Conformally rigid graphs

The purpose of this paper is to introduce a property, which we call conformal rigidity, that a finite, simple graph $G=(V,E)$ may or may not have. It is the discrete analogue of a property of interest in spectral geometry and can only be expected in the presence of structure. Roughly put, it says that among all possible ways one could weigh the edges, constant weight edges are already, in a suitable sense, the canonical choice. We start by defining things more formally. Let $G=(V,E)$ be a finite, simple, connected graph on $n$ vertices. A classical object associated to $G$ is its graph Laplacian $L=D-A$, where $D\in\mathbb{R}^{n\times n}$ is the diagonal matrix collecting the degrees of the vertices and $A\in\left\{0,1\right\}^{n\times n}$ is the adjacency matrix of $G$. The matrix $L$ is positive semidefinite (psd) and has eigenvalues

$$0=\lambda_{1}<\lambda_{2}\leq\dots\leq\lambda_{n}$$

which are fundamental objects in spectral graph theory. The theory generalizes to the case where edges are weighted $w_{ij}>0$; the weighted Laplacian $L_{w}\in{\mathbb{R}}^{n\times n}$ is the psd matrix defined as:

(4)

$\displaystyle(L_{w})_{ij}:=\left\{\begin{array}[]{rl}-w_{ij}&\textup{ if }(i,j)\in E\\ \sum_{j\sim i}w_{ij}&\textup{ in the }(i,i)\textup{ entry}\\ 0&\textup{ otherwise }\end{array}\right.$

with eigenvalues $0=\lambda_{1}(w)<\lambda_{2}(w)\leq\dots\leq\lambda_{n}(w)$. We use $j\sim i$ to denote that $(i,j)\in E$. We only work with undirected graphs, so there is no difference between $j\sim i$ and $i\sim j$. The constant vector $\mathbbm{1}:=(1,1,\ldots,1)$ spans the eigenspace of $\lambda_{1}(w)=0$. By the Rayleigh-Ritz Theorem [15, §4.2], we can write $\lambda_{2}(w)$ as

$$\lambda_{2}(w)=\min_{\sum_{v\in V}f(v)=0}\frac{\sum_{(i,j)\in E}w_{ij}(f(i)-f(j))^{2}}{\sum_{v\in V}f(v)^{2}}$$

where the minimum ranges over all $f:V\rightarrow\mathbb{R}$ that are not identically 0. The condition $\sum_{v\in V}f(v)=0$ forces orthogonality to the constant vector $\mathbbm{1}$. Likewise, and without orthogonality conditions, we have

$$\lambda_{n}(w)=\max_{f:V\rightarrow\mathbb{R}}\frac{\sum_{(i,j)\in E}w_{ij}(f(i)-f(j))^{2}}{\sum_{v\in V}f(v)^{2}}.$$

It is well known that $G$ is connected if and only if $\lambda_{2}(w)>0$. In fact, $\lambda_{2}(w)$ measures the connectivity of a graph: larger values correspond to a more connected graph. The largest eigenvalue $\lambda_{n}(w)$ is more intricate but corresponds roughly to the largest possible oscillation that can occur among the functions of the graph.

It is a natural question how, for any given graph $G=(V,E)$, one should choose weights $w$ to maximize $\lambda_{2}(w)$ or to minimize $\lambda_{n}(w)$, respectively. Since the eigenvalues scale linearly with $w$, we require both $w_{ij}\geq 0$ and the normalization

$$\sum_{(i,j)\in E}w_{ij}=|E|$$

which we do throughout the paper. Maximizing $\lambda_{2}(w)$ can be understood in several ways, including creating the ‘densest’ network, while minimizing $\lambda_{n}(w)$ is akin to a smoothness condition on the weights. It is a way of ‘regularizing’ the graph, finding its most symmetric representation. We are interested in graphs for which these optimization procedures cannot improve on the unweighted case.

This corresponds, in a suitable sense, to graphs which are already so symmetric that varying edge weights cannot further increase $\lambda_{2}$ or decrease $\lambda_{n}$. The definition can be phrased in terms of the spectrum of the Laplacian matrix as in Fig. 2. An alternative viewpoint is as follows: by changing the weights, we change the eigenvalues. As mentioned already, there is a linear scaling, doubling all the weights doubles all the eigenvalues, which makes it natural to normalize the sum of the weights. Conformal rigidity is concerned with the setting where the largest attainable second eigenvalue over all possible weights is simply the eigenvalue of the unweighted graph itself, meaning

$$\sup_{w>0}\frac{\lambda_{2}(w)}{\sum_{(i,j)\in E}w_{ij}}=\sup_{w>0}\inf_{f:V\rightarrow\mathbb{R}\atop\sum_{v\in V}f(v)=0}\frac{\sum_{(i,j)\in E}w_{ij}(f(i)-f(j))^{2}}{\left(\sum_{(i,j)\in E}w_{ij}\right)\left(\sum_{v\in V}f(v)^{2}\right)}=\lambda_{2}(\mathbbm{1})$$

and where, simultaneously, the same is true for the largest eigenvalue $\lambda_{n}(w^{\prime})$

$$\inf_{w^{\prime}>0}\frac{\lambda_{n}(w^{\prime})}{\sum_{(i,j)\in E}w^{\prime}_{ij}}=\inf_{w^{\prime}>0}\sup_{f:V\rightarrow\mathbb{R}}\frac{\sum_{(i,j)\in E}w^{\prime}_{ij}(f(i)-f(j))^{2}}{\left(\sum_{(i,j)\in E}w^{\prime}_{ij}\right)\left(\sum_{v\in V}f(v)^{2}\right)}=\lambda_{n}(\mathbbm{1}).$$

These equations show the implicit mathematical structure of the question: there is an entire additional optimization problem sitting on top of the Rayleigh-Ritz formulations of $\lambda_{2}$ and $\lambda_{n}$ which are optimization problems in themselves.

Let us quickly consider some examples. It may not come as a surprise that the complete graph $K_{n}$ is conformally rigid. An elementary proof can be found in § 5.1.

Conformal rigidity does not imply that $\mathbbm{1}$ is the unique weight vector that maximizes $\lambda_{2}(w)$ and minimizes $\lambda_{n}(w^{\prime})$: a conformally rigid graph $G$ may have non-constant weights $w,w^{\prime}$ with $\lambda_{2}(w)=\lambda_{2}(\mathbbm{1})$, an example being a circulant graph in Example 2.4, or $\lambda_{n}(w^{\prime})=\lambda_{n}(\mathbbm{1})$ with an example being the Haar Graph 565, §6.3.

Our definition of conformal rigidity in graphs has a number of different motivations, we describe three below.

In this section we collect together various results in the literature that are close to the notion of conformal rigidity.

In § 2 we state our main results beginning with a summary. In § 3 we explain how semidefinite programming can be used to both test and certify conformal rigidity. We also explain how the automorphisms of the graph can sometimes drastically reduce the computational effort and also provide structural insight. § 4 discusses the connection between conformal rigidity and graph embeddings into Euclidean space. Section 5 contains the proofs of all our main results. Section 6 surveys various ways of proving that an explicitly given graph is conformally rigid via semidefinite programming certificates.

Our main result is that the class of conformally rigid graphs form an interesting and highly nontrivial collection of graphs.

1. (1)

   All edge-transitive graphs are conformally rigid and this is true for a simple reason (see §2.1). The smallest example of a conformally rigid graph that is not edge-transitive (that we know of) is the Hoffman graph on 16 vertices.

2. (2)

   All distance-regular graphs and, as a special case, all strongly-regular graphs are conformally rigid.

3. (3)

   Some Cayley graphs are conformally rigid and others are not. We provide a sufficient condition for Cayley graphs to be conformally rigid (Theorem 2.3). The smallest Cayley graph that we found that is conformally rigid but not edge-transitive is a circulant graph on 18 vertices (see §2.2).

4. (4)

   As an application of the criterion for Cayley graphs, we found (empirically) that there seems to be a large number of circulant graphs that are conformally rigid (see §2.3). They seem to have a rich structure. There are also infinite families of circulants that are not conformally rigid.

5. (5)

   Conformal rigidity can be characterized in terms of the existence of special spectral embeddings of the graph on the eigenspaces of $\lambda_{2}(\mathbbm{1})$ and $\lambda_{n}(\mathbbm{1})$. These embeddings are edge-isometric, meaning that that all edge lengths must be equal (see § 4).

6. (6)

   For any given graph $G=(V,E)$ it can be decided whether it is conformally rigid with a finite amount of computation by phrasing it as a semidefinite program. We explain how this is done and use it to verify conformal rigidity of a number of seemingly isolated examples that do not seem to fall into any of the above categories

   • •

     the Hoffman graph on 16 vertices.

   • •

     the crossing number graph 6B on 20 vertices [24].

   • •

     the distance-2 graph of the Klein graph on 24 vertices.

   • •

     the $(20,8)$ accordion graph [16] on 40 vertices.

We summarize our results in the figure below. A solid arrow implies containment while a dashed arrow implies partial containment.

Throughout this paper we assume that the vertex set of $G=(V,E)$ is enumerated by the first $n$ integers, $V=[n]$, and that all graphs are connected, undirected and simple. Also, for simplicity, we sometimes write $\lambda_{2}$ for $\lambda_{2}(\mathbbm{1})$ and $\lambda_{n}$ for $\lambda_{n}(\mathbbm{1})$.

Recall that an automorphism of $G=(V,E)$ is a permutation $\pi$ of its vertices so that $(i,j)\in E$ if and only if $(\pi(i),\pi(j))\in E$. The set of all automorphisms of $G$, denoted $\textup{Aut}(G)$ and called the automorphism group of $G$, is a subgroup of $S_{n}$. The graph $G$ is vertex-transitive if all vertices of $G$ lie in a single orbit of $\textup{Aut}(G)$. The action of $\textup{Aut}(G)$ on $V$ induces an action on $E$ via $(i,j)\mapsto(\pi(i),\pi(j))$. The graph $G$ is edge-transitive if all edges of $G$ lie in a single orbit of $\textup{Aut}(G)$. Complete graphs are edge-transitive and so are many other common families of graphs such as cycles, complete bipartite graphs and edge graphs of hypercubes. All distance-transitive graphs are edge-transitive (see Fig. 3). The following is a first observation.

The proof of this Proposition is a simple corollary of the technique of symmetry reduction (see for example [3]) which we recall in § 3. All proofs of results in this section can be found in § 5. One way of interpreting Proposition 2.1 is as follows: edge-transitivity is a strong symmetry property that essentially says that all edges in $G$ are the “same” which points toward weighting them equally. Therefore, in some sense, the most interesting conformally rigid graphs are the ones that are not edge-transitive since they are conformally rigid for a different reason. In the rest of the paper, we will focus on such graphs.

Recall that a finite graph $G=(V,E)$ is distance-regular if for any two vertices $u,v$, the number of vertices that is distance $i$ from $u$ and distance $j$ from $v$ only depends on $i,j$ and the distance $d(u,v)$.

Note that, in particular, every strongly-regular graph is distance-regular. Strongly-regular graphs are distance-regular graphs of diameter two (where diameter is the largest distance between any pair of points). For example, the Shrikhande graph and its complement (see Fig. 1) are strongly-regular. The proof of Theorem 2.2 uses structural theory of distance-regular graphs (see [11]) and the duality theory of semidefinite programs which are developed in § 3 and § 4.

A natural class of graphs with a great deal of symmetry are Cayley graphs. The Cayley graph $\Gamma(G,S)$ has vertex set $V=G$ where $G$ a finite group and the edge set $E=\left\{(g,s\circ g):g\in G,s\in S\right\}$ where $S\subset G$ is a generating set. Since we only deal with undirected graphs, we want the edge set to be symmetric which is achieved by requiring a symmetry in the generating set $S$, meaning $s\in S\implies s^{-1}\in S$. We always assume this.

Cayley graphs need not be conformally rigid. The smallest example we found is shown on the left in Fig. 5. This unweighted graph has $\lambda_{2}=2$ and $\lambda_{6}=5$. As it turns out, both of these eigenvalues can be improved by changing the edge weights: by increasing the weights on the three edges connecting the two triangles (while decreasing the weight on the other edges to maintain the normalization), we can increase $\lambda_{2}$ and attain $\lambda_{2}(w)=18/7>2$. Likewise, by setting the three edge weights connecting the two triangles to 0 (and increasing the remaining weights uniformly) we can decrease the largest eigenvalue to $\lambda_{6}(w)=4.5<5$ (and thus, in particular, can achieve values smaller than 5 for small positive weights by continuity of the eigenvalues). On the other hand, the circulant graph shown on the right of Fig. 5 is conformally rigid but not edge-transitive. Its conformal rigidity is implied by the following Theorem.

Comments. Several comments are in order.

1. (1)

   The condition in (7) is simple to check for any given eigenvector $\varphi$: one has to compute $|S|$ different sums and check whether the values all coincide. This is illustrated below where we use Theorem 2.3 to prove that the circulant shown in Fig. 5 is conformally rigid.

2. (2)

   Recall that $S$ is symmetric ($s\in S\implies-s\in S$). By changing variables one sees that $\sum_{g\in\Gamma}\varphi(g)\varphi(g\circ s)=\sum_{g\in\Gamma}\varphi(g)\varphi(g\circ(-s)),$ and hence it suffices to check $|S|/2$ sums in (7).

3. (3)

   The difficulty in applying Theorem 2.3 is that one has to find suitable eigenvectors. What we have observed in practice is that when the eigenspace corresponding to, say, $\lambda_{2}$, has multiplicity larger than 1, then the property will typically only be true for a single element in the space.

4. (4)

   The result is sufficient. It is not clear to us whether it is necessary. In small-scale numerical experiments (carried out exclusively in the setting of circulants) we found that every time a circulant turned out to be conformally rigid, there indeed seemed to be a suitable choice of eigenvectors $\varphi_{2}$ and $\varphi_{n}$ which made the result applicable. It would be interesting to understand this better.

Denote a circulant with $n$ vertices and generators $S$ by $C_{n}(S)$. It is typical to list only one of $s$ or $-s$ in $S$, with the assumption that the other one is included. The set of vertices of $C_{n}(S)$ is $V={\mathbb{Z}}_{n}=\left\{0,1,2,\dots,n-1\right\}$ and its edges are $(v,v+s)$, as we vary over all $v\in V$ and $s\in S$ (mod $n$). Since circulants are Cayley graphs, Theorem 2.3 applies to them. We start with some examples.

1. (1)

   The cycle graph $C_{n}$ and the complete graph $K_{n}$ are circulants. These are edge-transitive and hence conformally rigid by Proposition 2.1.

2. (2)

   Paley graphs of prime order are circulants, edge-transitive and thus also conformally rigid.

3. (3)

   Cocktail Party graphs are circulants, distance-transitive and hence, edge-transitive, and therefore conformally rigid.

4. (4)

   The smallest circulant on two generators which is not edge-transitive but nonetheless conformally rigid is $C_{18}(\left\{1,5\right\})$ (see Fig. 5).

Circulants are rich and rather nontrivial. In particular, understanding when a circulant is edge-transitive appears to be a nontrivial problem, no definite characterization appears to be known. There are infinitely many circulants that are not conformally rigid.

The restriction $n\geq 7$ is necessary, $C_{5}(\left\{1,2\right\})=K_{5}$ is conformally rigid and so is $C_{6}(\left\{1,2\right\})$ (also known as the octahedral graph). The list of examples in (1)-(4) contains several infinite families of conformally rigid circulants, however, all of them are edge-transitive. It is a natural question whether there are infinitely many conformally rigid circulants that are not edge-transitive. We were unable to answer this question. Numerically, they appear to be fairly frequent. As an example, the complete list of conformally rigid circulants on $n=21$ vertices with 2 generators is listed below (with graphs that are not edge transitive underlined):

Questions. We conclude with a number of questions.

1. (1)

   Is it possible to characterize conformally rigid circulants $C_{n}(\left\{a,b\right\})$ in terms of $n,a,b$? While there seem to be a great many, we were unable to detect a simple pattern or even identify a single infinite family.

2. (2)

   Somewhat unrelated, but of independent interest in this context: is it possible to characterize when $C_{n}(\left\{a,b\right\})$ is edge-transitive in terms of $n,a,b$? Some partial results can be found in [2, 5, 6, 25, 28, 29].

3. (3)

   Theorem 2.3 provides a rigorous criterion that can be used to prove conformal rigidity of Cayley graphs and, in particular, circulants. Does the Theorem identify all conformally rigid circulants? Numerically this seems to be true for small conformally rigid circulants.

4. (4)

   Are there infinitely many conformally rigid circulants $C_{n}(\left\{a,b\right\})$ that are not edge-transitive? The smallest one appears to be $C_{18}(\left\{1,5\right\})$ described in Example 2.4, but there appear to be many more after that.

5. (5)

   All these questions would be also interesting in the context of circulants with three or more generators.

At this point, we have identified a number of reasons why a graph may be conformally rigid; most of the examples can be identified as belonging to a number of different groups, these being

1. (1)

   edge-transitive graphs,

2. (2)

   distance-regular graphs, in particular, strongly-regular graphs,

3. (3)

   and certain Cayley graphs, in particular, certain circulant graphs.

The purpose of this section is to describe some isolated sporadic examples of graphs that do not belong into any of these groups.

The graphs in this list are mostly well-known; the less well known examples (3), (5), (6), (8) are implemented in Mathematica 13.3 as (”Haar”, 565), (”ZeroTwoBipartite”, 7, 1),(”ZeroTwoBipartite”, 7, 2) and (”NoncayleyTransitive”, 24, 23). The graphs in Theorem 2.6 were all proven to be conformally rigid by identification of a suitable dual semidefinite programming certificate (see § 6). However, we also found one other example, the non-Cayley vertex-transitive graph (20,10) on $n=20$ vertices, that is not edge-transitive, not distance-regular, but very likely conformally rigid (beyond any reasonable numerical doubt); however, we were unable to find a certificate that would rigorously prove its conformal rigidity.

The conformal rigidity of a graph can be checked and certified using semidefinite programming which is a branch of convex optimization [4]. This connection provides both theoretical and computational tools as we will see in § 4 and § 6. We first explain the necessary background.

Assume that $G=([n],E)$ is an undirected connected graph with positive edge weights $w_{ij}$. Let $\mathcal{S}_{+}^{n}$ be the cone of $n\times n$ real symmetric psd matrices. We use $X\succeq 0$ to denote that $X\in\mathcal{S}^{n}_{+}$. For $A,B\in\mathcal{S}^{n}_{+}$, $A\succeq B$ stands for $A-B\succeq 0$.

Recall from § 1.3.1 that the problems of maximizing $\lambda_{2}(w)$ and minimizing $\lambda_{n}(w)$ with the sum of edge weights being bounded, have been studied in a number of papers such as [9, 12, 13, 27]. We follow the exposition in [27] where one observes

(8)

$\displaystyle\lambda_{2}^{\ast}:=\textup{ max }\left\{\lambda_{2}(w)\,:\,\sum w_{ij}=|E|,\,\,w\geq 0\right\}.$

is a convex optimization problem that is equivalent to

(9)

$\displaystyle p^{\ast}:=\textup{ min}\left\{\sum_{ij\in E}w_{ij}\,:\,\lambda_{2}(w)\geq 1,\,\,w\geq 0\right\}.$

Problem (8) has an optimal solution since its objective function is continuous and its feasible region is compact. Therefore, (9) also has an optimal solution. If $w^{*}$ is optimal for (9), then $\lambda_{2}(w^{*})=1$ and hence there is no harm in optimizing over $\lambda_{2}(w)\geq 1$. The two problems are off by a scaling; $\lambda_{2}^{*}=|E|/p^{*}$ and $w^{*}$ is optimal for (9) if and only if $(|E|w^{*})/p^{*}$ is optimal for (8). Using $I$ to denote the identity matrix and $J$ to denote the matrix filled with 1’s, the constraint $\lambda_{2}(w)\geq 1$ can be modeled by $L_{w}\succeq I-(1/n)J$ (the eigenvalues of $I-(1/n)J$ are $1,1,\ldots,1,0$) and (9) is the semidefinite program (SDP):

(10)

$\displaystyle\begin{split}p^{\ast}:=&\textup{ min}\sum_{ij\in E}w_{ij}\\ \textup{s.t. }&L_{w}\succeq I-\frac{1}{n}J\\ &w\geq 0.\end{split}$

SDPs are optimization problems in the space of symmetric matrices where we are optimizing a linear function over a (convex) region described by linear matrix inequalities, or equivalently, an affine slice of the cone of psd matrices, see [4]. Calling (10) the primal SDP, its dual SDP is derived in [27] as

(11)

$\displaystyle\begin{split}d^{\ast}:=&\textup{ max }\textup{Trace }X\\ \textup{s.t. }&X_{ii}+X_{jj}-2X_{ij}\leq 1\,\,\forall ij\in E\\ &\mathbbm{1}^{\top}X\mathbbm{1}=0,X\succeq 0.\end{split}$

Strong duality holds between the primal and dual SDPs and so $p^{*}=d^{*}$. A pair of primal/dual feasible solutions $w^{*}$ and $X^{*}$ are optimal for their respective problems if and only if they satisfy the complementary slackness conditions

(12)		$\displaystyle(1-(X_{ii}^{*}+X_{jj}^{*}-2X_{ij}^{*}))w_{ij}^{*}=0\,\,\forall ij\in E$
(13)		$\displaystyle L_{w^{*}}X^{*}=X^{*}$

Analogous to the above, the problem of minimizing $\lambda_{n}(w)$ can be written as

(14)

$\displaystyle\lambda_{n}^{\ast}:=\textup{ min }\left\{\lambda_{n}(w)\,:\,\sum w_{ij}=|E|,\,\,w\geq 0\right\}.$

which is equivalent to

(15)

$\displaystyle q^{\ast}:=\textup{ max}\left\{\sum_{ij\in E}w_{ij}\,:\,\lambda_{n}(w)\leq 1,\,\,w\geq 0\right\}.$

As before, if $w^{*}$ is optimal for (15) then $\lambda_{n}(w^{*})=1$, $\lambda_{n}^{*}=|E|/q^{*}$ and $w^{*}$ is optimal for (15) if and only if $(|E|w^{*})/q^{*}$ is optimal for (14). Problem (15) is the SDP:

(16)

$\displaystyle\begin{split}q^{\ast}:=&\textup{ max}\sum_{ij\in E}w_{ij}\\ &\textup{s.t. }L_{w}\preceq I-\frac{1}{n}J\\ &w\geq 0\end{split}$

whose dual is

(17)

$\displaystyle\begin{split}r^{\ast}:=&\textup{ min }\textup{Trace }Y\\ &\textup{s.t. }Y_{ii}+Y_{jj}-2Y_{ij}\geq 1\,\,\forall ij\in E\\ &\mathbbm{1}^{\top}Y\mathbbm{1}=0,\,\,Y\succeq 0.\end{split}$

Again, strong duality holds so that $q^{*}=r^{*}$, and a pair of primal/dual feasible solutions $(w^{*},Y^{*})$ are optimal for their respective problems if and only if they satisfy complementary slackness:

(18)		$\displaystyle(Y^{*}_{ii}+Y^{*}_{jj}-2Y^{*}_{ij}-1)w^{*}_{ij}=0\,\,\forall ij\in E$
(19)		$\displaystyle L_{w^{*}}Y^{*}=Y^{*}$

We can now test for conformal rigidity via the above optimization problems.

Under strong duality, the pair of primal and dual SDPs will have the same optimal value, which allows the following re-phrasing:

Usually, these observations cannot be used directly to certify conformal rigidity since SDPs are solved using numerical algorithms making it difficult to know precisely what their optimal values and solutions are. In § 6 we illustrate several methods to overcome these difficulties. A more computationally friendly certificate for conformal rigidity can be obtained from the complementary slackness conditions. Indeed, when $G$ is conformally rigid, the optimal weights are $\mathbbm{1}/\lambda_{2}(\mathbbm{1})$ and $\mathbbm{1}/\lambda_{n}(\mathbbm{1})$ and so complementary slackness implies the following.

The condition $LX=\lambda_{2}(\mathbbm{1})X$ is saying that the columns of $X$ are in the eigenspace of $L$ for $\lambda_{2}(\mathbbm{1})$. Therefore, the rank of $X$ is bounded above by the multiplicity of $\lambda_{2}(\mathbbm{1})$ as an eigenvalue of $L$. Analogously, the columns of $Y$ must come from the eigenspace of $L$ for $\lambda_{n}(\mathbbm{1})$ and so the rank of $Y$ cannot exceed the multiplicity of $\lambda_{n}(\mathbbm{1})$. In particular, if $\lambda_{2}(\mathbbm{1})$ (respectively, $\lambda_{n}(\mathbbm{1})$) has multiplicity $1$, then $X$ (respectively, $Y$) can be taken to be the rank one matrix

(22)

$\displaystyle p^{*}uu^{\top}\textup{}(\textup{ respectively, }q^{*}uu^{\top})$

where $u$ is a normalized eigenvector of $\lambda_{2}(\mathbbm{1})$ (respectively, $\lambda_{n}(\mathbbm{1})$). These observations were noted in [27]. A very short explanation why the notion of conformal rigidity simplifies when the eigenspace associated to $\lambda_{2}$ (or $\lambda_{n}$) has multiplicity 1 is as follows: in that case, we have

$$\lambda_{2}(\mathbbm{1})=\min_{\sum_{v\in V}f(v)=0}\frac{\sum_{(i,j)\in E}(f(i)-f(j))^{2}}{\sum_{v\in V}f(v)^{2}}$$

We note that introducing weights $w_{ij}$ (normalized to $\sum_{(i,j)\in E}w_{ij}=|E|$) and then continuously varying the weights leads to a continuously perturbation of the matrix. If $\lambda_{2}$ has multiplicity 1, then both the eigenvalue $\lambda_{2}(w)$ and the associated eigenvector depend continuously on the change in the weights (and, in particular, do not change much for small changes in the weight). Suppose now that, for the eigenvector in the unweighted case, not all terms in the sum $(f(i)-f(j))^{2}$ are of the same size: then we could consider a small perturbation of the weights, putting more emphasis on larger summands and less on smaller ones and arrive at a weight for which $\lambda_{2}(w)>\lambda_{2}(\mathbbm{1})$ which contradicts conformal rigidity.

We now do a small example to illustrate the above ideas, one in which all claims can be checked by hand.