Measurement Isomorphism of Graphs
  The d-measurement set of a graph is its set of possible squared edge lengths over all d-dimensional embeddings. In this note, we define a new notion of graph isomorphism called d-measurement isomorphism. Two graphs are d-measurement isomorphic if there is agreement in their d-measurement sets. A natural question to ask is "what can be said about two graphs that are d-measurement isomorphic?" In this note, we show that this property coincides with the 2-isomorphism property studied by Whitney. 
Steven J. Gortler and Dylan P. Thurston
math.MG cs.DM math.CO