Classifying the globally rigid edge-transitive graphs and distance-regular graphs in the plane

A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius characterised exactly which vertex-transitive graphs are globally rigid solely by their degree and maximal clique number, two easily computable parameters for vertex-transitive graphs. In this short note we will extend this characterisation to all graphs that are determined by their automorphism group. We do this by characterising exactly which edge-transitive graphs and distance-regular graphs are globally rigid by their minimal and maximal degrees.