Let be the weighted Laplacian matrix defined for a simple connected graph on vertices with edge set and edge weights defined by
(1)

where means . Let have eigenvalues
(2)

and let be the vector of length consisting of all 1's. Steinerberger and Thomas (2024) then call a graph conformally rigid if its 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 2024).
All connected edgetransitive graphs and distanceregular graphs are conformally rigid (Steinerberger and Thomas 2024). Since connected distanceregular graphs are strongly regular, connected strongly regular graphs are also conformally rigid.
There are no conformally rigid graphs that are edgetransitive or distanceregular on 10 or fewer vertices (E. Weisstein, Mar. 1, 2024). The smallest known conformally rigid graph that is not edgetransitive or distanceregular is the Hoffman graph on 16 vertices (Steinerberger and Thomas 2024). The following table, which extends the results of Steinerberger and Thomas (2024), lists all 13 known such exceptionally conformally rigid graphs (E. Weisstein, Feb. 23, 2024).
nonET, nonDR, CR graph  
16  Hoffman graph 
18  circulant graph 
20  smallest cubic crossing number graph CNG6B 
20  565Haar graph 
20  (10, 3)incidence graph 3 
20  (10, 3)incidence graph 4 
20  20noncayley vertextransitive graph 10 
24  distance2 graph of the 24Klein graph 
24  24noncayley vertextransitive graph 23 
40  (20, 8)accordion graph 
48  (0, 2)bipartite graph (7, 1) 
48  (0, 2)bipartite graph (7, 2) 
120  120Klein graph 
Some Cayley graphs are conformally rigid and others are not. Steinerberger and Thomas (2024) provide a sufficient condition for Cayley graphs to be conformally rigid.
Circulant graphs are not conformally rigid for (Steinerberger and Thomas 2024), meaning antiprism graphs (other than the octahedral graph) are also not conformally rigid.