Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = \lambda_1 < \lambda_2 \leq \dots \leq \lambda_n$. One can also consider the same graph equipped with positive edge weights $w:E \rightarrow \mathbb{R}_{> 0}$ normalized to $\sum_{e \in E} w_e = |E|$ and the associated weighted Laplacian matrix $L_w$. We say that $G$ is conformally rigid if constant edge-weights maximize the second eigenvalue $\lambda_2(w)$ of $L_w$ over all $w$, and minimize $\lambda_n(w')$ of $L_{w'}$ over all $w'$, i.e., for all $w,w'$, $$ \lambda_2(w) \leq \lambda_2(1) \leq \lambda_n(1) \leq \lambda_n(w').$$ Conformal rigidity requires an extraordinary amount of symmetry in $G$. Every edge-transitive graph is conformally rigid. We prove that every distance-regular graph, and hence every strongly-regular graph, is conformally rigid. Certain special graph embeddings can be used to characterize conformal rigidity. Cayley graphs can be conformally rigid but need not be, we prove a sufficient criterion. We also find a small set of conformally rigid graphs that do not belong into any of the above categories; these include the Hoffman graph, the crossing number graph 6B and others. Conformal rigidity can be certified via semidefinite programming, we provide explicit examples.

翻译：给定一个有限、简单、连通图 $G=(V,E)$，其中 $|V|=n$，我们考虑其关联的图拉普拉斯矩阵 $L = D - A$，特征值为 $0 = \lambda_1 < \lambda_2 \leq \dots \leq \lambda_n$。同样可考虑该图配备正边权 $w:E \rightarrow \mathbb{R}_{> 0}$（归一化为 $\sum_{e \in E} w_e = |E|$）及其关联的加权拉普拉斯矩阵 $L_w$。若在所有权值 $w$ 上常数边权最大化 $L_w$ 的第二特征值 $\lambda_2(w)$，且在所有 $w'$ 上最小化 $L_{w'}$ 的 $\lambda_n(w')$，即对所有 $w,w'$ 有 $$ \lambda_2(w) \leq \lambda_2(1) \leq \lambda_n(1) \leq \lambda_n(w')，$$ 则称 $G$ 为共形刚性。共形刚性要求图 $G$ 具有非凡的对称性。每个边传递图都是共形刚性的。我们证明每个距离正则图（从而每个强正则图）都是共形刚性的。某些特殊图嵌入可用于刻画共形刚性。凯莱图可能具有共形刚性但非必然，我们证明了一个充分判据。我们还发现了一小类不属于上述任何类别的共形刚性图，包括霍夫曼图、交叉数图 6B 等。共形刚性可通过半定规划验证，我们提供了显式实例。