  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.
