Conformally Rigid Graph

A conformally rigid graph is a simple connected graph whose uniform edge weights maximize the second smallest Laplacian matrix eigenvalue and minimize its largest eigenvalue among nonnegative edge weights with the same total weight (Steinerberger and Thomas 2025). For such a graph on vertices with edge set , let be the weighted Laplacian matrix defined by

(1)

where means . Let have eigenvalues

(2)

and let be the edge-weight vector of length consisting of all 1's. The condition for conformal rigidity is that the weighted Laplacian eigenvalues satisfy

(3)

for all edge weights and that are nonnegative and normalized such that

(4)

where is the edge count of .

Conformal rigidity reflects an extraordinary amount of symmetry in a graph (Steinerberger and Thomas 2025).

All connected edge-transitive graphs and distance-regular graphs are conformally rigid (Steinerberger and Thomas 2025). Since connected strongly regular graphs are distance-regular, they are also conformally rigid.

There are no conformally rigid graphs that are neither edge-transitive nor distance-regular on 10 or fewer vertices (E. Weisstein, Mar. 1, 2024). The smallest known conformally rigid graph that is neither edge-transitive nor distance-regular is the Hoffman graph on 16 vertices (Steinerberger and Thomas 2025). The following table, which extends the results of Steinerberger and Thomas (2025), lists all 13 known such exceptionally conformally rigid graphs (E. Weisstein, Feb. 23, 2024).

	non-ET, non-DR, CR graph
16	Hoffman graph
18	circulant graph
20	smallest cubic crossing number graph CNG6B
20	565-Haar graph
20	(10, 3)-incidence graph 3
20	(10, 3)-incidence graph 4
20	20-noncayley vertex-transitive graph 10
24	distance-2 graph of the 24-Klein graph
24	24-noncayley vertex-transitive graph 23
40	(20, 8)-accordion graph
48	(0, 2)-bipartite graph (7, 1)
48	(0, 2)-bipartite graph (7, 2)
120	120-Klein graph

Some Cayley graphs are conformally rigid and others are not. Steinerberger and Thomas (2025) provide a sufficient condition for Cayley graphs to be conformally rigid.

Circulant graphs are not conformally rigid for (Steinerberger and Thomas 2025), meaning antiprism graphs (other than the octahedral graph) are also not conformally rigid.