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

Abstract

AbstractA 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.