Quantitative Obata's theorem in discrete setting

TL;DR

This paper proves that certain weighted graphs with positive Ricci curvature and eigenvalues close to the curvature bound resemble hypercube graphs both structurally and in eigenfunction behavior, extending Obata's theorem to discrete settings.

Contribution

It establishes a quantitative discrete analogue of Obata's theorem, showing that graphs with specific spectral and curvature conditions are close to hypercube graphs.

Findings

Graphs with Ricci curvature > 0 and eigenvalues near the bound are structurally close to hypercubes.

Such graphs are close to hypercube graphs in Frobenius distance.

Eigenfunctions of these graphs resemble those of hypercube graphs.

Abstract

Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-\'Emery and the $d$-th non-zero Laplacian eigenvalue $\lambda_d$ close to $K$, with $d$ being the maximal combinatorial vertex degree of $G$, has an underlying combinatorial structure of the $d$-dimensional hypercube graph. Moreover, such a graph $G$ is close in terms of Frobenius distance to a properly weighted hypercube graph. Furthermore, we establish their closeness in terms of eigenfunctions. Our results can be viewed as discrete analogies of the almost rigidity theorem and quantitative Obata's theorem on Rimennian manifolds.

Equations

$$d={\lambda }_1(S_d)=\cdots ={\lambda }_{d+1}(S_d)<{\lambda }_{d+2}(S_d).$$

$${\Vert u-\sqrt{d+1}\cos {\mathrm{dist}}_p\Vert }_{L^2(M)}\le C(d)({\lambda }_1(M)-d{)}^{\frac{1}{8d+4}}.$$

$$2={\lambda }_1(H_d)=\cdots ={\lambda }_d(H_d)<{\lambda }_{d+1}(H_d).$$

$${\mathrm{dist}}_p(\cdot )-\frac{d}{2}$$

$${\mathrm{dist}}_F(G,H)\le C({\lambda }_d-K{)}^{\frac{1}{2}},$$

$$\Vert {\mathrm{dist}}_{x_0}-(d/2)-u{\Vert }_2\le C({\lambda }_d-K{)}^{\frac{1}{2}},$$

$$\Delta f(x)=\frac{1}{m(x)}\sum _{y}w(x,y)(f(y)-f(x)).$$

$$2\Gamma (f,g):$$

$$2{\Gamma }_2(f,g)$$

$$\Gamma (f)(x)=\frac{1}{m(x)}\sum _{y}w(x,y)(f(y)-f(x){)}^2.$$

$${\Gamma }_2(f)(x)\ge \frac{1}{N}(\Delta f{)}^2(x)+K\Gamma f(x).$$

$$\mathrm{diam}(G)\le \frac{2{\mathrm{Deg}}_{\max }}{K}.$$

$${\mathrm{dist}}_F(G,H):=\underset{\pi }{\min }\Vert A_G^{\pi }-A_H{\Vert }_F,$$

$$\Vert f{\Vert }_{(W,p)}:=\{\begin{array}{rlr}& {\left(\sum _{x\in W}|f(x){|}^{p}m(x)\right)}^{\frac{1}{p}}& \text{for }p\in [1,\infty );\\ & \underset{x\in W}{\sup }|f(x)|& \text{for }p=\infty .\end{array}$$

$$\frac{1}{\delta }\sum _{x\in V}|f(x){|}^p\le \sum _{x\in V}|f(x){|}^{p}m(x)\le \delta \sum _{x\in V}|f(x){|}^p;$$

$$\frac{1}{\delta }\le m(x)\le \delta ,\forall x\in V(G)\text{with}m(x)=1\text{for some vertex}x$$

$$\frac{m(x)}{m(y)}=\frac{q(x,y)}{q(y,x)}\le \delta ,\forall x\sim y.$$

$$\Vert \Gamma f{\Vert }_1\le C(D,d,\delta )\Vert f{\Vert }_2^2.$$

$$\Vert \Gamma f{\Vert }_1=$$

$$\le$$

$$⟨\Gamma u,1⟩\le \Vert u{\Vert }_1\Vert f{\Vert }_{\infty }.$$

$$⟨\Gamma u,1⟩=-⟨u,\Delta u⟩\le ⟨u,f⟩\le \Vert u{\Vert }_1\Vert f{\Vert }_{\infty }.$$

$${\Gamma }_2\phi \le K\Gamma \phi +(K+1)\Vert \phi {\Vert }_2^2\delta ϵ.$$

$$\left|\Gamma \phi -const.\right|\le C(D,K,d,\delta ,\eta )\Vert \phi {\Vert }_2^2\sqrt{ϵ}.$$

$$\left|\phi (z)+\phi (x)-\frac{2{\sum }_{y:x\sim y\sim z}\phi (y)w(x,y)w(y,z)/m(y)}{{\sum }_{y:x\sim y\sim z}w(x,y)w(y,z)/m(y)}\right|\le 2\Vert \phi {\Vert }_2(K+1{)}^{\frac{1}{2}}{\delta }^{\frac{3}{2}}{\eta }^{-1}\sqrt{ϵ}.$$

$$⟨{\Gamma }_2\phi -K\Gamma \phi ,1⟩=$$

$$=$$

$$\le$$

$$K\Gamma (\phi ,\sum _{i=1}^d{a}_i\phi )\le$$

$$=$$

$$\frac{1}{2}\Delta \Gamma (\phi )\ge$$

Full text

1 Introduction

There are quite some deep analogies between the $d$-dimensional round sphere $S_{d}$ and hypercube graph $H_{d}$. For example, in both cases, the concentration of measure converges to the Gaussian measure when the dimension $d$ tends to infinity [Gro99, Section 3$\frac{1}{2}$.21]; Cheng’s maximal diameter theorem for spheres also holds for hypercube graphs [LMP24]. However, Obata type eigenvalue rigidity theorem for hypercube graphs [LMP24] has certain essential differences from that for spheres.

In Riemannian geometry, the Lichnerowicz estimate relates Laplacian eigenvalues and Ricci curvature. In fact, for a $d$-dimensional complete Riemannian manifold $M$ with $\mathrm{Ric}\geq K>0$, its first non-zero eigenvalue $\lambda_{1}(M)\geq\frac{d}{d-1}K$. Obata’s rigidity theorem tells that the equality holds if and only if $M$ is a $d$-dimensional round sphere of suitable radius. Recall that the unit sphere $S_{d}$ has constant Ricci curvature $d-1$ and its first non-zero eigenvalue $\lambda_{1}(S_{d})$ is equal to $d$ with multiplicity $d+1$, i.e.,

[TABLE]

Improving previous works of Petersen [Pet99], Aubry [Aub05] showed that a $d$-dimensional complete Riemannian manifold with $\mathrm{Ric}\geq d-1$ and $\lambda_{d}(M)$ close to $d$ is both Gromov-Hausdorff close and diffeomorphic to the unit sphere $S_{d}$. This is known as almost rigidity theorem. For recent progress on almost rigidity results on Kähler manifolds, see [CWZ25].

Furthermore, we recall that a function $u:S_{d}\to\mathbb{R}$ is an eigenfunction to $\lambda_{1}(S_{d})$ with $\|u\|_{L^{2}(S_{d})}=\mathrm{vol}(S_{d})$ if and only if there exists a point $p\in S_{d}$ such that $u=\sqrt{d+1}\cos\mathrm{dist}_{p}$, where $\mathrm{dist}_{p}$ is the distance function to the point $p$ on $S_{d}$. Cavalletti, Mondino and Semola [CMS23] proved that if $u$ is a first eigenfunction of a $d$-dimensional complete Riemannian manifold $M$ with $\mathrm{Ric}\geq d-1$ and satisfies $\|u\|_{L^{2}(M)}=\mathrm{vol}(M)$, then there exists a point $p\in M$ such that

[TABLE]

This improves previous works in [Pet99, Ber07]. Such kind of results can be considered as a quantitative Obata’s theorem.

In discrete setting, an analogue of the Ricci curvature condition $\mathrm{Ric}\geq K$ is given by the Bakry-Émery curvature-dimension inequalities $CD(K,\infty)$ [BGL14, Elw91, LY10, Sch99]. Recall that the $d$-dimensional hypercube graph $H_{d}$ is a graph with vertex set $V=\{0,1\}^{d}$ and $\{u,v\}\in E$ is an edge if and only if they differ in exactly one coordinate. The graph $H_{d}$ satisfies $CD(2,\infty)$ and its first non-zero Laplacian eigenvalue is equal to $2$ with multiplicity $d$, i.e.

[TABLE]

However, a graph $G$ satisfying $CD(2,\infty)$ and $\lambda_{1}(G)=2$ is not necessarily a hypercube. That is, a directly analogue of Obata’s theorem in discrete setting cannot be true. By requiring in addition that the first non-zero eigenvalue has a high multiplicity, an Obata type theorem has been established for graphs in [LMP24], which will be recalled in details as Theorem 2 below.

Let us still use $\mathrm{dist}_{p}$ to denote the combinatorial distance to a given vertex $p$ of $H_{d}$. Then, for any vertex $p$, the function

[TABLE]

is an eigenfunction of $H_{d}$ to the eigenvalue $\lambda_{1}(H_{d})$. However, for $d>1$, there exist eigenfunctions of $H_{d}$ corresponding to $\lambda_{1}(H_{d})$ which can not be expressed as a function composed with the distance function, see Example 2. This provides another difference between hypercube graphs and round spheres.

Keeping those analogies and differences between hypercube graphs and round spheres in mind, we explore in this paper analogous results of the almost rigidity theorem and quantitative Obata’s theorem in discrete setting.

1.1 Statements of results

We work in the setting of weighted graphs. A weighted graph $G=(V,w,m)$ is a triple, where $V$ is the set of vertices, $w:V\times V\to[0,\infty)$ is symmetric and vanishes on the diagonal, and $m:V\to(0,\infty)$ is a vertex measure. There is an underlying combinatorial structure $\tilde{G}:=(V,E)$ of $G=(V,w,m)$, where the set $E$ of edges is defined as $E:=\{\{x,y\}\in V\times V:w(x,y)>0\}$. We say $G=(V,w,m)$ is connected if $\tilde{G}$ is connected. For any $x\in V$, we define its combinatorial vertex degree as $\mathrm{deg}(x):=\#\{y:\{x,y\}\in E\}$, and its weighted vertex degree as $\mathrm{Deg}(x):=\frac{1}{m(x)}\sum_{y\in V}w(x,y)$. Let us denote $\mathrm{Deg}_{\max}:=\sup_{x\in V}\mathrm{Deg}(x)$ and $\mathrm{deg}_{\max}:=\sup_{x\in V}\mathrm{deg}(x)$. We say the graph $G$ is locally finite if $\mathrm{deg}(x)<\infty$ holds for any $x\in V$.

For an edge $\{x,y\}\in E$, there are two corresponding oriented edges $(x,y)$ and $(y,x)$. We define the edge degree $q(x,y)$ of an oriented edge $(x,y)$ as $q(x,y):=w(x,y)/m(x)$. We say a weighted graph $G=(V,w,m)$ has constant edge degree $c$, if $q(x,y)=q(y,x)=c$ holds for any edge $\{x,y\}\in E$.

It is known that a connected locally finite weighted graph satisfying Bakry-Émery curvature condition $CD(K,\infty)$ with $K>0$ and $\mathrm{Deg}_{\max}<\infty$ is finite [LMP18].

Below is a discrete version of the Lichnerowicz eigenvalue estimate (see e.g. [Bau+17, LMP19]).

Theorem 1** (A discrete Lichnerowicz estimate).**

Let $G=(V,w,m)$ be a finite connected weighted graph satisfying $CD(K,\infty)$ for some $K>0$. Let $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{|V|-1}$ be the eigenvalues of the graph Laplacian $-\Delta$. Then we have $\lambda_{1}\geq K$.

Next, we recall the Obata type rigidity theorem proved in [LMP24, Theorem 2.12].

Theorem 2** (A discrete Obata’s rigidity theorem).**

Let $G=(V,w,m)$ be a finite connected weighted graph. Let $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{|V|-1}$ be the eigenvalues of the graph Laplacian $-\Delta$. Then, $G$ satisfies $CD(K,\infty)$ and $\lambda_{\mathrm{deg}_{\max}}=K$ for some $K>0$ if and only if $G$ is the $\mathrm{deg}_{\max}$-dimensional weighted hypercube graph with constant edge degree $K/2$.

For convenience, we denote by $H_{d}(K/2)$ the $d$-dimensional weighted hypercube graph with constant edge degree $K/2$. Notice that we can not replace the condition $\lambda_{\mathrm{deg}_{\max}}=K$ in Theorem 2 by $\lambda_{\ell}=K$ for some positive integer $\ell<\mathrm{deg}_{\max}$. Counterexamples can be found in [LMP24, Example 3.2]. This demonstrate some essential difference between the discrete and continuous Obata’s rigidity theorem.

In this paper, we aim at establishing quantitative versions of Theorem 2. In order to state our results, we introduce the following class of graphs.

Definition 1**.**

Let $D>0,\delta>1$ be two real numbers and $d$ be a positive integer. We define the set $\mathscr{G}(D,d,\delta)$ to be the collection of connected weighted graphs $G=(V,w,m)$ with $\mathrm{deg}_{\max}=d$ satisfying the following conditions:

• (i)

$\mathrm{Deg}_{\max}\leq D$;

• (ii)

$\delta^{-1}\leq m(x)\leq\delta$, for any $x\in V$.

A natural metric to measure the similarity between two weighted graphs is the Frobenius distance $\mathrm{dist}_{F}$ (see Definition 4 below). Our first result can be considered as an analogy to the almost rigidity result of Petersen [Pet99] and Aubry [Aub05].

Theorem 3**.**

For any $D>0,\delta>1,d\in\mathbb{Z}_{+}$ and $K>0$, there exists a positive constant $\epsilon_{0}(D,K,d,\delta)$, such that any graph $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ satisfying $CD(K,\infty)$ and $\lambda_{d}\leq K+\epsilon_{0}$ has a combinatorial structure $\tilde{G}$ coinciding with the $d$-dimensional hypercube $H_{d}$. Furthermore, the Frobenius distance between $G$ and $H:=H_{d}(K/2)$ satisfies

[TABLE]

where $C=C(d,K)$ is a constant depending only on $d$ and $K$.

Next, we prove an analogy of the quantitative Obata’s theorem [Pet99, Ber07, CMS23] in terms of the closeness of eigenfunctions.

Theorem 4**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$. Let $\Lambda(d)$ be the space spanned by a family of orthonormal eigenfunctions $\varphi_{1},\ldots,\varphi_{d}$ corresponding to $0<\lambda_{1}\leq\cdots\leq\lambda_{d}$, respectively. Then there exists a positive constant $\epsilon_{0}(D,K,d,\delta)$, such that if $\lambda_{d}\leq K+\epsilon_{0}$, then for any $x_{0}\in V$ there exists $u\in\Lambda(d)$ such that

[TABLE]

where $C=C(d,K)$ is a constant depending only on $d$ and $K$.

Notice that we cannot replace the assumption $\lambda_{d}\leq K+\epsilon_{0}$ in Theorem 4 by $\lambda_{\ell}\leq K+\epsilon_{0}$ for some $\ell<d$. Indeed, there exist graphs $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ satisfying $CD(K,\infty)$, such that $\lambda_{1}=\cdots=\lambda_{d-1}=K$ and none of the eigenfunctions to $\lambda_{1}=\cdots=\lambda_{d-1}$ can be expressed as a function composed with the distance function. See Example 1 below.

1.2 Organization of the paper

The paper is organized as follows: In Section 2, we collect basics on graph Laplacian, Bakry-Émery curvature, Frobenius distance, etc. In Section 3, we analyze the stability of eigenfunctions under perturbation of the first $d$ non-zero eigenvalues of a graph $G\in\mathscr{G}(D,d,\delta)$ satisfying $CD(K,\infty)$. In Section 4, we prove Theorem 3 and Theorem 4.

2 Preliminaries

Let $G=(V,m,w)$ be a locally finite weighted graph. For two vertices $x,y$, we write $x\sim y$ if $w(x,y)>0$. We define the combinatorial distance between two vertices $x$ and $y$ by $\mathrm{dist}(x,y):=\min\{n\ |\ \text{there exist }x=x_{0}\sim\cdots\sim x_{n}=y\}$. The diameter is defined as $\mathrm{diam}(G):=\sup_{x,y\in V}d(x,y)$. We denote by $S_{k}(x):=\{y\in V\ |\ \mathrm{dist}(x,y)=k\}$ and $B_{k}(x):=\{y\in V\ |\ \mathrm{dist}(x,y)\leq k\}$ the sphere and ball of radius $k$ round $x\in V$, respectively.

2.1 Bakry-Émery curvature

We define the function space $C(V):=\mathbb{R}^{V}$. The $graph~Laplacian$ $\Delta:C(V)\to C(V)$ is defined by

[TABLE]

Next, we introduce Bakry-Émery $\Gamma$-calculus.

Definition 2**.**

For any two functions $f,g\in C(V)$, we define

[TABLE]

We write $\Gamma(f)=\Gamma(f,f)$ and $\Gamma_{2}(f)=\Gamma_{2}(f,f)$, for simplification. Note that $\Gamma$ and $\Gamma_{2}$ are bilinear and it is direct to check

[TABLE]

Definition 3**.**

Let $G=(V,w,m)$ be a locally finite weighted graph. We say the graph $G$ satisfies $CD(K,N)$ for some $K\in\mathbb{R}$ and $N\in(0,\infty]$ if, for every functions $f\in C(V)$ and every vertex $x\in V$,

[TABLE]

Here we use the notation $1/\infty=0$. We only use the condition $CD(K,\infty)$ in this paper. A discrete version of the Bonnet–Myers theorem is proved by [LMP18], improving previous results in [Hor+19, FS18].

Theorem 5** ([LMP18]).**

Let $G=(V,w,m)$ be a locally finite connected weighted graph satisfying $CD(K,\infty)$ for some $K>0$ and $\mathrm{Deg}_{\max}<\infty$. Then the graph $G$ is finite. Moreover, we have

[TABLE]

2.2 Frobenius distance between weighted graphs

In this paper, we employ the Frobenius distance to measure similarities between weighted graphs. This distance is widely used, see, e.g., [GRW18]. Recall that the Frobenius norm for a matrix $M=(M_{ij})$ is given by $\|M\|_{F}:=\sqrt{\sum_{i,j}M_{ij}^{2}}$.

For a finite weighted graph $G=(V,m,w)$, we consider the following square matrix $A_{G}=(A_{xy})$, where $A_{x,y}=q(x,y)$ if $x\sim y$ and $A(x,y)=0$ otherwise. Notice that $A_{G}$ is not necessarily symmetric.

Definition 4** (Frobenius distance).**

The Frobenius distance between two $n$-vertex weighted graphs $G$ and $H$ is defined as

[TABLE]

where the minimization is taken over all all permutations $\pi$ of the vertex set of $G$, and the matrix $A_{G}^{\pi}$ is generated by permuting rows and columns of $A_{G}$ according to $\pi$.

The distance can also measure the similarity between two finite weighted graphs of different sizes by adding isolated vertices to the graph with fewer vertices.

2.3 Equivalence of $\ell_{p}$-norms

Definition 5**.**

Let $G=(V,w,m)$ be a locally finite weighted graph. Let $\emptyset\neq W\subset V$, we define $\ell_{p}(W):=\{f\in C(W):\|f\|_{(W,p)}<\infty\}$ with

[TABLE]

We simply write $\|f\|_{p}$ if $W=V$. The space $\ell_{2}(V)$ is a Hilbert space with the inner product $\langle f,g\rangle:=\sum_{x\in V}f(x)g(x)m(x)$.

Lemma 1**.**

(Equivalence of norms) Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$. Then all norms $\|\cdot\|_{p}$, $p\in[1,\infty]$ of $C(V)$ are equivalent with respect to a constant $C(D,K,d,\delta,p)$.

Proof.

Let $f$ be any function in $C(V)$. For $p\in[1,\infty)$, we have

[TABLE]

For $p=\infty$, the value of $\|f\|_{\infty}$ is irrelevant to $\delta$. By Theorem 5, we have $\mathrm{diam}(G)\leq\frac{2D}{K}$. Thus $\dim(C(V))=|V|\leq C(D,K,d).$

By the equivalence of norms in finite dimensional spaces, we prove Lemma 1. ∎

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph.Throughout this work, for convenience, we assume there exists a vertex $x\in V(G)$ such that $m(x)=1$. It does not make any restriction to the graph $G$, since, in calculation of spectrum and curvature, $m(\cdot)$ appear as denominators. In Definition 1, if we keep other conditions unchanged, condition

[TABLE]

can be replaced by

[TABLE]

3 Geometric stability of eigenfunctions

In Subsection 3.1, we study the stability of geometric properties of eigenfunctions. With this, in Subsection 3.2, we are able to approximate distance functions in $\Lambda_{0}(d)$. In Subsection 3.3, we give estimations on vertex measure and degree weight through the properties of the distance functions.

Note that the value of the constants $C(\cdot)$ may change line by line during the proof.

3.1 Perturbations of eigenvalues

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$. Let $\Lambda_{0}(d)$ be the space spanned by a family of orthonormal eigenfunctions $\varphi_{0},\varphi_{1},\ldots,\varphi_{d}$ corresponding to $0=\lambda_{0}<\lambda_{1}\leq\cdots\leq\lambda_{d}$ respectively. In this subsection, we study the stability of functions in $\Lambda_{0}(d)$ and give a quantitative estimate of part of [LMP24, Thoerem 3.4]’s results under small perturbation of the first $d$ non-zero eigenvalues.

First, we prove the following two lemmas which estimate the gradient.

Lemma 2**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph and $f\in C(V)$. There exists a constant $C(D,d,\delta)$ such that

[TABLE]

Proof.

For any $f\in C(V)$, we have

[TABLE]

We prove Lemma 2. ∎

Lemma 3**.**

(Gradient Estimation) Let $G=(V,w,m)$ be a weighted graph. Let $u,f\in C(V)$ be non-negative functions satisfying $\Delta u\geq-f$, then

[TABLE]

Proof.

By Hölder’s inequality, we have

[TABLE]

We prove Lemma 3. ∎

Now, we are ready to study the stability of eigenfunctions.

Theorem 6**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$. Let $\Lambda_{0}(d)$ be the space spanned by a family of orthonormal eigenfunctions $\varphi_{0},\varphi_{1},\ldots,\varphi_{d}$ corresponding to $0=\lambda_{0}<\lambda_{1}\leq\cdots\leq\lambda_{d}$, respectively. If $\lambda_{d}\leq K+\epsilon$ for some $\epsilon\in(0,1)$, we have for any $\varphi\in\Lambda_{0}(d)$,

[TABLE]

Furthermore, if $\mathop{\min}\limits_{w(x,y)>0}w(x,y)\geq\eta>0$, then the following hold true:

1. (a)

There exists a constant depending on $\varphi$ such that

[TABLE] 2. (b)

For all $x,z$ with $d(x,z)=2$, we have

[TABLE]

Proof.

First, we prove (3.3). Let $0=\lambda_{0}<\lambda_{1}=K+\epsilon_{1}\leq\cdots\leq\lambda_{d}=K+\epsilon_{d}\leq K+\epsilon$ be the eigenvalues of the graph Laplacian $-\Delta$ and $\varphi_{i}$ be eigenfunction corresponding to the $i$-th eigenvalue $\lambda_{i}$ with $\Braket{\varphi_{i},\varphi_{j}}=\delta_{ij}$. We spectrally decompose $\varphi=\sum_{i=0}^{d}a_{i}\varphi_{i}\in\Lambda_{0}(d)$. Then

[TABLE]

Since, for any $x\in V$, $(\Gamma_{2}\varphi-K\Gamma\varphi)(x)m(x)\leq\Braket{\Gamma_{2}\varphi-K\Gamma\varphi,1}$ and $(\Gamma_{2}\varphi-K\Gamma\varphi)(x)\geq 0$, we prove (3.3).

Now, we prove (3.4). Since $G$ satisfies $CD(K,\infty)$, we have $K\Gamma\varphi\leq\Gamma_{2}\varphi$. We rewrite it as

[TABLE]

Then,

[TABLE]

Applying Lemma 2, the Cauchy-Schwarz inequality, and the equivalence of norms,

[TABLE]

Applying Lemma 2 and Lemma 3, we obtain

[TABLE]

Thus, for any $x\in V$and $y\sim x$,

[TABLE]

Due to connectedness, we prove (3.4).

Next, we prove (3.5). The vector $\varphi_{x}\in\mathbb{R}^{\#B_{2}(x)}$ is defined by the restriction of the function $\varphi$ on $B_{2}(x)$. Let $\Gamma_{2}(x)$ be the $(\#B_{2}(x))\times(\#B_{2}(x))$ symmetric matrix such that $\Gamma_{2}\varphi(x)=\varphi_{x}^{\top}\Gamma_{2}(x)\varphi_{x}$. In fact, for a vertex $z\in S_{2}(x)$, the column $\Gamma_{2}(x)1_{\{z\}}$ of $\Gamma_{2}(x)$ is given as follows (see [CLP20, Section 2.3] and [CLP20, Section 10]):

[TABLE]

For any $y\in S_{1}(x)$, we have $(\Gamma_{2}(x))_{y,z}=-\frac{1}{2m(x)}w(x,y)w(y,z)/m(y)$ if $y\sim z$ and 0 otherwise. Finally, $(\Gamma_{2}(x))_{z^{\prime},z}=0$ for any $z^{\prime}\in S_{2}(x)$ different from z. Therefore, we have for any $r\in\mathbb{R}$

[TABLE]

(3.3) implies

[TABLE]

Since $\varphi$ and $\varphi+r1_{\{z\}}$ agree on $B_{1}(x)$, we derive $\Gamma(\varphi)(x)=\Gamma(\varphi+r1_{\{z\}})(x)$. We have

[TABLE]

stands for any $r\in\mathbb{R}$. By the discriminant formula of quadratic polynomials,

[TABLE]

which yields

[TABLE]

Note that

[TABLE]

Dividing each side of (3.6) by $(\Gamma_{2}(x))_{z,z}=\frac{1}{4m(x)}\sum_{y:x\sim y\sim z}w(x,y)w(y,z)/m(y)$, (3.5) yields. ∎

3.2 Restriction maps

The aim of this subsection is to find a function in $\Lambda_{0}(d)$ which is close to the distance function $f_{0}:=\mathrm{dist}(x_{0},\cdot)$ for some fixed vertex $x_{0}\in V(G)$. The strategy is to find a function $\tilde{f}_{0}\in\Lambda_{0}(d)$ equal to $f_{0}$ on $B_{1}(x_{0})$ and exploit the previous estimates of functions in $\Lambda_{0}(d)$ in Theorem 6 to show that such $\tilde{f}_{0}$ is exactly the function we want.

Definition 6**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$. Let $\Lambda_{0}(d)$ be the space spanned by a family of orthonormal eigenfunctions $\varphi_{0},\varphi_{1},\ldots,\varphi_{d}$ corresponding to $0=\lambda_{0}<\lambda_{1}\leq\cdots\leq\lambda_{d}$, respectively. A restriction map $L_{x}$ with respect to $x\in V$ is defined as:

[TABLE]

The following theorem states that the restriction map $L_{x}$ is bijective and its inverse has bounded norm when the spectral gap deficit is sufficiently small.

Theorem 7**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$ and $\mathop{\min}\limits_{w(x,y)>0}w(x,y)\geq\eta>0$. For any $x\in V(G)$, let $L_{x}$ be the restriction map from Definition 6. Then there exists $\epsilon_{1}(D,K,d,\delta,\eta)>0$, such that for any $0<\epsilon<\epsilon_{1}$, if $\lambda_{d}\leq K+\epsilon$, $L_{x}$ is bijective. i.e. $deg(x)=d$. Moreover, $\|L_{x}^{-1}\|_{2}\leq C(D,K,d,\delta,\eta)$.

Proof.

In fact, we will show that there exists a constant $C(D,K,d,\delta,\eta)>0$ such that

[TABLE]

It is direct to see that $\|L_{x}(\varphi)\|_{2}\leq 1$. Since $dim(\Lambda_{0})=d+1\geq\#B_{1}(x)$ and $L_{x}$ is injective, we prove the theorem. Now, we prove (3.8).

For any $\varphi\in\Lambda_{0}(d)$, we assume $\|L_{x}(\varphi)\|_{\infty}=\|\varphi\|_{(B_{1}(x),\infty)}=:\delta_{0}$. For $z\in B_{2}(x)$, due to (3.5), we have

[TABLE]

For any integer $k\geq 3$, suppose we have $|\varphi(w)|\leq(3^{k-1}-1)\|\varphi\|_{2}(K+1)^{\frac{1}{2}}\delta^{\frac{3}{2}}\eta^{-1}\sqrt{\epsilon}+3^{k-1}\delta_{0}$, for any $w\in B_{k}(x)$. For any $z\in S_{k+1}(x)$, pick any $x_{0}\in S_{k-1}(x)$ such that $x_{0}\sim y\sim z$ for some $y\in S_{k}(x)$. By (3.5), we derive

[TABLE]

Since, graph is connected and $\mathrm{diam}(G)\leq\frac{2Deg_{max}}{K}\leq\frac{2D}{K}$. Iterating the above process, we obtain

[TABLE]

By the equivalence of norms, we have

[TABLE]

Picking $\sqrt{\epsilon}\leq\frac{1}{2}C^{-1}(D,K,d,\delta,\eta)$, there holds

[TABLE]

Again, by the the equivalence of norms, we prove (3.8). ∎

The next Lemma states that it is available to approximate distance functions in $\Lambda_{0}(d)$ when the spectral gap deficit is sufficiently small. It is crucial for the proof of Theorem 4 and build a bridge between eigenfunctions and distance functions.

Lemma 4**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$ and $\mathop{\min}\limits_{w(x,y)>0}w(x,y)\geq\eta>0$. For any fixed vertex $x_{0}\in V(G)$, define the distance function $f_{0}(\cdot):=\mathrm{dist}(\cdot,x_{0})$.

Then, there exist constants $\epsilon_{1}=\epsilon_{1}(D,K,d,\delta,\eta)>0$ and $C=C(D,K,d,\delta,\eta)$ such that for all $0<\epsilon<\epsilon_{1}$, if $\lambda_{d}\leq K+\epsilon$, we have $\tilde{f_{0}}:=L_{x_{0}}^{-1}(f_{0}|_{B_{1}(x_{0})})$ exists and

[TABLE]

Proof.

First, let $\epsilon_{1}$ be the $\epsilon_{1}(D,d,\delta,K,\eta)$ in Theorem 7. Then, $\tilde{f_{0}}=L_{x_{0}}^{-1}(f_{0}|_{B_{1}(x_{0})})$ exists. For all $0<\epsilon<\epsilon_{1}$, we prove the Theorem as follows.

Note that, for distance function $f_{0}(\cdot):=\mathrm{dist}(\cdot,x_{0})$, it is direct to check

[TABLE]

where $z\in B_{k+1}(x_{0})$ and $x\in B_{k-1}(x_{0})$, $k\in\mathbb{N}^{+}$.

Thus, for $z\in B_{2}(x_{0})$, by (3.5), we have

[TABLE]

Since $f_{0}$ and $\tilde{f_{0}}$ agree on $B_{1}(x_{0})$, we obtain

[TABLE]

Note that graph is connected and $\mathrm{diam}(G)\leq\frac{2Deg_{max}}{K}\leq\frac{2D}{K}$. By induction,

[TABLE]

It is straightforward to check that $\|f_{0}\|_{(B_{1}(x),2)}\leq\sqrt{d\delta}$. Then due to Theorem 7, we prove (3.12). ∎

Here, we are able to give a discrete analogy of Bertrand’s result. Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$. If we only assume the first $l<d$ non-zero eigenvalues close to Lichnerowicz bound, recalling examle 1, it is possible that no eigenfunctions corresponding to $\lambda_{1}\dots\lambda_{l}$ can not be expressed as a function composed with the distance function; on the other hand, the following theorem shows, as long as $deg(x_{0})\leq l$, for some $x_{0}\in V(G)$, distance function $\mathrm{dist}_{x_{0}}$ can be expressed as a combination of corresponding eigenfunctions of the first $l$ non-zero eigenvalues up to a constant.

Theorem 8**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$ and

[TABLE]

Let $0<l\leq d$ be an integer. Let $\Lambda_{0}(l)$ be the space spanned by a family of orthonormal eigenfunctions $\varphi_{0},\varphi_{1},\ldots,\varphi_{l}$ corresponding to $0=\lambda_{0}<\lambda_{1}\leq\cdots\leq\lambda_{l}$, respectively. Then for any $x_{0}\in V$ with $deg(x_{0})\leq\ell$ there exists $u\in\Lambda_{0}(l)$ such that

[TABLE]

Proof of Theorem 8.

By the same proof of Theorem 6, If $\lambda_{l}<K+\epsilon$ for some $\epsilon\in(0,1)$, (3.4) and (3.5) hold for $\varphi\in\Lambda_{0}(l)$. For any $x_{0}\in V(G)$ with $deg(x_{0})\leq l$, we define the restriction map $L_{x_{0}}^{l}$ as:

[TABLE]

Note that here $dim(\Lambda_{0}(l))=l+1\geq\#B_{1}(x_{0})$. Paralleling the proof of Theorem 7, it can be shown that $L_{x_{0}}^{l}$ is bijective and has a bounded inverse when $\epsilon$ is sufficiently small. Finally, Theorem 8 follows by Lemma 4’s argument. Note that, for large $\epsilon$, we simply pick $u\equiv 0$ and estimate $\|\mathrm{dist}_{x_{0}}\|_{\infty}\leq{2D}/{K}.$

Thus, we prove Theorem 8.

∎

Notice that for a positive integer $\ell<d$, a graph $G\in\mathscr{G}(D,d,\delta)$ satisfying $CD(K,\infty)$ with $\lambda_{1}=\cdots=\lambda_{\ell}=K$ is not necessarily a hypercube, see [LMP24, Example 3.2]. Therefore, it is natural that the constant $C$ in Theorem 8 might depend on more parameters. We remark that the condition (3.13) is also necessary. Since, if an edge with arbitrary small weight is added to a graph $G$, its curvature and eigenvalues only perturb a little while its combiatorial distance can vary dramatically.

3.3 Estimations of edge degree and vertex measure

In this subsection, we prove Theorem 9 which states that the edge degree and vertex measure is quantitatively close to a constant in terms of spectral gap deficit.

Theorem 9**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$ and $\mathop{\min}\limits_{w(x,y)>0}w(x,y)\geq\eta>0$. For any fixed vertex $x_{0}\in V(G)$, define the distance function $f_{0}(\cdot):=\mathrm{dist}(\cdot,x_{0})$. For simplicity, we write $\mathrm{Deg}_{max}(G)$ as $D(G)$.

There exist constants $\epsilon_{1}=\epsilon_{1}(D,K,d,\delta,\eta)>0$ and $C=C(D,K,d,\delta,\eta)$ such that for all $0<\epsilon<\epsilon_{1}$, if $\lambda_{d}\leq K+\epsilon$, then the following hold:

1. (i)

$\left|\Gamma f_{0}(x)-D(G)/2\right|\leq C(D,K,d,\delta,\eta)\sqrt{\epsilon}$, for any $x\in V(G)$. 2. (ii)

$\left|\mathrm{Deg}(x)-D(G)\right|\leq C(D,K,d,\delta,\eta)\sqrt{\epsilon}$, for any $x\in V(G)$. 3. (iii)

$\left|q(x,y)-K/2\right|\leq C(D,K,d,\delta,\eta)\sqrt{\epsilon}$, for any $q(x,y)>0$. 4. (iv)

$\left|m(x)-1\right|\leq C(D,K,d,\delta,\eta)\sqrt{\epsilon}$, for any $x\in V(G)$.

Proof.

First, let $\epsilon_{1}$ be the $\epsilon_{1}(D,d,\delta,K,\eta)$ in Theorem 7. Then, $\tilde{f_{0}}=L_{x_{0}}^{-1}(f_{0}|_{B_{1}(x_{0})})$ exists. We first prove (i) and (ii) when $x_{0}\in V(G)$ such that $Deg(x_{0})=D(G)$.

To simplify the calculation, we shall adopt the following notation: Let $f,g$ be functions defined on a set $E$ with values in $\mathbb{R}$ and $x\in E$. For any $\epsilon>0$, we write $f(x)=g(x)+C[\epsilon]$, if $|f(x)-g(x)|\leq C\epsilon$ for some constant $C$.

(i)’. By Lemma 4, we have

[TABLE]

On the other hand, applying (3.4), we have

[TABLE]

Since $D(G)=2\Gamma f_{0}(x_{0})$, we prove (i) in such $x_{0}\in V(G)$.

(ii)’. By (i), for any $x\in V(G)$,

[TABLE]

For convenience, we define the inner and outer degree w.r.t. $x_{0}\in V$ via

[TABLE]

and spherical-degree

[TABLE]

We remark $\mathrm{Deg}(z)=d_{+}^{x_{0}}(z)+d_{0}^{x_{0}}(z)+d_{-}^{x_{0}}(z)$ and note that $D(G)\geq\mathrm{Deg}(x)$. By (3.14), we have for any $x\in V(G)$

[TABLE]

and

[TABLE]

Since $\mathrm{Deg}(x)=d_{+}^{x_{0}}(x)+d_{0}^{x_{0}}(x)+d_{-}^{x_{0}}(x)$, we obtain

[TABLE]

Thus we prove (i) and (ii) when $x_{0}\in V(G)$ such that $\mathrm{Deg}(x_{0})=D(G)$. Due to (3.15), for any $x_{0}\in V(G)$, the value of $\mathrm{Deg}(x_{0})$ only differs from $D(G)$’s by $C_{0}(D,K,d,\delta,\eta)\sqrt{\epsilon}$. Repeat the above proofs of (i)’and (ii)’. Then the above formulas hold for arbitrary $x_{0}\in V(G)$.

(iii). For any $x_{0}\in V(G)$, by Theorem 7 and Lemma 4, we have

[TABLE]

where $\varphi\in\Lambda(d)$ with

[TABLE]

and $C$ is a constant such that

[TABLE]

Therefore,

[TABLE]

Thus,

[TABLE]

Just take C= $\frac{D(G)}{K}$. Therefore, for any $x\in V(G)$,

[TABLE]

We obtain

[TABLE]

Combined with (3.16),

[TABLE]

For any $x,y\in V(G)$ with $x\sim y$, we have

[TABLE]

(iv). Then, we have

[TABLE]

which yields

[TABLE]

On the other hand, we always assume that there exists some $x_{0}\in G$ such that $m(x_{0})=1$. Since $G$ is connected and $\mathrm{diam}(G)\leq\frac{2Deg_{max}}{K}\leq\frac{2D}{K}$, we prove (iv). ∎

4 Quantitative Obata’s Theorem

First, we prove the almost-rigidity theorem by contradiction.

Theorem 10** (Almost-rigidity theorem).**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ be a graph satisfying $CD(K,\infty)$ for some $K>0$ . There exists an $\epsilon_{0}(D,K,d,\delta)$, such that if $\lambda_{d}(G)\leq K+\epsilon_{0}$, then

1. (i)

$G$* is a hypercube in the combinatorial sense;* 2. (ii)

there exists an  $\eta(D,K,d,\delta)>0$, such that $\mathop{\min}\limits_{w(x,y)>0}w(x,y)\geq\eta(D,K,d,\delta)$, for any $w(x,y)\in w(G)$.

Proof.

(1)First, we prove (i). Suppose not. There exists a sequence of graphs

[TABLE]

such that $\lambda_{d}(G_{n})\leq K+\frac{1}{n}$, while $\{G_{n}\}_{n=1}^{\infty}$ are not hypercubes in the combinatorial sense. Since $\{G_{n}\}_{n=1}^{\infty}$ are connected and $diam(G_{n})\leq\frac{2Deg_{max}(G_{n})}{K}\leq\frac{2D}{K}$, there are only finite many combinatorial structures among all $\{G_{n}\}_{n=1}^{\infty}$. Thus, we may pick an infinite subsequence of $\{G_{n}\}_{n=1}^{\infty}$( still denoted as $\{G_{n}\}_{n=1}^{\infty}$) sharing the same combinatorial structure, say $G=(V,E)$. We label vertices and edges in $\{G_{n}\}_{n=1}^{\infty}$ with $G=(V,E)$.

By the definition of $\mathscr{G}$, $w_{n}$ and $m_{n}$ are bounded. There exists a subsequence of $\{G_{n}\}_{n=1}^{\infty}$ ( still denoted as $\{G_{n}\}_{n=1}^{\infty}$), such that

[TABLE]

and

[TABLE]

exist, for any $x\sim y$ and $x\in V$.

Thus, we define the weighted graph $G_{0}:=(V,w_{0},m_{0})$. By continuity of spectrum, $\lambda_{d}(G_{0})=K$. By the Rigidity Theorem (see Theorem 2) , $G_{0}$ is a hypercube. Note that for any $\{x,y\}\in E$, $w_{0}(x,y)$ will not vanish, otherwise $deg(x)\leq d-1$. This contradicts with $G_{0}$ being a hypercube. Thus G is isomorphic to $G_{0}$, a hypercube. This contradicts the assumption. Hence, we prove that such $\epsilon_{0}$ exists.

(2)Now, we prove (ii). For the above $\epsilon_{0}$ in (i), suppose not. Since, for any $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ satisfying $CD(K,\infty)$ for some $K>0$ with $\lambda_{d}(G)<K+\epsilon_{0}$, $G$ is already a hypercube combinatorially, we may pick a subsequence of $\{G_{n}\}_{n=1}^{\infty}$ such that ${w_{n}(x^{\prime},y^{\prime})}<\frac{1}{n}$, for some $\{x^{\prime},y^{\prime}\}\in E$. By the similar argument of (i), $G_{0}$ fails to be a hypercube, a contradiction. Thus, such $\eta(D,K,d,\delta)$ exists. ∎

Now, we are ready to prove our main theorem.

Proof of Theorem 3.

We employ a self-improving argument to prove the constant is irrelevant to $\delta,\eta$. By Theorem 10, there exist $\epsilon_{0}(D,K,d,\delta)$ and $\eta(D,K,d,\delta)$, if $\lambda_{d}(G)<K+\epsilon_{0}$, then $G$ is a hypercube combinatorially and $\min_{w(x,y)>0}w(x,y)\geq\eta(D,K,d,\delta)$.

Applying Theorem 9 for the above $\eta(D,K,d,\delta)$, there exists an

[TABLE]

such that when $\lambda_{d}=K+\epsilon\leq K+\epsilon_{2}$, we obtain

[TABLE]

and

[TABLE]

Set

[TABLE]

We have

[TABLE]

and

[TABLE]

It yields $D(G)\leq\frac{3dK}{4}$. Applying Theorem 9 again, there exists

[TABLE]

When $\lambda_{d}(G)=K+\epsilon\leq K+\epsilon_{4}(D,K,d,\delta)$, we have

[TABLE]

and

[TABLE]

Since $G$ and $H$ share the same combinatorial structure, there holds

[TABLE]

We prove the theorem. ∎

Proof of Theorem 4.

For any fixed $x_{0}\in V$, set $f_{0}:=\mathrm{dist}_{x_{0}}$. By the equivalence of norms in $\mathscr{G}(D,K,d,\delta)$, we actually show

[TABLE]

in Lemma 4. Picking $\epsilon_{4}(D,d,\delta,K)$ in the proof of Theorem 3, we have

[TABLE]

when $\lambda_{d}=K+\epsilon<K+\epsilon_{4}(D,d,\delta,K)$. Due to the property of Hilbert spaces, there exists a function $u\in\Lambda(d)$ such that

[TABLE]

By (4.1), we have

[TABLE]

By the triangle inequality and the equivalence of norm, we obtain the theorem. ∎

4.1 Examples

Given two weighted graphs $G_{1}=(V_{1},w_{1},m_{1}\equiv 1)$ and $G_{2}=(V_{2},w_{2},m_{2}\equiv 1)$, their Cartesian product $G_{1}\times G_{2}=(V(G_{1})\times V(G_{2}),w,m)$ is a weighted graph with vertex set $V(G_{1})\times V(G_{2})$,

[TABLE]

and $m\equiv 1$. Let $\{\alpha_{i}\}_{i=0}^{|V_{1}|-1}$ be the sequence of eigenvalues of $G_{1}$ and $\{\beta_{j}\}_{j=0}^{|V_{2}|-1}$ be the sequence of eigenvalues of $G_{2}$. The eigenvalues of $G_{1}\times G_{2}$ are given by $\{\alpha_{i}+\beta_{j},i=0,1\ldots|V_{1}|-1,j=0,1\ldots|V_{2}|-1\}$ with corresponding eigenfunctions $\{h_{\alpha_{i},\beta_{j}}\}$ in the form of

[TABLE]

where $f_{\alpha_{i}}$ is the corresponding eigenfunction of $\alpha_{i}$ and $g_{\beta_{j}}$ is the corresponding eigenfunction of $\beta_{j}$.

Further, if $G_{1}$ and $G_{2}$ satisfy $CD(K_{1},\infty)$ and $CD(K_{1},\infty)$, respectively. By [LP18, Theorem 1.3]), their Cartesian product $G_{1}\times G_{2}$ satisfies $CD(\min\{K_{1},K_{2}\},\infty)$. To clarify the examples, we state the following lemma first.

Lemma 5**.**

Let $G_{1}=(V_{1},w_{1},m_{1}\equiv 1)$ and $G_{2}=(V_{2},w_{2},m_{2}\equiv 1)$ be two connected weighted graphs with $\left|V(G_{1})\right|\geq 2$. If

[TABLE]

can be expressed as a composite function of distance from any fixed vertex $(x_{0},y_{0})$ in $G_{1}\times G_{2}$, that is, $h(\cdot)=w(\mathrm{dist}_{(x_{0},y_{0})}(\cdot))$, then $h\equiv w(0)$, a constant.

Proof.

We set $e_{1}:=\max_{x\in V(G_{1})}{\mathrm{dist}(x,x_{0})}\geq 1$ and $e_{2}:=\max_{y\in V(G_{2})}{\mathrm{dist}(y,y_{0})}$. For any integer $0\leq k\leq e_{1}+e_{2}$, it is direct to find a vertex $(x,y)$ such that $\mathrm{dist}_{G_{1}}(x_{0},x)=\min\{k,e_{1}\}$ and $\mathrm{dist}_{G_{2}}(y_{0},y)=\max\{k-e_{1},0\}$. We have

[TABLE]

Since $G_{1}\times G_{2}$ is connected, it yields $w(k)\equiv w(0)$. ∎

Now, we are ready to give some examples.

Example 1**.**

Let $G=(V,w,m)\in\mathscr{G}(D,d,\delta)$ satisfying $CD(K,\infty)$. If we only assume the first $l\leq d-1$ eigenvalues reach the Lichnerowicz sharpness, there exist graphs such that the first eigenfunction cannot be expressed as a function of distance. Examples are illustrated as follows.

The Cartesian product $G:=H_{n}(2)\times H_{l}(1)$, where $n,l\geq 1$ be integers. Note that $H_{d}(K/2)$ satisfies $CD(K,\infty)$. Therefore, the graph $G$ satisfies $CD(2,\infty)$ with $deg_{max}(G)=n+l$ and its first non-zero Laplacian eigenvalue is equal to $2$ with multiplicity $l$, i.e.

[TABLE]

Since the non-zero eigenvalues of $H_{n}(2)$ are greater than 2, the corresponding eigenfunction $h$ of $\lambda=2$ in $H_{n}\times H_{l}$ could only be of the form $h=g(y)$, where $g(y)$ is an eigenfunction of non-zero eigenvalue 2 of $H_{l}(1)$. Suppose $h$ can be expressed as a function of distance. By Lemma 5, $h$ is a constant, a contradiction.

Example 2**.**

Let $d>1$ be an integer. There exist eigenfunctions of $H_{d}$ corresponding to $\lambda_{1}(H_{d})=2$ which can not be expressed as a function of distance.

We write $H_{d}=H_{d-l}\times H_{l}$, where $0<l<d$ is an integer. $H_{d}$ satisfies $CD(2,\infty)$. Then, by Lemma 5, any eigenfunction of $H_{d}$ in the form of $h=g(y)$, where $g(y)$ is an eigenfunction of eigenvalue 2 in $H_{l}$, cannot be a function of distance.