior point x with x ∼ p satisfiesthat d re ( x, z ) (cid:54) d re ( p, z ) (cid:54) t − ( τ +1) by Lemma 3.3. Since we have already showedthat u W ( x, τ + 1) = 0 for any x ∈ G satisfying d re ( x, z ) (cid:54) t − ( τ + 1), it followsthat u W ( p, τ + 1) = 0. In the case of p = z , for any interior point x connected to z ,we have u W ( x, τ + 1) = 0 if 1 (cid:54) t − ( τ + 1). This is applicable to all our inductionsteps since τ (cid:54) t −
2, and therefore we have u W ( z, τ + 1) = 0.For the second line in (3.5), let p ∈ ∂G satisfying d re ( p, z ) = t − ( τ + 1) + 1 = t − τ . In particular p = z since τ < t . As in the previous case, we see that anyinterior point x with x ∼ p satisfies d re ( x, z ) (cid:54) d re ( p, z ) = t − τ . Since we havealready showed that u W ( x, τ + 1) (cid:62) x , we get u W ( p, τ + 1) (cid:62)
0. Thisconcludes the proof of (3.5) by induction.Next, we prove that (3.6) holds for τ + 1. The induction assumption gives that u W ( p, τ ) (cid:62) p ∈ G ∪ ∂G satisfying d re ( p, z ) = t − τ + 1. Moreover, thereexists one such p ∈ G , denoted by x τ , so that u W ( x τ , τ ) >
0. Let γ be a shortestpath of length t − τ + 1 from x τ to z in the reduced graph. Since τ (cid:54) t − x τ +1 be the second vertex along this path,14nd then d re ( x τ +1 , z ) = t − τ (cid:62)
2. Observe that x τ +1 is also an interior point: ifnot, then Lemma 3.3 implies that d re ( x τ , z ) (cid:54) d re ( x τ +1 , z ) = t − τ as x τ ∼ x τ +1 ,contradiction.To prove (3.6), it remains to prove that u W ( x τ +1 , τ + 1) >
0. We considerthe formula (3.7) with p = x τ +1 . The induction assumption for (3.5) shows that u W ( x τ +1 , τ ), u W ( x τ +1 , τ −
1) and q ( x τ +1 ) u W ( x τ +1 , τ ) are all equal to zero, since d re ( x τ +1 , z ) (cid:54) t − τ . Thus by (3.7), u W ( x τ +1 , τ + 1) = ∆ G u W ( x τ +1 , τ ) = 1 µ x τ +1 X y ∼ x τ +1 y ∈ G ∪ ∂G g yx τ +1 u W ( y, τ ) . For a point y ∈ G ∪ ∂G connected to x τ +1 , we have d re ( y, z ) (cid:54) t − τ + 1, andtherefore the induction assumption for (3.5) gives u W ( y, τ ) (cid:62)
0. Notice that one ofthe points y in the sum above is x τ , for which u W ( x τ , τ ) >
0. Hence the whole sumis positive. This concludes the proof of (3.6) by induction.Now we turn to the statement of the lemma, with (3.5) and (3.6) in hand. Wesee that u W ( z, t ) = 0 for all t < t by (3.5). At time t , the Neumann boundarycondition gives u W ( z, t ) = X x ∼ zx ∈ G g xz u W ( x, t ) .X x ∼ zx ∈ G µ z g xz . (3.9)Let x ∈ G be an arbitrary point satisfying x ∼ z . The formula (3.7) gives u W ( x, t ) = 2 u W ( x, t − − u W ( x, t −
2) + ∆ G u W ( x, t − − q ( p ) u W ( x, t − . Since u W ( x, t −
1) = u W ( x, t −
2) = 0 by (3.5), we have∆ G u W ( x, t −
1) = 1 µ x X y ∼ xy ∈ G ∪ ∂G g xy u W ( y, t − . Note that d re ( y, z ) (cid:54) d re ( y, x ) + d re ( x, z ) = 1 + 1 = 2. If d re ( y, z ) = 0, then y = z and u W ( y, t −
1) = u W ( z, t −
1) = 0. If d re ( y, z ) = 1, then u W ( y, t −
1) = 0 bythe first line of (3.5). If d re ( y, z ) = 2, then u W ( y, t − (cid:62) y for which u W ( y, t − > G u W ( x, t − > u W ( x, t ) >
0. The latter and (3.9) yield that u W ( z, t ) > W ( x ) = 0 simply follows from the sameproof as above but without the need for (3.6), and by replacing instances of W (cid:62) , W > W = 0 and those of u W (cid:62) , u W > u W = 0.
4. The Inverse Spectral Problem
In this section, we reconstruct the graph structure and the potential from the Neu-mann boundary spectral data, and prove Theorem 1 and Theorem 2. Since thestructure of the boundary is a priori given, it suffices to reconstruct the reducedgraph G re (recall Definition 2.4). The assumption that G is strongly connected isequivalent to G re being connected. Due to Lemma 2.5 and the fact that removingedges between boundary points does not affect the boundary spectral data, withoutloss of generality, we assume G = G re throughout this section . In other words, weassume that there are no edges between boundary points in G .The full reconstruction process is divided into two main parts.15 .1 Characterization by boundary data In this subsection, we construct a characterization of the boundary distance func-tions by boundary data. We mention that the related constructions on partiallyordered lattices that contain boundary distance functions as maximal elements havebeen used to study inverse problems on manifolds in [61].Let s : ∂G → Z + . We equip the set of such functions with the following partialorder ∀ z ∈ ∂G : s ( z ) (cid:54) s ( z ) = ⇒ s (cid:54) s . (4.1)We consider the set of initial values for which the corresponding waves are notobserved at the boundary before time s ( · ), W ( s ) = { W : G ∪ ∂G → R | ∂ ν W | ∂G = 0 , u W ( z, t ) = 0 for all z ∈ ∂G, t < s ( z ) } . Let N = | G | be the number of interior vertices, and we define the set U = { s : ∂G → Z + | (cid:54) s ( · ) (cid:54) N, dim( W ( s )) = 0 } . (4.2)The set W ( s ) is a linear space over R , so dim( W ( s )) is simply its dimensionas a vector space. The condition dim( W ( s )) = 0 simply means that there existsa nonzero initial value such that the corresponding wave satisfies the conditions of W ( s ). Observe that the conditions of W ( s ) indicate that any initial value W ∈W ( s ) vanishes on the whole boundary since u W ( z,
0) = W ( z ). Then the condition u W ( z, t ) = 0 for t = 0 implies the same condition for t = 1 due to the initialconditions of (3.1). Hence we only need to consider s (cid:62) U is a set of functions equipped with the partial order (4.1). We areinterested in its maximal elements with respect to the partial order, denoted bymax( U ). Lemma 4.1.
Let G be a finite connected weighted graph with boundary satisfyingAssumption 1. Then we have { r x | x ∈ G − N ( ∂G ) } ⊆ max( U ) . (4.3) Furthermore, for any nonzero initial value W ∈ W ( r x ) where x ∈ G − N ( ∂G ) , wehave supp( W ) = { x } . Proof.
For an arbitrary point x ∈ G − N ( ∂G ), we first show that for any nonzeroinitial value W ∈ W ( r x ), we have supp( W ) = { x } and consequently r x ∈ U byLemma 3.4.From the condition W ∈ W ( r x ), we see that the wave u W corresponding to thisinitial value satisfies u W ( z, t ) = 0 when z ∈ ∂G and t < r x ( z ). Let S = { y ∈ G | W ( y ) = 0 } ∪ { x } . If | S | (cid:62)
2, the Two-Points Condition in Assumption 1 implies that there exist x ∈ S − { x } and z ∈ ∂G , such that x is the unique nearest point in S from z , whichin particular yields d ( x , z ) (cid:54) d ( x, z ) − r x ( z ) −
1. But by the propagation ofthe wavefront (Lemma 3.4), we see that u W ( z , t ) = 0 for t = d ( x , z ). This isa contradiction because W ( r x ) requires u W ( z , t ) = 0 for t < r x ( z ). Therefore | S | (cid:54) W ) ∩ G ⊆ { x } . Since x / ∈ N ( ∂G ), we see by the Neumannboundary condition that W = 0 on ∂G . Hence supp( W ) = { x } . Precisely, r x is the boundary distance function with respect to d re on the reduced graph G re .Recall that we assumed G = G re . This is the case throughout Section 4.
16e next show that the boundary distance functions are maximal elements in U .Let x ∈ G − N ( ∂G ) and suppose there exists an element s ∈ U such that s (cid:62) r x .By definition of U , there exists a nonzero initial value W such that u W ( z, t ) = 0 , ∀ z ∈ ∂G, t < s ( z ) . (4.4)Since r x (cid:54) s , the same vanishing conditions hold for all t < r x ( z ). Then the sameargument above yields supp( W ) = { x } . If r x ( z ) < s ( z ) for some boundary point z ∈ ∂G then Lemma 3.4 shows that u W ( z , t ) = 0 for t = d ( x, z ) = r x ( z ). Thiscontradicts (4.4). Therefore if r x (cid:54) s , then r x = s and therefore r x is a maximalelement in U .Next, we recover the boundary distance functions corresponding to points in G ∩ N ( ∂G ). We define W b ( s ) = { W : G ∪ ∂G → R | ∂ ν W | ∂G = 0 , u W ( z, t ) = 0 when z ∈ ∂G, s ( z ) (cid:62) t < s ( z ) } , and for y ∈ G ∩ N ( ∂G ), define the set U b ( y ) = { s : ∂G → Z + | dim( W b ( s )) = 0 , s ( · ) (cid:54) N, s ( z ) = 1 only if z ∼ y } . Recall that N = | G | is the number of interior points. Functions s ∈ U b ( y ) can have s ( z ) > z ∼ y .One can show the following lemma by a similar argument as Lemma 4.1. Lemma 4.2.
Let G be a finite connected weighted graph with boundary satisfyingAssumption 1. Then for any y ∈ G ∩ N ( ∂G ) , we have r y ∈ max (cid:0) U b ( y ) (cid:1) . (4.5) Furthermore, for any nonzero initial value W ∈ W b ( r y ) where y ∈ G ∩ N ( ∂G ) , wehave supp ( W ) ∩ G = { y } .Proof. Following the argument in Lemma 4.1, for any nonzero initial value W ∈W b ( r y ), consider the set S = { x ∈ G | W ( x ) = 0 } ∪ { y } . If | S | (cid:62)
2, we can find anextreme point y ∈ G − { y } of S with respect to some z ∈ ∂G . The extreme pointcondition implies that z cannot be connected to y , and hence W ( z ) = u W ( z ,
0) =0 by the condition W ∈ W b ( r y ). The assumptions of Lemma 3.4 are satisfied for W and the pair of points y , z , so u W ( z , t ) = 0 for t = d ( y , z ). But bythe definition of the extreme point, we have d ( y , z ) < d ( y, z ) = r y ( z ). Thiscontradicts the condition u W ( z , t ) = 0 for t < r y ( z ) of W ∈ W b ( r y ), considering z y . Hence | S | = 1 and supp( W ) ∩ G = { y } .Let y ∈ G ∩ N ( ∂G ). We first show that r y ∈ U b ( y ). Clearly r y (cid:54) N and r y ( z ) = 1only at boundary points z connected to y . It remains to show that there exists anonzero initial value in W b ( r y ). Consider an initial value W satisfying W ( x ) = 1at x = y and W ( x ) = 0 otherwise in G . The values of W on ∂G are determinedby the Neumann boundary condition (3.2). By the definition of W b ( r y ), it sufficesto show that u W ( z, t ) = 0 for all t < r y ( z ) when r y ( z ) (cid:62)
2. At such boundarypoints z satisfying r y ( z ) (cid:62) z y ), the Neumann condition gives W ( z ) = 0.Moreover, we have W = 0 at all points in N ( z, d ( y, z )) ∩ G except for y at which W ( y ) >
0. Hence Lemma 3.4 yields that u W ( z, t ) = 0 for all t < d ( y, z ) = r y ( z ).Thus r y ∈ U b ( y ).Next, we show that r y is maximal. Let s ∈ U b ( y ) with r y (cid:54) s . By the definitionof U b ( y ), we have s ( · ) (cid:54) N and s ( z ) > z y . Furthermore, there is a nonzeroinitial value W ∈ W b ( s ) satisfying u W ( z, t ) = 0 , ∀ z ∈ ∂G, s ( z ) (cid:62) , t < s ( z ) . (4.6)17f s ( z ) = 1 occurs, it follows from the definition of U b ( y ) that z ∼ y , i.e. r y ( z ) = 1.Since r y (cid:54) s , the wave u W satisfies the following possibly less strict set of conditions u W ( z, t ) = 0 , ∀ z ∈ ∂G, r y ( z ) (cid:62) , t < r y ( z ) . This exactly means W ∈ W b ( r y ). Then the same argument above yields supp( W ) ∩ G = { y } . Assume that r y ( z ) < s ( z ) for some z ∈ ∂G . This indicates that s ( z ) (cid:62) r y >
0. Hence (4.6) implies that u W ( z , t ) = 0 for t < s ( z ), and inparticular W ( z ) = u W ( z ,
0) = 0. Then Lemma 3.4 shows that u W ( z , t ) = 0 for t = d ( y, z ) < s ( z ), which is a contradiction. Hence r y ( z ) = s ( z ) for all z ∈ ∂G ,and therefore r y is maximal.To uniquely determine G ∩ N ( ∂G ), we need to find all maximal elements of U b ( y ) for every y ∈ G ∩ N ( ∂G ). Then this set of maximal elements contains theset of boundary distance functions { r y } y ∈ G ∩ N ( ∂G ) , which corresponds to the initialvalues supported only at one single point of G ∩ N ( ∂G ) in the interior. Howeverin general, as with Lemma 4.1, there are more maximal elements than just theboundary distance functions.To reconstruct the graph structure, we need to single out the actual boundarydistance functions from the whole set of maximal elements. We will spend the restof this subsection to address it. Definition 4.3.
We define the arrival time of a wave with an initial value W at aboundary point z ∈ ∂G , to be the earliest time t (cid:62) u W ( z, t ) = 0. Denotethe arrival time at z by t Wz . Definition 4.4.
Denote by A the set of all the L ( G )-normalized initial values W satisfying the following three conditions:(1) W : G ∪ ∂G → R , ∂ ν W | ∂G = 0, i.e. W is an initial value;(2) W ∈ W ( s ) for some s ∈ max( U ), or W ∈ W b ( s ) for some s ∈ max( U b ( y )) andsome y ∈ G ∩ N ( ∂G ), i.e. W corresponds to a maximal element;(3) for all z ∈ ∂G , we have u W ( z, t Wz ) >
0, i.e. the first arrival of the wave at anyboundary point is with a positive sign.Finally, define the set A as the initial values supported at one single point, A = (cid:26) W x k W x k L ( G ) (cid:12)(cid:12)(cid:12)(cid:12) x ∈ G, ∂ ν W x | ∂G = 0 , supp( W x ) ∩ G = { x } , W x ( x ) > (cid:27) . Lemma 4.5.
Let G be a finite connected weighted graph with boundary satisfyingAssumption 1. Then A ⊆ A .Proof. Let W ∈ A . Then W is L ( G )-normalized and it satisfies property 1 inDefinition 4.4. Furthermore W = W x for some x ∈ G . We claim that t Wz = r x ( z )for all z ∈ ∂G .This claim follows directly from the propagation of the wavefront (Lemma 3.4) if x ∈ G − N ( ∂G ), which yields that u W ( z, t Wz ) > z ∈ ∂G . If x ∈ G ∩ N ( ∂G )we see that t Wz = r x ( z ) when r x ( z ) (cid:62) r x ( z ) = 1, then z ∼ x and u W ( z, t ) is determined by the Neumann boundary condition (3.2), which gives u W ( z,
0) = u W ( z, >
0. Hence t Wz = 1 in this case by Definition 4.3. In conclusion, t Wz = r x ( z ) and u W ( z, t Wz ) > z ∈ ∂G , i.e. the property 3 of Definition 4.4.Moreover, Lemmas 4.1 and 4.2 imply that t W · is a maximal element, i.e. the property2 with s ( z ) = t Wz .Observe that A is an orthonormal basis of the linear span of A with respect tothe L ( G )-inner product. 18 emma 4.6. Let G be a finite connected weighted graph with boundary satisfyingItem 2 of Assumption 1.(1) Given any initial value W satisfying ∂ ν W | ∂G = 0 and the property 3 of Defini-tion 4.4, if x is an extreme point of supp( W ) ∩ G , then W ( x ) > .(2) Given any nonzero initial value W satisfying ∂ ν W | ∂G = 0 and W | G (cid:62) , forany z ∈ ∂G , we have t Wz = min x ∈ supp( W ) ∩ G t W x z . As a consequence, if
W, W are two nonnegative initial values satisfying the Neu-mann boundary condition and supp( W ) ∩ G ⊆ supp( W ) ∩ G , then t W z (cid:62) t Wz forany z ∈ ∂G .Proof. For the first claim, let z ∈ ∂G be a boundary point realizing the extremepoint condition of x . If d ( x , z ) (cid:62)
2, the arrival time t Wz = d ( x , z ) due toLemma 3.4. If d ( x , z ) = 1, then t Wz = 1 due to the Neumann boundary conditionfor W . Hence the property 3, Lemma 3.4 and the Neumann boundary conditionyield W ( x ) > W | G (cid:62)
0, the initial value W restrictedto G can be written as W | G = X x ∈ supp( W ) ∩ G α x W x | G for some positive numbers α x , where W x ( x ) = 1 and W x ( y ) = 0 if y ∈ G − { x } .Since W and each W x determine their boundary values uniquely and linearly fromtheir values on G by (3.2), the form above extends to the whole graph G ∪ ∂G . Bylinearity and the uniqueness of the solution of (3.1), the wave u W has the followingform at any z ∈ ∂G and t ∈ N , u W ( z, t ) = X x ∈ supp( W ) ∩ G α x u W x ( z, t ) . Since u W x ( z, t W x z ) > z ∈ ∂G by Lemma 3.4 and all α x are positive, weknow that the earliest time u W ( z, · ) becomes nonzero is the earliest time when anyof u W x ( z, · ) becomes nonzero. This shows that for any z ∈ ∂G , t Wz = min x ∈ supp( W ) ∩ G t W x z . The last part of the lemma follows from the condition that supp( W ) ∩ G ⊆ supp( W ) ∩ G , since a minimum over a smaller set can only be larger. Lemma 4.7.
Let G be a finite connected weighted graph with boundary satisfyingAssumption 1. If an initial value W satisfies W | G (cid:62) and | supp( W ) ∩ G | (cid:62) ,then W / ∈ A .Proof.
Denote S = { x ∈ G | W ( x ) = 0 } and we have | S | (cid:62) W ∈ W ( s ) forsome s ∈ max( U ). By the Two-Points Condition, there exists x ∈ S and z ∈ ∂G ,such that x is the unique nearest point in S from z . Since W ∈ W ( s ) and s ( z ) (cid:62) s ∈ U , we have W ( z ) = u W ( z ,
0) = 0. Then Lemma 3.4 implies that t Wz = d ( x , z ) and 2 (cid:54) s ( z ) (cid:54) t Wz . We consider the following modified function s : ∂G → Z + defined by s ( z ) = s ( z ) + 1 and equal to s at all other boundarypoints. 19ow we prove that s ∈ U and consequently s cannot be maximal in U . On onehand, we have s (cid:62) s (cid:62)
2. On the other hand, we have s (cid:54) N . This is becauseif s ( z ) = N then d ( x , z ) (cid:62) N , which means that all y ∈ S − { x } are at leastdistance N +1 from z . But this is impossible since there are only N interior points,considering that distances (precisely d re on the reduced graph G re ) are realized bypaths passing through interior points by Lemma 3.3. Hence 2 (cid:54) s (cid:54) N and itremains to show that W ( s ) is nontrivial. Define another initial value W to be W ( x ) = 0 and equal to W elsewhere on G . By the propagation of the wavefront(Lemma 3.4), we have u W ( z , t ) = 0 for t (cid:54) d ( x , z ). Since s ( z ) (cid:54) t Wz = d ( x , z )and s ( z ) = s ( z ) + 1, we have u W ( z , t ) = 0 for t < s ( z ). Since W | G (cid:62)
0, thearrival time of the wave u W at any other boundary point is no earlier than thatof u W by Lemma 4.6. This shows W ∈ W ( s ) and it is a nontrivial element since | S | (cid:62)
2. Hence s ∈ U and s cannot be maximal.Next we consider W ∈ W b ( s b ) for some s b ∈ U b ( y ) and show that s b cannotbe maximal. Following the previous argument, we can find x ∈ S and z ∈ ∂G ,such that x is the unique nearest point in S from z . If s b ( z ) (cid:62)
2, then theprevious argument applies. Otherwise if s b ( z ) = 1, then z ∼ y . We define s b : ∂G → Z + by s b ( z ) = 2 and equal to s b at all other boundary points. Asbefore, we see that s b (cid:54) N , and s b ( z ) = 1 implies z ∼ y . It remains to showthat there is a nontrivial initial value W ∈ W b ( s b ). We choose W ( x ) = 0 andequal to W elsewhere on G . Since x is an extreme point of S with respect to z and W ( x ) = 0, we have W ( z ) = 0 by the Neumann boundary condition. Thisimplies that u W ( z ,
0) = u W ( z ,
1) = W ( z ) = 0, and hence u W ( z , t ) = 0 for t < s b ( z ). Then the same argument as for the earlier case shows that u W ( z, t ) = 0for t < s b ( z ) when s b ( z ) (cid:62)
2. Thus we find a nontrivial initial value W ∈ W b ( s b ).Therefore s b cannot be maximal in U b ( y ).Finally, we use the following criteria to distinguish A from A . Lemma 4.8.
Let G be a finite connected weighted graph with boundary satisfyingAssumption 1. Then a subset ˜ A ⊆ A satisfies the following two properties(1) ˜ A is an orthogonal basis of the linear span of A in L ( G ) ;(2) for any W ∈ A − ˜ A , there exists ˜ W ∈ ˜ A such that h W, ˜ W i L ( G ) < ,if and only if ˜ A = A .Remark. Elements of A are normalized, so we are actually searching for an or-thonormal basis satisfying Property (2). Property (1) can also be formulated asfollows: ˜ A has cardinality equal to | G | , and its elements are mutually orthogonalwith respect to the L ( G )-inner product. Proof.
First, we show that A satisfies these two properties. The set A satisfiesProperty (1) as a direct consequence of Lemma 4.5. Since every function in A − A is supported at multiple interior points by the definition of A , Lemma 4.7 impliesthat any function W ∈ A − A must have a negative value at some interior point,say at x ∈ G . Then the condition h W, ˜ W i L ( G ) < W = W x .Hence Property 2 is satisfied for A .Next, we prove the “only if” direction. We claim that if ˜ A (cid:42) A , then theProperties (1) and (2) cannot be satisfied at the same time. Suppose ˜ A (cid:42) A andProperty (1) is true. The set ˜ A consists of two types of initial values: a) initialvalues supported at one single point in the interior (corresponding to the boundarydistance functions), and b) initial values supported at multiple points in the interior(where interactions occur). Note that ˜ A may not contain the former type of initialvalues, but it must contain the latter type of initial values since ˜ A − A = ∅ by20ssumption. Property (1) implies that the support of these two types of initialvalues does not intersect.Consider the union of the support (intersected with G ) of all the initial valuesof type b) in ˜ A , denoted by S = [ W ∈ ˜ A , W of type b) supp( W ) ∩ G. By the Two-Points Condition, we can find an extreme point ˜ x ∈ G of S . Thenwe consider the L ( G )-normalized initial value W ˜ x ∈ A supported at ˜ x ∈ G with W ˜ x (˜ x ) >
0. Orthogonality implies that W ˜ x / ∈ ˜ A , or equivalently W ˜ x ∈ A − ˜ A .For any ˜ W ∈ ˜ A with h W ˜ x , ˜ W i L ( G ) = 0, we know that ˜ W is supported at multiplepoints containing ˜ x . The condition that ˜ x is an extreme point of S implies that ˜ x isalso an extreme point of its subset supp( ˜ W ) ∩ G . Then ˜ W (˜ x ) > h W ˜ x , ˜ W i L ( G ) = µ ˜ x W ˜ x (˜ x ) ˜ W (˜ x ) > . Hence h W ˜ x , ˜ W i L ( G ) (cid:62) W ∈ ˜ A . This contradicts Property (2), and thereforeproves the claim.The claim shows that ˜ A ⊆ A for any subset ˜ A ⊆ A satisfying both properties.The set A is an orthogonal basis of the linear span of A , and the only subset of A also forming a basis is A itself. Hence Property (1) yields ˜ A = A . In this subsection, we will tie in the previous subsection’s objects to the spectral andboundary data of a graph. We will show that if two graphs have the same a priori data, then the spectral characterization of various objects, such as U , A from theprevious subsection, of these two graphs coincide. This leads to the conclusion thatthe inverse spectral problem is solvable.Without loss of generality, we still assume G = G re throughout this section . Lemma 4.9.
Let G be a finite connected weighted graph with boundary, and q be areal-valued potential function on G . Let { φ j } Nj =1 be the orthonormalized Neumanneigenfunctions of ( G , q ) . For any function W : G ∪ ∂G → R , denote c W ( j ) = h W, φ j i L ( G ) , c W = (cid:0)c W (1) , · · · , c W ( N ) (cid:1) ∈ R N . If W is an initial value for (3.1) , i.e. satisfying ∂ ν W | ∂G = 0 , we have W ( x ) = N X j =1 c W ( j ) φ j ( x ) , ∀ x ∈ G ∪ ∂G. Conversely, given any ( c j ) Nj =1 ∈ R N , P j c j φ j gives an initial value for (3.1) .Proof. Since G is connected and there are no edges between boundary points byassumption, every boundary point is connected to the interior. Then the claims area direct consequence of the orthonormality of the eigenfunctions in L ( G ), (3.2) and ∂ ν φ j | ∂G = 0. Notation.
Given V ⊆ L ( G ), we denote b V = { b f ∈ R N | f ∈ V } . If G is anotherfinite weighted graph with boundary, then we denote by V a subset of L ( G ). Inthis case, c V is defined the same as above, but the hat-notation itself is definedusing the eigenfunctions φ j of G rather than those of G .The following lemma enables us to calculate a wave at any boundary pointand any time, if we know the Neumann boundary spectral data and the Fouriertransform (or the spectral representation) of the initial value of the wave.21 emma 4.10. Let G be a finite connected weighted graph with boundary, and q bea real-valued potential function on G . Let ( λ j , φ j ) Nj =1 be the Neumann eigenvaluesand orthonormalized Neumann eigenfunctions of ( G , q ) . Suppose W is the initialvalue of some wave u W satisfying the wave equation (3.1) . Then u W ( x, t ) = X { j | λ j =0 } c W ( j ) φ j ( x ) + X { j | λ j =4 } c W ( j )( − t + 1)( − t φ j ( x )+ X { j | λ j / ∈{ , }} c W ( j ) µ t +1 j, − µ t +1 j, − ( µ tj, − µ tj, ) µ j, − µ j, φ j ( x ) , where µ j, = − λ j − s(cid:18) λ j − (cid:19) − , µ j, = − λ j s(cid:18) λ j − (cid:19) − . (4.7) Conversely, given any c W ∈ R N , then the wave u W defined as above solves (3.1) with the initial value W = P Nj =1 c W ( j ) φ j .Proof. By assumption, every boundary point is connected to the interior. The wavesatisfies ∂ ν u W | ∂G × N = 0, so the orthonormality of { φ j } Nj =1 in L ( G ) and (3.2) implythat u ( x, t ) = N X j =1 a j ( t ) φ j ( x )on ( G ∪ ∂G ) × N for some functions a j : N → R . The wave equation (3.1) and theeigenvalue problem (1.10) yield that a j ( t + 1) + ( λ j − a j ( t ) + a j ( t −
1) = 0 (4.8)for all t ∈ Z + . The solutions to the associated characteristic equation µ j + ( λ j − µ j + 1 = 0 are shown in the lemma statement. The characteristic equation hastwo identical solutions if λ j = 0 or 4, in which case the solutions are 1 or −
1. Hence a j has the following form: a j ( t ) = b j t + c j , λ j = 0 , ( b j t + c j )( − t , λ j = 4 ,b j µ tj, + c j µ tj, , λ j / ∈ { , } , t ∈ N . (4.9)Recall that u W ( · ,
0) = u W ( · , a j implies that X { j | λ j =0 } c j φ j + X { j | λ j =4 } c j φ j + X { j | λ j / ∈{ , }} ( b j + c j ) φ j = X { j | λ j =0 } ( b j + c j ) φ j − X { j | λ j =4 } ( b j + c j ) φ j + X { j | λ j / ∈{ , }} ( b j µ j, + c j µ j, ) φ j . Taking the inner product with any φ j , and the orthonormality of φ j allows us tosolve b j as a function of c j , µ j, and µ j, for each j = 1 , . . . , N . This gives b j = , λ j = 0 , − c j , λ j = 4 , − c j µ j, − µ j, − , λ j / ∈ { , } . Note that µ j, , µ j, = ± λ j / ∈ { , } . Hence we obtain the formula for the wave: u W ( x, t ) = X { j | λ j =0 } c j φ j ( x ) + X { j | λ j =4 } c j ( − t + 1)( − t φ j ( x )++ X { j | λ j / ∈{ , }} c j (cid:18) − µ j, − µ j, − µ tj, + µ tj, (cid:19) φ j ( x ) , x ∈ G ∪ ∂G and t ∈ N . This satisfies the second initial condition, the Neumannboundary condition and the wave equation in (3.1). By Lemma 4.9, the first initialcondition u W ( x,
0) = W ( x ) gives that c j = c W ( j ) , λ j = 0 , c W ( j ) , λ j = 4 , c W ( j ) µ j, − µ j, − µ j, , λ j / ∈ { , } . The first claim of the lemma follows after plugging these into the formula for thewave. The converse claim is a straightforward calculation whose details are actuallyscattered in the proof of the first claim.In our setting, we are working with two graphs having the same boundary andthe same Neumann boundary spectral data. For convenience, we make use of thefollowing pullback notation.
Notation.
Given two finite weighted graphs with boundary G , G and a boundary-isomorphism Φ (Definition 1.6), we define the following notation.• For f : ∂G → R , we denote Φ ∗ f = f ◦ (Φ | ∂G ).• If S is a set of functions on ∂G , denote Φ ∗ S = { Φ ∗ f | f ∈ S } .This notation defines Φ ∗ f : ∂G → R , and Φ ∗ S as a set of functions on ∂G .We consider initial values not just as functions on the graph, but also as abstractpoints in R N using their Fourier series representation in Lemma 4.9. Lemma 4.10shows that the spectral (Fourier) coefficients of an initial value W uniquely deter-mine the boundary values of the corresponding wave u W . Lemma 4.11.
Let G , G be two finite connected weighted graphs with boundary,and q, q be real-valued potential functions on G, G . Suppose ( G , q ) is spectrallyisomorphic to ( G , q ) with a boundary-isomorphism Φ , namely ( λ j , φ j | ∂G ) Nj =1 = ( λ j , Φ ∗ ( φ j | ∂G )) Nj =1 . Let ( c j ) Nj =1 ∈ R N , W = P Nj =1 c j φ j , W = P Nj =1 c j φ j , and u W , u W be the corre-sponding solution to the wave equation (3.1) in G , G . Then u W ( z, t ) = u W (Φ ( z ) , t ) for all z ∈ ∂G and t ∈ N .Proof. This is a direct consequence of the representation formula for u W , u W inLemma 4.10, since c W ( j ) = c j = c W ( j ), λ j = λ j and φ j = φ j ◦ Φ on ∂G .We remark that a full boundary-isomorphism is not needed for this lemma; asimple bijection ∂G → ∂G which makes the boundary spectral data equivalent isenough.Our next few tasks are to show that various objects from Section 4.1 are equiva-lent, or that their spectral representations are the same for two spectrally isomorphicgraphs. Recall the definitions of the various objects W , W b , U , U b which were de-fined just before Lemmas 4.1 and 4.2. The following lemma shows that knowing theNeumann boundary spectral data leads to the knowledge of the sets U , U b . Lemma 4.12.
Let G , G be two finite connected weighted graphs with boundary,and q, q be real-valued potential functions on G, G . Suppose ( G , q ) is spectrallyisomorphic to ( G , q ) with a boundary-isomorphism Φ . Then c W ( s ) = c W ( s ) and c W b ( s ) = c W b ( s ) for all s : ∂G → Z + and s = s ◦ (Φ | ∂G ) − . As a consequence, U = Φ ∗ U , U b ( y ) = Φ ∗ (cid:0) U b ( y ) (cid:1) , for all y ∈ G ∩ N ( ∂G ) and y = Φ ( y ) . roof. We use the notation s = s ◦ (Φ | ∂G ) − throughout this proof. By symmetry,it suffices to prove that c W ( s ) ⊆ c W ( s ). Suppose c W = ( c j ) Nj =1 ∈ R N for some W ∈ W ( s ). This gives u W ( z, t ) = 0 for all z ∈ ∂G and t < s ( z ). Then take thefunction W = P Nj =1 c j φ j with the same Fourier coefficients. By Lemma 4.11, wesee that u W (Φ ( z ) , t ) = u W ( z, t ) = 0for z ∈ ∂G and t < s ( z ) = s (Φ ( z )). As z runs through ∂G , the point Φ ( z ) runsthrough the whole ∂G . Hence W ∈ W ( s ) and ( c j ) Nj =1 = c W ∈ c W ( s ). The sameargument shows that c W b ( s ) ⊆ c W b ( s ).For the claim on U , U , notice thatdim( W ( s )) = dim( c W ( s )) = dim( c W ( s )) = dim( W ( s )) . This is because the map W c W from L ( G ) to R N is an invertible linear map byLemma 4.9. Thus s ∈ U if and only if s ∈ U .If s ∈ U b ( y ) for some y ∈ G ∩ N ( ∂G ), then dim( W b ( s )) = dim( W b ( s )) = 0. Bydefinition, s ( z ) = 1 only if z ∼ y . By the definition of Φ , Φ ( z ) ∼ Φ ( y ) holds ifand only if z ∼ y . Hence s satisfies all conditions of U b ( y ).The set A contains the initial values for which the corresponding wavefrontreaches the boundary with positive values everywhere and as late as possible (RecallDefinition 4.4). The following lemma shows that knowing the boundary spectraldata leads to the knowledge of the spectral data of all such initial values. Lemma 4.13.
Let G , G be two finite connected weighted graphs with boundary,and q, q be real-valued potential functions on G, G . Suppose ( G , q ) is spectrallyisomorphic to ( G , q ) . Then b A = c A .Proof. Let Φ be a boundary isomorphism making the graphs spectrally isomorphic.Let W ∈ A , i.e. k W k L ( G ) = 1 and it satisfies the three conditions in Definition 4.4.Let W = P j c W ( j ) φ j . Then ∂ ν W | ∂G = 0 and k W k L ( G ) = N X j =1 | c W ( j ) | = k W k L ( G ) = 1 . Since u W ( z, t ) = u W (Φ ( z ) , t ) for all z ∈ ∂G and t ∈ N by Lemma 4.11, we have u W ( z , t W z ) > z ∈ ∂G . It remains to verify Condition (2) for W , i.e.that W corresponds to a maximal element s : ∂G → N .By Lemma 4.12, we have for any y ∈ G ∩ N ( ∂G ),max( U ) = max(Φ ∗ U ) , max( U b ( y )) = max (cid:0) Φ ∗ U b (Φ ( y )) (cid:1) . If W ∈ W ( s ) for some s ∈ max( U ), then c W = c W ∈ c W ( s ) = c W ( s )for s = s ◦ (Φ | ∂G ) − by Lemma 4.12. Hence W ∈ W ( s ) for the given s which isa maximal element of U . Next we show that s is a maximal element of U . SinceΦ ∗ s = s , s ∈ max( U ) and U = Φ ∗ U by Lemma 4.12, we see that Φ ∗ s ∈ max(Φ ∗ U ).The pullback does not affect the partial order, and hence s ∈ max( U ).Similarly, if W ∈ W b ( s ) for some s ∈ max( U b ( y )) and y ∈ G ∩ N ( ∂G ), then wesee that W ∈ W b ( s ) for Φ ∗ s = s . Then Φ ∗ s ∈ max (cid:0) Φ ∗ U b (Φ ( y )) (cid:1) , which impliesthat s ∈ max( U b (Φ ( y ))). Moreover, by the definition of Φ (Definition 1.6), wehave Φ ( y ) ∈ G ∩ N ( ∂G ). Hence W ∈ W b ( s ) with s ∈ max( U b ( y )) for some y ∈ G ∩ N ( ∂G ), just as required in Condition (2) for A . Thus W ∈ A impliesthat W ∈ A , where c W = c W . A symmetric proof shows c A ⊆ b A .24ith Lemma 4.13, we can finally apply Lemma 4.8 to deduce the set of Fouriercoefficients corresponding to initial values supported at one single interior point.These initial values correspond to individual points of the graph. Proposition 4.14.
Let G , G be two finite connected weighted graphs with bound-ary satisfying Assumption 1, and q, q be real-valued potential functions on G, G .Suppose ( G , q ) is spectrally isomorphic to ( G , q ) . Then c A = c A .Proof. We need to define a subset of L ( G ), such that its Fourier transform is equalto c A and satisfies the conditions in Lemma 4.8. To write relevant notations clearly,we use F to denote the Fourier transform in this proof. Let f A = (cid:26) N X j =1 c j φ j (cid:12)(cid:12)(cid:12)(cid:12) ( c j ) Nj =1 ∈ F A (cid:27) . For f, g ∈ L ( G ), we know that h f, g i L ( G ) = P j F f ( j ) F g ( j ) = F f · F g ,where · is the inner product in R N . Furthermore, F span A = span F A . Since A is an orthonormal basis of span A , these two observations indicate that F A is an orthonormal basis of span F A . The latter is equal to span F A byLemma 4.13. Hence we can deduce that F f A = F A is an orthonormal basis ofspan F A . Thus f A is an orthonormal basis of span A . Condition (1) has beenverified.Next, let us verify Condition (2) in Lemma 4.8. Let W ∈ A − f A and W = P j F W ( j ) φ j . Since F A = F A by Lemma 4.13, we have F W = F W ∈ F A − F f A = F A − F A , which shows W ∈ A − A . By Lemma 4.8, there exists f W ∈ A such that h W, f W i L ( G ) <
0. Take f W = P j F f W ( j ) φ j . Then f W ∈ f A by the latter’s defini-tion. Moreover, h W , f W i L ( G ) = F W · F f W = F W · F f W = h W, f W i L ( G ) < , which shows that Condition (2) in Lemma 4.8 holds for f A . Hence Lemma 4.8 yields f A = A , and the lemma follows.Recall that A is the set of normalized initial values supported at one singleinterior point. Since the Fourier transforms of these sets are the same, it makessense to identify interior vertices via their Fourier transforms. We will show thatthis identification gives the desired bijection Φ in Theorem 1. Lemma 4.15.
Let G , G be two finite connected weighted graphs with boundary sat-isfying Assumption 1, and q, q be real-valued potential functions on G, G . Suppose ( G , q ) is spectrally isomorphic to ( G , q ) . We define a relation ≡ on G × G by x ≡ x ⇐⇒ d W x = d W x , d W x ∈ c A , d W x ∈ c A . Then ≡ is a one-to-one correspondence.Proof. Let us verify that ≡ satisfies the conditions for a one-to-one correspondence.We make use of Proposition 4.14 which gives c A = c A .“Every x ∈ G is paired with exactly one x ∈ G ”: Let x ∈ G . Then d W x ∈ c A = c A . The latter set consists of all elements of the form d W x for x ∈ G . Since theFourier transform is invertible, there is a unique x ∈ G such that d W x = d W x .“For any x ∈ G , there exists a unique x ∈ G such that x ≡ x ”: the same proofas above. 25 efinition 4.16. With the assumptions of Lemma 4.15, given a boundary-isomorphismΦ which makes ( G , q ) and ( G , q ) spectrally isomorphic, we define a bijective mapΦ : G ∪ ∂G → G ∪ ∂G byΦ( x ) = ( Φ ( x ) , x ∈ ∂G,x , x ∈ G, x ≡ x . With this map Φ, we have the point-equivalence between these two graphs. Nowwe show that Φ also preserves the edge structure. From what we have done in thissection, the boundary spectral data provide the knowledge of the Fourier transformof initial values, boundary values of waves, and the inner product of waves. We usethis information to determine if there is an edge between two points.
Lemma 4.17.
Let G be a finite weighted graph with boundary, and x ∈ G , z ∈ ∂G .Then x ∼ z if and only if W x ( z ) = 0 .Proof. This directly follows from ∂ ν W x | ∂G = 0 and (3.2). Lemma 4.18.
Let G be a finite weighted graph with boundary satisfying Item 2 ofAssumption 1, and x, y ∈ G . Then x ∼ y if and only if min (cid:8) t ∈ N (cid:12)(cid:12) h u W x ( · , t ) , W y i L ( G ) = 0 (cid:9) = 2 . Proof.
By Item 2 of Assumption 1 and calculating (3.1) up to time t = 2, we seethat G ∩ supp (cid:0) u W x ( · , t ) (cid:1) ⊆ ( { x } , t (cid:54) ,G ∩ N ( x, , t = 2 . Moreover, we have u W x ( · , t ) > G ∩ N ( x, − { x } at t = 2.If x ∼ y , then y ∈ G ∩ N ( x, − { x } and hence the minimum in question is equalto 2. If x y , then either x = y in which case the minimum is 0, or d ( x, y ) (cid:62) y / ∈ G ∩ supp( u W x ( · , t )) for t (cid:54)
2, and the minimum is more than 2.Finally, we are ready to prove the main theorems.
Proof of Theorem 1.
The first property of Φ | ∂G being identical to Φ | ∂G followsby definition. It remains to verify the second property that the edge relations arepreserved by Φ and its inverse. Let p , p ∈ G ∪ ∂G .If p , p ∈ G , then by Lemma 4.18, p ∼ p ⇐⇒ min (cid:8) t ∈ N (cid:12)(cid:12) h u W p ( · , t ) , W p i L ( G ) = 0 (cid:9) = 2 . Let us write the inner product using the Fourier transform of the initial values W p , W p . By Lemma 4.10, u W p ( · , t ) = X { j | λ j =0 } d W p ( j ) φ j + X { j | λ j =4 } d W p ( j )( − t + 1)( − t φ j + X { j | λ j / ∈{ , }} d W p ( j ) µ t +1 j, − µ t +1 j, − ( µ tj, − µ tj, ) µ j, − µ j, φ j . Taking the inner product with W p = P j d W p φ j yields h u W p ( · , t ) , W p i L ( G ) = X { j | λ j =0 } d W p ( j ) d W p ( j ) + X { j | λ j =4 } d W p ( j ) d W p ( j )( − t + 1)( − t + X { j | λ j / ∈{ , }} d W p ( j ) d W p ( j ) µ t +1 j, − µ t +1 j, − ( µ tj, − µ tj, ) µ j, − µ j, = h u W Φ( p ( · , t ) , W Φ( p ) i L ( G ) . d W p = (cid:92) W Φ( p ) , d W p = (cid:92) W Φ( p ) , considering that p ≡ Φ( p ), p ≡ Φ( p ) and the eigenvalues are the same for G , G . Hence theminimal time in question is equal to 2 for p , p in G if and only if it is so forΦ( p ) , Φ( p ) in G .Next, consider the edge between an interior point and a boundary point. If p ∈ G , p ∈ ∂G , then by Lemma 4.17, p ∼ p ⇐⇒ W p ( p ) = 0 . Since p ≡ Φ( p ), p ≡ Φ( p ) and φ j = φ j ◦ Φ = φ j ◦ Φ on ∂G , we have W p ( p ) = N X j =1 d W p ( j ) φ j ( p ) = N X j =1 (cid:92) W Φ( p ) ( j ) φ j (cid:0) Φ( p ) (cid:1) = W Φ( p ) (cid:0) Φ( p ) (cid:1) . Hence p ∼ p if and only if Φ( p ) ∼ Φ( p ).Finally, the case of p , p ∈ ∂G is trivial, because Φ( p ) , Φ( p ) ∈ ∂G and theedge structure on the boundary is a priori given.In Theorem 2, we assume that the isomorphic structure is already known andthe vertices of G have been identified with vertices of G via the Φ-correspondence.In terms of notations, a vertex x of G can also denote a vertex of G , which exactlyrefers to the vertex Φ( x ) of G . Proof of Theorem 2.
Recall from Definition 4.4 that W x ∈ A , W x ∈ A are definedas the L -normalized initial values satisfying ∂ ν W x | ∂G = ∂ ν W x | ∂G = 0 with G ∩ supp( W x ) = G ∩ supp( W x ) = { x } . Then W x ( x ) = µ − / x and W x ( x ) = µ / x . Now let us prove (1). Assume µ = µ . First, we prove g = g .For x, y ∈ G with x ∼ y , we have (cid:10) ( − ∆ G + q ) W x , W y (cid:11) L ( G ) = µ y W y ( y ) (cid:0) ( − ∆ G + q ) W x (cid:1) ( y ) = µ y W y ( y )( − ∆ G W x )( y )= − W y ( y ) X p ∼ yp ∈ G ∪ ∂G g yp (cid:0) W x ( p ) − W x ( y ) (cid:1) = − W y ( y ) X p ∼ yp ∈ G ∪ ∂G g yp W x ( p ) . (4.10)On the other hand, this inner product can be determined from the spectral data.Namely by (1.10), we have (cid:10) ( − ∆ G + q ) W x , W y (cid:11) L ( G ) = (cid:28) X j d W x ( j ) λ j φ j , X j c W y ( j ) φ j (cid:29) L ( G ) = X j λ j d W x ( j ) c W y ( j ) = X j λ j d W x ( j ) c W y ( j ) = (cid:10) ( − ∆ G + q ) W x , W y (cid:11) L ( G ) . (4.11)since λ j = λ j , d W x = d W x and c W y = c W y , due to Lemma 4.15 and Definition 4.16.We consider two cases from here. (i) If x ∈ G − N ( ∂G ) in (4.10), then for all p ∈ G ∪ ∂G , we have W x ( p ) = 0unless p = x . Then (4.10) and (4.11) yield − W y ( y ) W x ( x ) g xy = (cid:10) ( − ∆ G + q ) W x , W y (cid:11) L ( G ) = (cid:10) ( − ∆ G + q ) W x , W y (cid:11) L ( G ) = − W y ( y ) W x ( x ) g xy . This implies that g xy = g xy , since W y ( y ) = µ − / y = µ / y = W y ( y ) and similarly W x ( x ) = W x ( x ). 27 ii) It remains to consider the case where x, y ∈ G ∩ N ( ∂G ). In this case, (4.10)and (4.11) reduce to (4.12) W y ( y ) W x ( x ) g xy + W y ( y ) X z ∼ y,z ∼ xz ∈ ∂G g yz W x ( z ) = W y ( y ) W x ( x ) g xy + W y ( y ) X z ∼ y,z ∼ xz ∈ ∂G g yz W x ( z ) . Since ∂ ν W x ( z ) = ∂ ν W x ( z ) = 0, by (3.2) we see that W x ( z ) = g xz W x ( x ) P p ∼ z,p ∈ G g pz = g xz W x ( x ) P p ∼ z,p ∈ G g pz = W x ( z ) , where we have used that W x ( x ) = W x ( x ) and g = g on edges from the boundary.By (4.12), we see that g xy = g xy . This concludes the unique determination of g .Next, we prove q = q , assuming µ = µ . Let x ∈ G . The spectral datadetermines the following inner product: h ( − ∆ G + q ) W x , W x i L ( G ) = µ x W x ( x ) (cid:0) ( − ∆ G + q ) W x (cid:1) ( x )= − W x ( x ) X p ∼ xp ∈ G ∪ ∂G g xp (cid:0) W x ( p ) − W x ( x ) (cid:1) + q ( x ) . Observe that all relevant quantities have already been uniquely determined as above,during the proof for the unique determination of g . Hence q = q .At last, we prove (2). Assume q = q = 0. For the graph Laplacian (with zeropotential), there exists j such that λ j = λ j = 0 and φ j = φ j = c for someconstant c ∈ R . Given any x ∈ G , we have h W x , φ j i L ( G ) = µ x W x ( x ) c , where W x ( x ) = µ − / x . Then µ x = h W x , φ j i L ( G ) c = (cid:0)d W x ( j ) (cid:1) c = (cid:0)d W x ( j ) (cid:1) c = h W x , φ j i L ( G ) c = µ x . Hence µ = µ . Then the assumption of (1) is satisfied and therefore g = g .In particular, if µ = deg G , µ = deg G , then µ = µ since Φ preserves the edgestructure by Theorem 1. Hence the conclusion follows from (1). References [1] M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, M. Taylor,
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