Conformally rigid graphs

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 = λ_1 < λ_2 ≤…≤λ_n. One can also consider the same graph equipped with positive edge weights w:E →ℝ_> 0 normalized to ∑_e ∈ 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 λ_2(w) of L_w over all w, and minimize λ_n(w') of L_w' over all w', i.e., for all w,w', λ_2(w) ≤λ_2(1) ≤λ_n(1) ≤λ_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.