Spatio–Spectral Limiting on Replacements of Tori by Cubes

Abstract

A class of graphs is defined in which each vertex of a discrete torus is replaced by a Boolean hypercube in such a way that vertices in a fixed subset of each replacement cube are adjacent to corresponding vertices of a neighboring replacement cube. Bases of eigenvectors of the Laplacians of the resulting graphs are described in a manner suitable for quantifying the concentration of a low-spectrum vertex function on a single vertex replacement. Functions that optimize this concentration on these graphs can be regarded as analogues of Slepian prolate functions that optimize concentration of a bandlimited signal on an interval in the classical setting of the real line. Comparison to the case of a simple discrete cycle shows that replacement allows for higher concentration.

1. Introduction

Spatio–spectral limiting refers to the composition of a spatial cutoff, meaning multiplication by the characteristic function of a set, with a spectral cutoff, meaning projection onto the span of eigenfunctions indexed by a subset of eigenvalues. In the classical case of the real line this is known as time and band limiting or duration–bandwidth limiting and arises in applications such as super-resolution [1,2], channel estimation [3] and spectrum estimation [4] among others (see, e.g., [5]). In this setting, it is known that the number of eigenvalues of the so-called time and band limiting operator close to 1 is essentially the time–bandwidth product (the product of the lengths of the time localization interval and of the frequency localization band). No clear analogue of this fact has been established in ${\mathbb{R}}^n$, where geometry is more complicated (e.g., [6]).

More recently, some authors have taken up the study of spatio–spectral limiting on graphs, addressing the behavior of operators that first multiply a vertex function by the characteristic function of a specific set of vertices, then project onto some part of the spectrum defined in terms of the graph Laplacian. This has been studied both for specific types of graphs [7] as well as in relation to graph signal processing more generally (e.g., [8,9,10,11,12]). In [9], a connection is made between spatio–spectral limiting and clustering. However, none of these works provide concrete methods to predict the number of independent low-spectrum graph signals that can have a specified fraction of their energies localized on a particular set of nodes—the analogue in this setting of the number of eigenvalues of time and band limiting that are close to 1 [13], or larger than $1/2$ [14]—a hallmark of the classical theory [15].

The primary purpose of this work is to study a specific family of graphs for which addressing this question about counting independent spatially concentrated and low-spectrum limited modes is possible, but not simply by referring to the finite version of the one-dimensional theory of time and band limiting [16]. The graphs under consideration capture locally high-dimensional behavior on the one hand and globally cyclical structure on the other, and they can be viewed as (very symmetric) models for networks that contain highly connected clusters in which a fraction of vertices are connected to other clusters. In such settings, it is of interest to quantify the extent to which low-spectrum vertex functions in the span of Laplacian eigenvectors having small eigenvalues can be localized or concentrated on a single cluster or group of clusters. Various practical problems in high dimensions are addressed in [17,18,19,20]. The graphs at hand amount to replacements of discrete tori by hypercubes. When regarded as graphs whose vertices are elements of ${\mathbb{Z}}_m^d$ and ${\mathbb{Z}}_2^N$, respectively, one replaces each element of ${\mathbb{Z}}_m^d$ by a copy of ${\mathbb{Z}}_2^N$ ($d\le N$), in such a way that certain distinguished antipodal pairs of elements of ${\mathbb{Z}}_2^N$ become boundary elements that are adjacent to corresponding replacements of vertices of ${\mathbb{Z}}_m^d$, while the remaining vertices of each copy of ${\mathbb{Z}}_2^N$ remain insulated.

The Laplacian eigenmodes can be described in terms of two types: those that are supported in a single copy of ${\mathbb{Z}}_2^N$ (and typically vanish on the boundary vertices) and those that pass through replacement boundary vertices. We will show that the latter can be described in terms of eigenfunctions of an augmented Laplacian on ${\mathbb{Z}}_2^N$ that extend via modulation by values of global eigenvectors of ${\mathbb{Z}}_m^d$. Numerical examples in Section 4 indicate that corresponding eigenvectors of spatio–spectral limiting on the replacement graphs can be more concentrated than eigenvectors of a comparable cycle.

The rest of this work is outlined as follows. In the next section, we review discrete tori $T_m^d$ (with vertices indexed by ${\mathbb{Z}}_m^d$) and hypercube graphs $B_N$ (with vertices ${\mathbb{Z}}_2^N$) and define the replacement graphs $T_m^dⓡB_N$. Our main contributions are found in Section 3 and Section 4. In Section 3, we characterize the Laplacian spectrum of $T_m^dⓡB_N$ in terms amenable to spatio–spectral analysis (Theorem 2). Then, in Section 4, we study the ability to localize or concentrate low-spectrum vertex functions on a single copy of $B_N$ within $T_m^dⓡB_N$, focusing primarily on the case $d=1$, comparing to the case of concentration for a simple cycle $C_{m\times (N+1)}$, but also considering a numerical example for $d=2$ (see Figure 1 and Figure 2). In the case $d=1$ (with $C_m=T_m^1$), we establish that there is a basis of vertex functions for the space of low-spectrum vertex functions on $C_mⓡB_N$ (for a suitable value of the spectrum width parameter), such that each basis element has at least half of its squared norm concentrated in a single copy of $B_N$ (Proposition 1 and Theorem 3).

2. Background

2.1. Discrete Tori and (Hyper)Cube Graphs

We denote by $T_m^d$ the (simple Cayley) graph whose vertices are elements of ${\mathbb{Z}}_m^d$, such that two vertices $v=({\ell }_1,\dots ,{\ell }_d)$ and $v^{\prime }=({\ell }_1^{\prime },\dots ,{\ell }_d^{\prime })$, ${\ell }_i,{\ell }_i^{\prime }\in {\mathbb{Z}}_m$ (the integers modulo m) are adjacent if $v-v^{\prime }=\pm e_i$, where $e_i$ is the vertex with entry 1 in the ith coordinate and 0s in the other coordinates. ${\mathbb{Z}}_m^d$ is sometimes called a discrete torus. In the case $m=2$, we instead refer to the corresponding graph $B_N$ ($=T_2^N$) with vertices ${\mathbb{Z}}_2^N$ as a (Boolean) hypercube, or cube for short.

For a finite, undirected, unweighted graph G with vertices V and edges $E\subset V\times V$, $\left|V\right|=n$, we assume an ordering $v_1,\dots ,v_n$ of V. We denote the space of vertex functions $f:V\to \mathbb{R}$ (or to $\mathbb{C}$) as ${\ell }^2\left(G\right)$ with inner product $\langle f,g\rangle ={\sum }_{i=1}^{n}f\left(v_i\right)\overline{g}\left(v_i\right)$. We define the $n\times n$ adjacency matrix A by $A_{ij}=1$ if $(v_i,v_j)\in E$ (denoted $v_i\sim v_j$) and by $A_{ij}=0$ otherwise. Let D be the diagonal matrix with $D_i={\sum }_j{A}_{ij}$. The (unnormalized) Laplacian $L\left(G\right)$ of G is $L=D-A$. The (Laplacian) spectrum of G consists of the eigenvalues of L (with multiplicity). The spectrum of the cycle $C_m$ with vertices ${\mathbb{Z}}_m$ is ${\left\{4{sin}^2(\pi \nu /m)\right\}}_{\nu \in {\mathbb{Z}}_m}$. The corresponding eigenvectors are (up to normalization) ${\{cos(2\pi \nu /m),sin(2\pi \nu /m)\}}_{\nu =0}^{\left[\right(m-1)/2]}$, the real and imaginary parts of the complex eigenvector $e^{2\pi i\nu /m}$, $\nu \in {\mathbb{Z}}_m$. As $T_m^d$ is a d-fold Cartesian product of $C_m$, its eigenvalues are d-term sums of the eigenvalues of $C_m$ and its corresponding eigenvectors are $E_{({\nu }_1,\dots ,{\nu }_d)}(\ell )={\prod }_{j=1}^d{e}^{2\pi i{\ell }_j{\nu }_j/m}$, $\ell =({\ell }_1,\dots ,{\ell }_d)\in {\mathbb{Z}}_m^d$.

The eigenvectors of $L\left(B_N\right)$ can be indexed by $\gamma =({\gamma }_1,\dots ,{\gamma }_N)\in {\{0,1\}}^N$ and have the form (up to normalization) $h_{({\gamma }_1,\dots ,{\gamma }_N)}({ϵ}_1,\dots ,{ϵ}_N)={(-1)}^{\sum {ϵ}_i{\gamma }_i}$ ($({ϵ}_1,\dots ,{ϵ}_N)\in {\{0,1\}}^N$). One has $L\left(B_N\right)h_{({\gamma }_1,\dots ,{\gamma }_N)}=2\left|\gamma \right|h_{({\gamma }_1,\dots ,{\gamma }_N)}$ where $\left|\gamma \right|=\sum {\gamma }_i\in \{0,\dots ,N\}$. In particular, the spectrum of $L\left(B_N\right)$ is $2\times \{0,1,\dots ,N\}$. It is simple to verify that the vectors $h_{\gamma }$ are mutually orthogonal and that ${||h_{\gamma }||}^2=2^N$.

One can re-organize the eigenvectors of $L\left(B_N\right)$ as follows. For each $k=0,\dots ,N$, set $h_{k,\mathrm{rad}}={\left(\genfrac{}{}{0pt}{}{N}{k}\right)}^{-1}{\sum }_{\left|\gamma \right|=k}{h}_{\gamma }$. Then, also, $L\left(h_{k,\mathrm{rad}}\right)=2k{h}_{k,\mathrm{rad}}$ while $h_{k,\mathrm{rad}}\left(\mathbf{0}\right)={\left(\genfrac{}{}{0pt}{}{N}{k}\right)}^{-1}{\sum }_{\left|\gamma \right|=k}1=1$ and $h_{k,\mathrm{rad}}\left(\mathbf{1}\right)={\left(\genfrac{}{}{0pt}{}{N}{k}\right)}^{-1}{\sum }_{\left|\gamma \right|=k}{(-1)}^{\sum {\gamma }_i}={(-1)}^k$. Here, $\mathbf{0}=(0,\dots ,0)$ and $\mathbf{1}=(1,\dots ,1)$. One can check that the functions $h_{k,\mathrm{rad}}$ are constant on Hamming spheres composed of points equidistant from $\mathbf{0}$, and we refer to them as radial eigenvectors. Suppose now that $\psi$ lies in the $2k$-eigenspace of $L\left(B_N\right)$. Then, one can write $\psi ={\sum }_{\left|\gamma \right|=k}{c}_{\gamma }{h}_{\gamma }$. One then has

$$\langle \psi ,h_{k,\mathrm{rad}}\rangle ={\left(\genfrac{}{}{0pt}{}{N}{k}\right)}^{-1}〈\sum _{\left|\gamma \right|=k}{c}_{\gamma }{h}_{\gamma },\sum _{|{\gamma }^{\prime }|=k}{h}_{{\gamma }^{\prime }}〉={\left(\genfrac{}{}{0pt}{}{N}{k}\right)}^{-1}{2}^N\sum _{\left|\gamma \right|=k}{c}_{\gamma }.$$

Thus, if $\langle \psi ,h_{k,\mathrm{rad}}\rangle =0$ then $\sum c_{\gamma }=0$. As $h_{\gamma }\left(\mathbf{0}\right)=1$ and $h_{\gamma }\left(\mathbf{1}\right)={(-1)}^k$ when $\left|\gamma \right|=k$, it follows that $\psi \left(\mathbf{0}\right)=\sum c_{\gamma }=0$ and $\psi \left(\mathbf{1}\right)={(-1)}^k\sum c_{\gamma }=0$. Thus, we have proved the following.

Lemma 1.

For $1\le k\le N$ let $h_{k,\mathrm{rad}}={\left(\genfrac{}{}{0pt}{}{N}{k}\right)}^{-1}{\sum }_{\left|\gamma \right|=k}{h}_{\gamma }$ and let $\psi ={\sum }_{\left|\gamma \right|=k}{c}_{\gamma }{h}_{\gamma }$. If $\langle \psi ,h_{k,\mathrm{rad}}\rangle =0$ then $\psi \left(\mathbf{0}\right)=0=\psi \left(\mathbf{1}\right)$.

We will refer to vectors $\psi ={\sum }_{\left|\gamma \right|=k}{c}_{\gamma }{h}_{\gamma }$, such that $\sum c_{\gamma }=0$ as Dirichlet eigenvectors of $L\left(B_N\right)$. This is consistent with terminology defining Dirichlet eigenvectors as ones of an induced subgraph, in this case induced from ${\mathbb{Z}}_2^N\setminus \{\mathbf{0},\mathbf{1}\}$ and vanishing at boundary vertices, in this case $\{\mathbf{0},\mathbf{1}\}$, (e.g., [21] (Chap. 8)). (The functions $\psi$ also satisfy ${\sum }_{\left|ϵ\right|=r}\psi \left(ϵ\right)=0$ on ${\mathbb{Z}}_2^N$ for each $r=0,\dots ,N$ and could thus be referred to, instead, as spherical functions on $B_N$).

For fixed $v_0\in V\left(B_N\right)$ we refer to the mapping that sends $v\in V\left(B_N\right)$ to $v+v_0$ as a rotation of $B_N$. Rotations of Laplacian eigenvectors on $B_N$ are also Laplacian eigenvectors. Suppose that f is a $\lambda$-eigenvector of $L\left(B_N\right)$, and set $g\left(v\right)=f(v_0+v)$. Then,

$$\begin{array}{c}\left(Lg\right)\left(v\right)=\sum _{w\sim v}[g\left(v\right)-g\left(w\right)]=\sum _{w\sim v}[f(v+v_0)-f(w+v_0)]\hfill \\ =\sum _{w+v_0\sim v+v_0}[f(v+v_0)-f(w+v_0)]=\sum _{u\sim v+v_0}[f(v+v_0)-f\left(u\right)]=\left(Lf\right)(v+v_0)\\ \hfill =\lambda f(v+v_0)=\lambda g\left(v\right).\end{array}$$

As the vectors ${\left\{h_{\gamma }\right\}}_{\left|\gamma \right|=k}$ form an orthogonal basis for the $2k$-eigenspace of $L\left(B_N\right)$, so do the vectors ${\left\{h_{\gamma }(·+v_0)\right\}}_{\left|\gamma \right|=k}$ for fixed $v_0$. In what follows, for a vertex $v_0\in V\left(B_N\right)$, we refer to ${\tilde{v}}_0=v_0+\mathbf{1}$ as the antipodal vertex of v.

Corollary 1.

For $1\le k\le N$, if ψ is an element of the $2k$-eigenspace of $B_N$ that is orthogonal to $h_{k,\mathrm{rad}}(·+v_0)$ then $\psi \left(v_0\right)=0=\psi \left({\tilde{v}}_0\right)$.

Proof. 

The proof is the same as that of Lemma 1, replacing $\mathbf{0}$ and $\mathbf{1}$ by $v_0$ and ${\tilde{v}}_0$, respectively. As in the proof of Lemma 1, $\psi \left(v\right)={\sum }_{\left|\gamma \right|=k}{c}_{\gamma }{h}_{\gamma }(v+v_0)$ lies in the $2k$-eigenspace of $L\left(B_N\right)$, and if $\langle \psi ,h_{k,\mathrm{rad}}(·+v_0)\rangle =0$ then $\psi \left(v_0\right)=0=\psi \left({\tilde{v}}_0\right)$. □

2.2. Replacements of Tori by Cubes

Replacement graphs are usually defined as $(r+1)$-regular graphs $G\circledR H$, where G is d-regular, H is r-regular with $\left|V\right(H\left)\right|=d$ and each vertex $v\in V\left(G\right)$ is replaced by a copy $H_v$ of H with the adjacencies between the replaced vertices being consistent with the adjacencies of G (e.g., [22,23]). Previte [24] considered a version of replacement graphs according to more general substitution rules that allow vertices v of G to be replaced by H having more than ${\mathrm{deg}}_G\left(v\right)$ vertices, by designating a boundary set $\partial H$ of $V\left(H\right)$ that accounts for the adjacencies among the different replaced vertices of G. We consider a version of this here, in which each vertex of G (in our case, G is a discrete torus) is replaced by the same substitution graph H (in our case, H is a cube $B_N$).

For toral dimension d fixed, one distinguishes $\partial B_N=\{v_1,{\tilde{v}}_1,\dots ,v_d,{\tilde{v}}_d\}\subset V\left(B_N\right)$, where ${\tilde{v}}_i=v_i+\mathbf{1}$ is the antipodal vertex of $v_i$ and defines $T_m^dⓡB_N$ as the graph obtained by replacing each vertex $\ell =({\ell }_1,\dots ,{\ell }_d)$ of ${\mathbb{Z}}_m^d$ by a copy $B_N^{\ell }$ of $B_N$, in such a way that the antipodal vertices in successive copies are made adjacent. Specifically, ${\tilde{v}}_i^{\ell }\sim v_i^{\ell +e_i}$. Vertices of $T_m^dⓡB_N$ will be denoted $v^{\ell }$ where $\ell =({\ell }_1,\dots ,{\ell }_d)\in {\mathbb{Z}}_m^d$ and $v\in V\left(B_N\right)$. See Figure 3 for a representation of $C_3ⓡB_2$ and Figure 4 for a partial representation of $T_m^2ⓡB_2$. We will suppress in the sequel any dependence of $T_m^dⓡB_N$ on the particular choice of $\partial B_N$. We will always assume that $d\le N$. This means that the global structure of $T_m^d$ is relatively low dimensional compared to the local structure of $B_N$ and facilitates statements of some results here. The following terminology, justified by Lemma 1 and Corollary 1, will be used in what follows.

Definition 1.

For each $k=0,\dots ,N$, the Hadamard sum $h_{k,\mathrm{rad}}={\left(\genfrac{}{}{0pt}{}{N}{k}\right)}^{-1}{\sum }_{\left|\gamma \right|=k}{h}_{\gamma }$ will be referred to as a radial eigenvector of $L\left(B_N\right)$. Given a distinguished set $\partial B_N=\{v_1,\dots ,v_d;{\tilde{v}}_1,\dots ,{\tilde{v}}_d\}$, $d\le N$, one refers to the rotations $h_{k,\mathrm{rad}}(v+v_i)$ as radial-type eigenvectors of $L\left(B_N\right)$ and to any vector in the orthogonal complement of the span of the radial-type eigenvectors as a Dirichlet-type vector on $B_N$. Any vertex function on $T_m^dⓡB_N$ whose restriction to each $B_N^{\ell }$ ($\ell \in {\mathbb{Z}}_d^m$) is in the span of the radial-type eigenvectors will be called a radial-type vector on $T_m^dⓡB_N$. Any vector in the orthogonal complement of the radial-type vectors on $T_m^dⓡB_N$ will be called a Dirichlet-type vector on $T_m^dⓡB_N$.

Evidently, ${\ell }^2(T_m^dⓡB_N)=R\oplus D$, where R consists of radial-type vectors and D of Dirichlet-type vectors. As a consequence of Corollary 1, Dirichlet vectors vanish at the boundary $\partial B_N^{\ell }$ for each $\ell \in {\mathbb{Z}}_d^m$.

3. Laplacian Spectrum of ${\mathit{T}}_{\mathit{m}}^{\mathit{d}}\mathbf{ⓡ}{\mathit{B}}_{\mathit{N}}$

3.1. Augmented Laplacian

The structure of $T_m^dⓡB_N$ ($d\le N$) allows certain eigenfunctions of $L(T_m^dⓡB_N)$ to be produced by a type of forcing from global eigenvectors of $L\left(T_m^d\right)$ in their complex exponential form, $E_{({\nu }_1,\dots ,{\nu }_d)}({\ell }_1,\dots ,{\ell }_d)={\prod }_{j=1}^d{e}^{2\pi i{\nu }_j{\ell }_j/m}$. The ratio of successive values of $E_{({\nu }_1,\dots ,{\nu }_d)}$ in the jth coordinate is ${\omega }_{{\nu }_j}=e^{2\pi i{\nu }_j/m}$. The ratio of an eigenvector of $L(T_m^dⓡB_N)$ at the jth antipodal boundary pair in successive replacements $B_N^{\ell }$ is also ${\omega }_{{\nu }_j}$: $f\left({\tilde{v}}_j^{\ell }\right)={\omega }_{{\nu }_j}f\left({\tilde{v}}_j^{\ell -{\mathit{e}}_j}\right)$, if $f\left({\tilde{v}}_j^{\mathbf{0}}\right)\ne 0$. We use this to express eigenvectors of $L(T_m^dⓡB_N)$ in terms of those of $L\left(B_N\right)$ augmented by operators $C_{{\omega }_{{\nu }_j}}$ that account for boundary adjacencies: if $v_j^{\ell }\in \partial B_N^{\ell }$ then $\left(L(T_m^dⓡB_N)f\right)\left(v_j^{\ell }\right)=\left(L\left(B_N\right)f\right)\left(v_j^{\ell }\right))+[f\left(v_j^{\ell }\right)-f\left({\tilde{v}}_j^{\ell -{\mathit{e}}_j}\right)]$. In what follows, we let $n\left(v\right)\in \{1,\dots ,2^N\}$ be an indexing of the vertices of $B_N$.

Definition 2.

For $\alpha \in \mathbb{C},\left|\alpha \right|=1$ and antipodal pair $(v,\tilde{v})\in V\left(B_N\right)$, let $C_{\alpha ,v}$ be the $2^N\times 2^N$ matrix, such that $C_{\alpha ,v}(n\left(v\right),n\left(v\right))=1=C_{\alpha ,v}(n\left(\tilde{v}\right),n\left(\tilde{v}\right))$, $C_{\alpha ,v}(n\left(v\right),n\left(\tilde{v}\right))=-1/\alpha =1/C_{\alpha ,v}(n\left(\tilde{v}\right),n\left(v\right))$ and $C_{\alpha ,v}(n,m)=0$ otherwise. For $v_1,\dots ,v_d\in V\left(B_N\right)$ and ${\alpha }_1,\dots ,{\alpha }_d\in \mathbb{C}$, $\left|{\alpha }_j\right|\hspace{0.17em}=\hspace{0.17em}1$, one defines an augmented Laplacian $L(v_1,\dots ,v_d;{\alpha }_1,\dots ,{\alpha }_d)=L\left(B_N\right)+{\sum }_{j=1}^d{C}_{{\alpha }_j,v_j}$.

Theorem 1.

Let $1\le d\le N$, set ${\omega }_{{\nu }_j}=e^{2\pi i{\nu }_j/m}$, and let $(v_1,{\tilde{v}}_1),\dots ,(v_d,{\tilde{v}}_d)$ be a collection of d distinct antipodal pairs of vertices in $V\left(B_N\right)$. Then, any radial-type eigenvector ϕ of the augmented Laplacian $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ can be extended to an eigenvector of $L(T_m^dⓡB_N)$ by setting $\varphi \left(v^{\ell }\right)=\left({\prod }_{j=1}^d{\omega }_{{\nu }_j}^{{\ell }_{\nu }}\right)\varphi \left(v^{\mathbf{0}}\right)$.

Observe that if $\varphi$ is of Dirichlet type, meaning that $\varphi$ vanishes at each of the antipodal pairs, then $\varphi$ is in the kernel of $C_{{\omega }_{\nu },v_{\nu }}$ for each $\nu =1,\dots ,d$ and, therefore, $L(T_m^dⓡB_N)\varphi =L\left(B_N\right)\varphi$. In general, if v is an insulated vertex of $B_N$ then $\left(L(T_m^dⓡB_N)f\right)\left(v\right)=L\left(B_N\right)f\left(v\right)$ for any vertex function f.

Proof. 

If $v_j$ is a boundary vertex and $\varphi \left(v^{\ell }\right)=\left({\prod }_{j=1}^d{\omega }_{{\nu }_j}^{{\ell }_{\nu }}\right)\varphi \left(v^{\mathbf{0}}\right)$ then

$$\begin{array}{c}\left(L(T_m^dⓡB_N)\varphi \right)\left(v_j^{\ell }\right)=\left(L\left(B_N\right)\phi \right)\left(v_j^{\ell }\right)+[\varphi \left(v_j^{\ell }\right)-\varphi \left({\tilde{v}}_j^{\ell -{\mathit{e}}_j}\right)]\hfill \\ =\left(L\left(B_N\right)\phi \right)\left(v_j^{\ell }\right)+[\varphi \left(v_j^{\ell }\right)-{\omega }_{{\nu }_j}^{-1}\varphi \left({\tilde{v}}_j^{\ell }\right)]\\ \hfill =\left(L\left(B_N\right)\phi \right)\left(v_j^{\ell }\right)+\left(C_{{\omega }_{{\nu }_j},v_j}\varphi \right)\left(v_j^{\ell }\right)=\left(L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})\varphi \right)\left(v_j^{\ell }\right),\end{array}$$

where, in the last identity, we use the fact that $\left(C_{{\omega }_{{\nu }_i},v_i}\varphi \right)\left(v_j^{\ell }\right)=0$ unless $i=j$ (the nonzero entries of the matrices $C_{{\omega }_{{\nu }_j},v_j}$ are in different rows and columns for different j). Thus, $\varphi$ is an eigenvector of $L(T_m^dⓡB_N)$ if and only if $\varphi$ restricted to $B_N$ is an eigenvector of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$. □

Corollary 2.

If λ is an eigenvalue of $L(T_m^dⓡB_N)$ then it is also an eigenvalue of the augmented Laplacian $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ for specific ${\omega }_{{\nu }_j}=e^{2\pi i{\nu }_j/m}$.

The corollary is subsumed in Theorem 2 below.

3.2. Laplacian Spectrum of $T_m^dⓡB_N$, $d\le N$

A complete set of eigenvectors of $L(T_m^dⓡB_N)$ is described by the following.

Theorem 2.

Fix integers $m>2$, $d\ge 1$ and $N\ge d$. Let $T_m^dⓡB_N$ be equipped with boundary vertex pairs $v_1,\dots ,v_d$ and ${\tilde{v}}_1,\dots ,{\tilde{v}}_d$, where $\widetilde{v_i}=v_i+\mathbf{1}\in {\mathbb{Z}}_2^N$. The eigenvectors of $L(T_m^dⓡB_N)$ are of one of the following two types:

(i) Dirichlet type. These are vertex functions φ on $T_m^dⓡB_N$ that vanish on the boundary vertices $v_i^{\ell }$ and ${\tilde{v}}_i^{\ell }$ for each $\ell \in {\mathbb{Z}}_m^d$ and each $i=1,\dots ,d$. The span of the $2k$-Dirichlet eigenvectors ($k=1,\dots ,N-1$) has dimension $m^d\left(\left(\genfrac{}{}{0pt}{}{N}{k}\right)-d\right)$.

(ii) Radial type. These are vertex functions φ on $T_m^dⓡB_N$ whose restrictions to any $B_N^{\ell }$ are eigenvectors of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ for some of $({\nu }_1,\dots ,{\nu }_d)\in {\mathbb{Z}}_m^d$. For fixed k, $1\le k<N$ there are $d{m}^d$ radial-type eigenvectors having eigenvalues in $[2k,2k+2)$ and there are $m^d$ such with eigenvalues in $[0,2)$ or in $2[N,N+1)$.

As k ranges from 0 to N, the number of Dirichlet eigenvalues plus the number of radial eigenvalues (with multiplicity) is equal to $m^d{\sum }_{k=0}^N\left(\genfrac{}{}{0pt}{}{N}{k}\right)=2^N{m}^d$.

The last statement indicates that the Dirichlet-type and radial-type eigenvectors together form a basis for ${\ell }^2(T_m^dⓡB_N)$. Before proceeding to the proof of Theorem 2, we provide some preliminary results.

Lemma 2.

Let $N\ge 3$. For each $1\le d\le N$ there are vertices $v_1,\dots ,v_d$, such that for each $k=1,\dots ,N-1$ the rotated radial vectors $h_{k,\mathrm{rad}}(v_j+·)$ are linearly independent.

Henceforth, we shall always assume that boundary vertex pairs $(v_1,{\tilde{v}}_1),\dots ,(v_d,{\tilde{v}}_d)$ are chosen such that the vectors $h_{k,\mathrm{rad}}(v_j+·)$, $j=1,\dots ,d$ ($d\le N$) are linearly independent. The proof of Lemma 2 can be found in Appendix A.

Lemma 3.

The augmented Laplacian $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ ($d\le N$) has $d(N-1)+2$ linearly independent radial-type eigenvectors.

Proof. 

The Dirichlet-type eigenvectors of $L\left(B_N\right)$ and of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ are the same. Denote by ${\mathcal{D}}_N$ the span of all Dirichlet-type eigenvectors. ${\mathcal{D}}_N$ has dimension ${\sum }_{k=1}^{N-1}(\left(\genfrac{}{}{0pt}{}{N}{k}\right)-d)=2^N-2-(N-1)d$, so its orthogonal complement has dimension $(N-1)d+2$. By Lemma 2, the span of the radial-type eigenvectors $h_{k,\mathrm{rad}}(v_j+·)$, $j=1,\dots ,d$; $k=1,\dots ,N-1$, together with the constant vector and the vector $h_N$, has dimension $d(N-1)+2$ and, therefore, forms a basis for the orthogonal complement of ${\mathcal{D}}_N$ inside ${\ell }^2\left(B_N\right)$. Consequently, the eigenvectors of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ that are orthogonal to ${\mathcal{D}}_N$ are all in the span of the radial-type eigenvectors of $L\left(B_N\right)$. As $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ has full rank $2^N$, these vectors also span the orthogonal complement of ${\mathcal{D}}_N$.□

Proof of Theorem  2.

The dimension of the $2k$-eigenspace of $L\left(B_N\right)$ is $\left(\genfrac{}{}{0pt}{}{N}{k}\right)$. Vanishing of an eigenvector at antipodal points $(v,\tilde{v})$ imposes a linear constraint $\sum c_k=0$ when $\phi ={\sum }_{\left|\gamma \right|=k}{c}_{\gamma }{h}_{\gamma }(·+v)$ is an expansion in rotated Hadamard vectors. This implies that there are $\left(\genfrac{}{}{0pt}{}{N}{k}\right)-d$ linearly independent $2k$-Dirichlet eigenvectors of $L\left(B_N\right)$. Such vectors are also $2k$-eigenvectors of $L(T_m^dⓡB_N)$ that are supported on a single copy $B_N^{\ell }$ of $B_N$. As $T_m^dⓡB_N$ contains $m^d$ replacements by $B_N$, the dimension of the $2k$-Dirichlet eigenspace of $L(T_m^dⓡB_N)$ is $m^d\left(\left(\genfrac{}{}{0pt}{}{N}{k}\right)-d\right)$.

The augmented Laplacians $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ have full rank. As the Hadamard vectors are complete in ${\ell }^2\left(B_N\right)$, the orthogonal complement of the span of the Dirichlet eigenvectors of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ is spanned by radial eigenvectors of $L\left(B_N\right)$. For $k=1,\dots ,N-1$ there are d linearly independent radial-type $2k$-eigenvectors of $L\left(B_N\right)$. The Hermitian matrices $C_{\alpha ,v}$ are positive semidefinite of norm two. For $v\ne v^{\prime }\in V\left(B_N\right)$ (and $\tilde{v}\ne v^{\prime }$) the matrices $C_{\alpha ,v}$ and $C_{{\alpha }^{\prime },v^{\prime }}$ have their nonzero entries in different rows and columns. Therefore, also, ${\sum }_{\nu =1}^d{C}_{v_{\nu },{\omega }_{\nu }}$ is positive semidefinite of norm two. It is then a straightforward consequence of the Courant–Fischer min-max theorem (e.g., [25]) that $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ has (at least) d eigenvalues in $[2k,2k+2)$ for $k=1,\dots ,N-1$ and that the corresponding eigenvectors lie in the span of the radial-type eigenvectors of $L\left(B_N\right)$. For the same reason, for $k=0$ and $k=N$ there is a single eigenvector of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ in $[0,2)$ and in $[2N,2N+2)$.

The proof of Lemma 3 shows that the collection ${\mathcal{B}}_{\mathrm{rad}}$ consisting of radial-type eigenvectors of $L\left(B_N\right)$ of the form $h_{k,\mathrm{rad}}(v_j+·)$, $j=1,\dots ,d$, $k=1,\dots N-1$, together with $h_0$ (the constant vector on $B_N$) and $h_N$ ($L\left(B_N\right)h_N=2N{h}_N$), forms a basis for the span of the radial-type eigenvectors of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ for each choice of $({\nu }_1,\dots ,{\nu }_N)\in {\mathbb{Z}}_m^d$. In particular, $\mathrm{span}\left({\mathcal{B}}_{\mathrm{rad}}\right)$ is equal to the orthogonal complement of ${\mathcal{D}}_N$, the span of the Dirichlet-type eigenvectors of $L\left(B_N\right)$ and, hence, by the proof of Lemma 3, equal to the span of the radial-type eigenvectors of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$, regardless of $({\nu }_1,\dots ,{\nu }_N)$. As the vectors $E_{({\nu }_1,\dots ,{\nu }_N)}(\ell )={\prod }_{j=1}^d{\omega }_{{\nu }_j}^{{\ell }_j}={\prod }_{j=1}^d{e}^{2\pi i{\nu }_j{\ell }_j/m}$ form a basis for ${\ell }^2\left({\mathbb{Z}}_m^d\right)$ (the Fourier basis), it follows that the tensor products of elements of ${\mathcal{B}}_{\mathrm{rad}}$ and the vectors $E_{({\nu }_1,\dots ,{\nu }_N)}$ form a tensor product basis for the span of the radial-type eigenvectors of $L(T_m^dⓡB_N)$ when regarding $V(T_m^dⓡB_N)$ as the product ${\mathbb{Z}}_m^d\times {\mathbb{Z}}_2^N$. Therefore, the dimension of this span is the number of elements in the tensor product basis, which is $m^d((N-1)d+2)$ by Lemma 3. As above, for each of the $m^d$ choices of $({\nu }_1,\dots ,{\nu }_d)$ there are d eigenvalues of $L(v_1,\dots ,v_d;{\omega }_{{\nu }_1},\dots ,{\omega }_{{\nu }_d})$ in $[2k,2k+2)$ ($k=1,\dots ,N-1$) that by Corollary 2 are also (radial-type) eigenvalues of $L(T_m^dⓡB_N)$. Thus, there are $d{m}^d$ radial-type eigenvalues of $L(T_m^dⓡB_N)$ in $[2k,2k+2)$ for each $k=1,\dots ,N-1$. Similarly, there are $m^d$ eigenvalues of $L(T_m^dⓡB_N)$ each in $[0,2)$ and in $[2N,2N+2)$.

As the number of Dirichlet-type eigenvectors of $L(T_m^dⓡB_N)$ is $m^d{\sum }_{k=1}^{N-1}\left(\left(\genfrac{}{}{0pt}{}{N}{k}\right)-d\right)=m^d(2^N-2-(N-1)d)$, the number of radial-type eigenvectors of $L(T_m^dⓡB_N)$ is $m^d((N-1)d+2)$, and as the spans of the Dirichlet eigenvectors and radial-type eigenvectors of $L(T_m^dⓡB_N)$ are orthogonal to one another, it follows that the sum of these spans has dimension $m^d{2}^N$ and, therefore, the Dirichlet and radial-type eigenvectors of $L(T_m^dⓡB_N)$ together span ${\ell }^2(T_m^dⓡB_N)$. This completes the proof.□

4. Spatio–Spectral Limiting

As they vanish at boundary vertices, Dirichlet-type eigenvectors of $L(T_m^dⓡB_N)$ can be spatially supported in a single replacement $B_N^{\ell }$. For small k, the $2k$-Dirichlet eigenvectors can be regarded as low-spectrum eigenvectors corresponding to small Laplacian eigenvalues. This phenomenon of localized low-spectrum eigenmodes is very different from the traditional Euclidean setting in which the Paley–Wiener theorem (e.g., [26]) prohibits bandlimited signals from having compact support and which has analogues in the finite setting of a discrete cycle. Similar uncertainty principles apply in discrete settings, such as tori $T_m^d$ (e.g., [27], cf., [16]). However, localization of a global Laplacian eigenvector on a single cluster is less surprising when the full graph can be viewed as a collection of weakly connected clusters whose eigenvectors may vanish at cluster boundaries.

Definition 3.

For a replacement graph $GⓡH$ and $v\in G$, we define by $Q_v$ the operator, such that $\left(Q_{v}f\right)\left(w\right)=f\left(w\right)$ if $w\in V\left(H_v\right)\subset V(GⓡH)$ is a vertex in the replacement of v by H, and $f\left(w\right)=0$ otherwise. We define by $P_K$ the projection onto the Paley–Wiener space ${\mathrm{PW}}_K$ spanned by eigenvectors of $L(GⓡH)$ having eigenvalues less than or equal to K. When choices of v and K are clear from context we will abbreviate $Q_v$ as Q and $P_K$ as P.

4.1. Radial-Type Eigenvectors of $PQ$, d = 1 Case

We focus here mainly on the case of replacement of a cycle ($d=1$) because the analogue of time and band limiting in Euclidean space (and tori) itself is substantially more complicated than the one-dimensional theory. For a vertex function f on $C_mⓡB_N$ we let $Qf\left(v\right)=f\left(v\right)$ if $v\in B_N^0$ and $Qf\left(v\right)=0$ otherwise. We then let $\left(Pf\right)\left(v\right)={\sum }_{\lambda \le K}\left({\pi }_{\lambda }f\right)\left(v\right)$, where ${\pi }_{\lambda }f$ is the orthogonal projection onto the $\lambda$-eigenspace of $C_mⓡB_N$. Here, ${\pi }_{\lambda }f={\sum }_{{\phi }_{\lambda }}\langle f,{\phi }_{\lambda }\rangle {\phi }_{\lambda }$, where the sum runs over an orthonormal basis of the $\lambda$-eigenspace of $L(C_mⓡB_N)$. We can write ${\mathrm{PW}}_K(C_mⓡB_N)=D_K\oplus R_K$, where $D_K$ is spanned by those Laplacian eigenvectors of Dirichlet type with eigenvalue at most K and $R_K$ is spanned by corresponding eigenvectors of radial type. By Theorem 2, $\dim \left(D_K\right)=m{\sum }_{k=1}^K\left(\left(\genfrac{}{}{0pt}{}{N}{k}\right)-1\right)$ while $\dim \left(R_K\right)=m(K+1)$.

For N fixed, let ${\left\{h_{k,\mathrm{rad}}\right\}}_{k=0}^N$ be the orthonormal radial eigenvectors on $B_N$ defined as above. For fixed $\nu \in {\mathbb{Z}}_m$, let ${\left\{{\phi }_{k,\nu }\right\}}_{k=0}^N$ be orthonormal radial-type eigenvectors of the augmented Laplacian $L(\mathbf{0};{\omega }_{\nu })$. Let $c_{\kappa ,k,\nu }=\langle h_{\kappa ,\mathrm{rad}},{\phi }_{k,\nu }\rangle$ be the coefficients of the change of basis with respect to $\left\{h_{\kappa ,\mathrm{rad}}\right\}$ and $\left\{{\phi }_{k,\nu }\right\}$ on the radial space of ${\ell }^2\left(B_N\right)$ whose kth column is ${[c_{0,k,\nu },\dots ,c_{N,k,\nu }]}^T$. That is, ${\phi }_{k,\nu }={\sum }_{\kappa =0}^N{c}_{\kappa ,k,\nu }{h}_{\kappa ,\mathrm{rad}}$. The radial-type eigenvectors ${\phi }_{k,\nu }$ of $L(\mathbf{0};{\omega }_{\nu })$ may be viewed as perturbations of those of $L\left(B_N\right)$, as indicated by the following hypothesis. The corresponding eigenvectors of $L(C_mⓡB_N)$ have the form

$$\phi \left(v_n^{\ell }\right)={\phi }_{k,\nu }\left(v_n\right)e^{2\pi i\nu \ell /m},0\le \ell <m;n=0,\dots ,2^N-1,$$

(1)

where ${\left\{v_n\right\}}_{n=0}^{2^N-1}$ now refers to an indexing of the vertices of $B_N$. We state the following as a hypothesis, as technical details do little to illuminate the consequences of interest here.

Hypothesis 1.

For fixed integers $N\ge 3$ and m odd, the matrix of the change of basis from $\left\{{\phi }_{k,\nu }\right\}$ to $\left\{h_{\kappa ,\mathrm{rad}}\right\}$ on the radial subspace of ${\ell }^2\left(B_N\right)$ is diagonally dominant. Specifically, $|c_{k,k,\nu }{|}^2>2/3$ for each $k=0,\dots ,N$ and $\nu =0,\dots ,m-1$, whereas ${\sum }_{\kappa =0}^N{\left|c_{\kappa ,k,\nu }\right|}^2=1$.

Here, m is odd simply to avoid the degenerate case ${\omega }_{\nu }=-1$ when $\nu =m/2$. The hypothesis can be verified for computable N using tools mentioned in our Data Availability Statement. It can be shown that as $N\to \infty$, $|c_{k,k,\nu }|\to 1$. A proof of this fact requires detailed analysis of the values $h_{k,\mathrm{rad}}\left(\mathbf{0}\right)$, which tend to zero as $N\to \infty$, along with the trace identity $\mathrm{tr}L(\mathbf{0};{\omega }_{\nu })=N(N+1)+2=\mathrm{tr}L\left(B_N\right)+2$.

Proposition 1.

Assuming Hypothesis 1, there exist $N+1$ linearly independent vectors in the span of the extensions to $C_mⓡB_N$ of the augmented radial-type eigenvectors ${\left\{{\phi }_{k,\nu }\right\}}_{k=0,\nu =0}^{N,m-1}$ that have at least half of their squared ${\ell }^2$-norms concentrated in the $\ell =0$ block $B_N^0$ of $C_mⓡB_N$.

Proof. 

The proof boils down to identifying suitable linear combinations ${\Phi }_k$ of radial-type eigenvectors on $C_mⓡB_N$. Set

$${\phi }_{k,\nu }\left(v_n\right)=c_{k,k,\nu }{h}_{k,\mathrm{rad}}\left(v_n\right)+\sum _{\kappa \ne k}{c}_{\kappa ,k,\nu }{h}_{\kappa ,\mathrm{rad}}\left(v_n\right)=c_{k,k,\nu }{h}_{k,\mathrm{rad}}\left(v_n\right)+{\phi }_{k,\nu }^{\prime }\left(v_n\right),$$

where ${\phi }_{k,\nu }^{\prime }$ is the sum of the terms with $\kappa \ne k$. Abbreviating $c_{k,k,\nu }=c_{k,\nu }$, set

$$\begin{array}{c}{\Phi }_k\left(v_n^{\ell }\right)=\sum _{\nu =0}^{m-1}\frac{1}{c_{k,\nu }}{\phi }_{k,\nu }\left(v_n\right)e^{2\pi i\ell \nu /m}=\sum _{\nu =0}^{m-1}{h}_{k,\mathrm{rad}}\left(v_n\right)e^{2\pi i\ell \nu /m}+\sum _{\nu =0}^{m-1}\frac{1}{c_{k,\nu }}{\phi }_{k,\nu }^{\prime }\left(v_n\right)e^{2\pi i\ell \nu /m}\hfill \\ \hfill =m{h}_{k,\mathrm{rad}}\left(v_n\right){\mathbf{1}}_{B_N^0}+\sum _{\nu =0}^{m-1}\frac{1}{c_{k,\nu }}{\phi }_{k,\nu }^{\prime }{e}^{2\pi i\ell \nu /m}=m{h}_{k,\mathrm{rad}}\left(v_n\right){\mathbf{1}}_{B_N^0}+{\Phi }_k^{\prime }\left(v_n^{\ell }\right),\end{array}$$

where ${\mathbf{1}}_{B_N^{\ell }}$ is the indicator function of $B_N^{\ell }$. Assuming Hypothesis 1, one has

$$\begin{array}{c}{||{\Phi }_k^{\prime }||}_{{\ell }^2(C_mⓡB_N)}^2=\sum _{n=0}^{2^N-1}\sum _{\ell =0}^{m-1}{\left|\sum _{\nu =0}^{m-1}\frac{1}{c_{k,\nu }}{\phi }_{k,\nu }^{\prime }\left(v_n\right)e^{2\pi i\ell \nu /m}\right|}^2\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=m\sum _{\nu =0}^{m-1}\sum _{n=0}^{2^N-1}{\left|\frac{1}{c_{k,\nu }}{\phi }_{k,\nu }^{\prime }\left(v_n\right)\right|}^2=m\sum _{\nu }\frac{1}{|c_{k,\nu }{|}^2}\sum _{\kappa \ne k}{\left|c_{\kappa ,k,\nu }\right|}^2<m^2\frac{1}{3}/\frac{2}{3}=\frac{m^2}{2},\end{array}$$

where we use Plancherel’s theorem for Fourier series on ${\mathbb{Z}}_m$ in the second identity and the orthonormal expansion of ${\phi }_{k,\nu }^{\prime }$ in $h_{\kappa ,\mathrm{rad}}$’s in the third. As $\parallel m{h}_{k,\mathrm{rad}}\left(v_n\right){\mathbf{1}}_{B_N^0}{\parallel }^2=m^2$, it follows that ${||{\Phi }_k{\mathbf{1}}_{B_N^0}||}^2/{\parallel {\Phi }_k\parallel }^2>1/2$ in ${\ell }^2(C_mⓡB_N)$. This applies to each $k=0,\dots ,N$ and the proposition follows. □

Theorem 3.

Assuming Hypothesis 1, let $0<K\le N$. Let $ϵ>0$ be such that, for each $\nu =0,\dots ,m-1$, the extensions ${\phi }_{K-1,\nu }$ have Laplacian eigenvalues smaller than $2K-ϵ$. Then, there is a basis of ${\mathrm{PW}}_{2K-ϵ}(C_mⓡB_N)$ of functions $\Psi$, such that for each basis element there is some $\ell \in \{0,\dots ,m-1\}$, such that $\parallel \Psi {\mathbf{1}}_{B_N^{\ell }}{\parallel }^2>\frac{1}{2}{\parallel \Psi \parallel }^2$ in ${\ell }^2(C_mⓡB_N)$.

Proof. 

The vectors ${\Phi }_k$ in the proof of Proposition 1 form a set of $N+1$ linearly independent vectors satisfying ${||{\Phi }_k{\mathbf{1}}_{B_N^0}||}^2/{\parallel {\Phi }_k\parallel }^2>1/2$. The shifted vectors ${\Phi }_k^{\ell }\left(v_n^{{\ell }^{\prime }}\right)={\Phi }_k\left(v_n^{{\ell }^{\prime }-\ell }\right)$ (where $\ell -{\ell }^{\prime }$ is defined modulo m) then satisfy ${||{\Phi }_k{\mathbf{1}}_{B_N^{\ell }}||}^2/{\parallel {\Phi }_k\parallel }^2>1/2$. They are linearly independent from one another and also from the vectors ${\Phi }_k^{{\ell }^{\prime }}$ for ${\ell }^{\prime }\ne \ell$ as a consequence of their concentrations. Thus, as k ranges over $0,\dots ,N$ and ℓ ranges over ${\mathbb{Z}}_m$ the vectors ${\Phi }_k^{\ell }$ form a collection of $m(N+1)$ linearly independent vectors. As ${\Phi }_k$ is a linear combination of extensions of augmented Laplacian eigenvectors with eigenvalues in $[2k,2k+2)$, it follows that each ${\Phi }_k^{\ell }\in {\mathrm{PW}}_{2k+2}(C_mⓡB_N)$.

By Theorem 2, for each ℓ there is a collection of ${\sum }_{k=1}^{K-1}(\left(\genfrac{}{}{0pt}{}{N}{k}\right)-1)$ linearly independent eigenvectors of $L(C_mⓡB_N)$ of Dirichlet type supported in $B_N^{\ell }$. These are also linearly independent from the vectors ${\Phi }_k^{\ell }$. As $(k,\ell )$ ranges over $\{0,\dots ,K-1\}\times \{0,\dots ,m-1\}$, the vectors ${\Phi }_k^{\ell }$ together with the corresponding Dirichlet eigenvectors thus form a collection of

$$m\left[K+\left(\sum _{k=1}^{K-1}\left(\genfrac{}{}{0pt}{}{N}{k}\right)-1\right)\right]=m\sum _{k=0}^{K-1}\left(\genfrac{}{}{0pt}{}{N}{k}\right)$$

linearly independent vectors, which is equal to the dimension of ${\mathrm{PW}}_{2K-ϵ}(C_mⓡB_N)$ for $ϵ>0$, sufficiently small that ${\mathrm{PW}}_{2K-ϵ}(C_mⓡB_N)$ contains all extended radial-type eigenvectors ${\phi }_{k,\nu }$, $k=0,\dots ,K-1$. □

4.2. Examples

Here, we illustrate the facts outlined above by three examples, chosen so that in each case the operator $PQ$, where Q denotes truncation to a single vertex replacement and P denotes projection onto a suitably chosen low spectrum, has precisely three radial-type eigenvalues of at least $1/2$. The examples are: (i) $C_{21\times 8}\simeq C_{21}ⓡP_8$ ($P_8$ is the path on eight vertices), (ii) $C_{21}ⓡB_7$, and (iii) $T_{11}^2ⓡB_4$. The distribution of eigenvalues of $PQ$ operators for a cycle $C_m$ was characterized in [28]. The example of $C_{21\times 8}\simeq C_{21}ⓡP_8$ is a special case. Distribution of eigenvalues of $PQ$ operators for general graphs $T_m^dⓡB_N$ has not been studied previously. The Laplacian eigenvalues of these graphs are plotted on the right (upper graphs) in Figure 2. The eigenvalues of the corresponding $PQ$ operators are plotted in Figure 5. In each case, Q truncates to one instance of vertex replacement: to the first copy of $P_8$, $B_7$ and $B_4$, respectively. Also in each case, P denotes projection onto a low-spectrum span of the Laplacian eigenvectors with n smallest eigenvalues for suitable n.

In the case of $C_{21\times 8}$ we choose $n=63=3\times 21$. Here, we define Q to be the cutoff to the first eight vertices of the cycle and P to be the projection onto the span of the first 63 Laplacian eigenvalues of the cycle. The eigenvalues are shown in the top plot of Figure 5. It is known in the case of a cycle $C_m$ that the number of eigenvalues of $PQ$ larger than $1/2$ (3 here) is the normalized time–bandwidth product: the length of the support of truncation Q (8 here) times the number of points in the truncated spectrum ($3\times 21$ here) divided by m ($m=8\times 21$ here) (see, e.g., [5,14,28,29,30]).

In the case of $C_{21}ⓡB_7$, we take Q to be the cutoff on $B_7^0$ and P is the projection onto the Laplacian eigenvectors of $L(C_{21}ⓡB_7)$ whose eigenvalues are less than or equal to 6. The eigenvalues are shown in the middle plot of Figure 5. There are 60 eigenvalues of $PQ$ equal to 1. These correspond to the span of the Dirichlet eigenvectors supported in $B_7^0$ whose Laplacian eigenvalues are 2 (with multiplicity 6), 4 (20) or 6 (34), according to Theorem 2. The case of $C_{21}ⓡB_7$ also admits 3 eigenvalues of $PQ$ whose eigenvectors are in the span of the radial-type eigenvectors with Laplacian eigenvalues smaller than 6, as explained by Hypothesis 1 above. For other spectrum cutoffs $2K$, the number of radial-type eigenvectors of $PQ$ is equal to K, which is the number of radial-type eigenvectors of $L\left(B_N\right)$ ($K\le N$) with an eigenvalue smaller than K.

In the case of $T_{11}^2ⓡB_4$, we take Q to be the cutoff to $B_4^{(0,0)}$ and P is the projection onto the span of the Laplacian eigenvectors whose eigenvalues are at most 4. The eigenvalues are shown in the bottom plot of Figure 5. $PQ$ admits 2 eigenvalues equal to 1 corresponding to the pair of Dirichlet 2-eigenvectors of $L\left(B_4\right)$ supported in $B_4^{(0,0)}$. This case also admits three eigenvectors of $PQ$ whose eigenvalues lie in $(1/2,1)$. These eigenvectors of $PQ$ lie in the span of the radial-type eigenvectors of $L(T_{11}^2ⓡB_4)$ that extend augmented Laplacian eigenvectors with eigenvalues smaller than 4 (there are $121={11}^2$ such eigenvalues smaller than 2 and 242 between 2 and 4; see Figure 2).

The radial-type eigenvalues of $PQ$ for these three cases are plotted in Figure 1. The eigenvalues are equal to the concentrations (${\parallel Q\phi \parallel }^2/{\parallel \phi \parallel }^2$) of the vectors on a single replacement. The case of $C_{21}ⓡB_7$ admits the most concentrated radial-type vectors, followed by $T_{11}^2ⓡB_4$ and then $C_{21\times 8}$. In the case of $C_{21}ⓡB_7$, radial-type vectors are constant on vertices $v^{\ell }$ that are equidistant from ${\mathbf{0}}^{\ell }$. This case can thus be viewed as a cycle of length $21\times 8$ with periodically weighted edges. The weighting allows for more effective concentration than in the case of an unweighted cycle of equal length.

The case of $T_{11}^2ⓡB_4$ is more complicated, but comparison with $C_{21}ⓡP_8$ in Figure 1 suggests that $B_4$ provides sufficient degrees of freedom for low-spectrum oscillations to be concentrated more than in the case of a simple cycle.

In each of the three cases ($C_{21}ⓡP_8$, $C_{21}ⓡB_7$ and $T_{11}^2ⓡB_4$) considered, there are three radial-type eigenvectors of $PQ$ concentrated on $P_8$, $B_7$ and $B_4$, respectively. For each case, we plot the third-most concentrated radial-type eigenvector of $PQ$ in Figure 2. Specifically, the third radial-type eigenvector of $PQ$ corresponding to the third eigenvalue shown in Figure 1 is plotted on the left in Figure 2 for cases of $C_{21\times 8}$ (top), $C_{21}ⓡB_7$ (middle) and $T_{11}^2ⓡB_4$ (bottom). Their Fourier coefficients (inner products with Laplacian eigenvectors) are plotted on the right in Figure 2 below plots of the corresponding Laplacian eigenspectra. One can verify that, together with the corresponding Dirichlet-type eigenvectors of $PQ$, counting an equal number of eigenfunctions of $PQ$ for a shifted version $Q_v$ of the cutoff Q corresponding to each vertex replacement $B_N^v$, one obtains bases of the corresponding Paley–Wiener space ${\mathrm{PW}}_K$ for each case consisting of eigenvectors that are concentrated on a single vertex replacement. This is an analogue of the shifted prolate bases of the Paley–Wiener subspaces of $L^2\left(\mathbb{R}\right)$ studied in [31].

5. Discussion

The graphs $T_m^dⓡB_N$ serve as simple models for networks that have a high degree of local connectivity and relatively low-dimensional global geometry. These graphs are highly structured. But what we lose in generality we gain in precision, formulating and verifying our results in terms of classical techniques of harmonic analysis. We have described the Laplacian spectrum of these graphs in a manner that facilitates analysis of spatio–spectral limiting operators $PQ$ that first truncate to a cluster (i.e., instance of $B_N$ inside $T_m^dⓡB_N$), then project onto the low part of the Laplacian spectrum, on these graphs. A relatively high-dimensional Dirichlet-type subspace lies in the span of the Laplacian eigenvectors with eigenvalues smaller than $2K$, $K<N$ when $d\ll N$. We have also quantified radial-type eigenvectors of $PQ$ that are concentrated on a single cluster. In the specific case of cycle replacements $C_mⓡB_N$, we have shown that there are K linearly independent eigenvectors of $PQ$ with eigenvalues larger than $1/2$ that lie in the span of radial-type Laplacian eigenvectors having Laplacian eigenvalues smaller than $2K$. Shifts of these concentrated vectors onto other clusters, together with those of the corresponding Dirichlet eigenvectors, form a basis of the corresponding Paley–Wiener space of $C_mⓡB_N$. Numerical results show that these basis elements are more concentrated than those of a comparable basis of shifted eigenvectors of the corresponding $PQ$ operator in the case of a simple cycle.

Developing methods to identify suitable bases or frames for low-spectrum Paley–Wiener spaces is an important area of research in graph signal processing, particularly in the case of clustered but otherwise general graphs (e.g, [32]). Analogues of Slepian-type eigenfunctions on graphs [8,9] are a relatively new aspect of this. While the techniques developed here cannot be brought directly to bear on general graphs, we anticipate that versions of the techniques used herein can be developed for certain tinker toy models: graphs in which one of a family $\left\{B_{\nu }\right\}$ of structured cluster graphs can be substituted for each vertex v of a global graph S, allowing low-spectrum basis functions spatially concentrated on a single cluster. We also anticipate a parallel numerical study of the concentration of low-spectrum vertex functions on clusters in more general graphs, expressed primarily in terms of local and global connectivity parameters.