Isometric embeddings of graphs into Riemannian manifolds

An isometric embedding of a graph into a metric space is an embedding of the vertices such that the smallest number of edges connecting any two vertices equals to the distance in the metric space between the images. In this paper, we study isometric embeddings of graphs into Riemannian manifolds. We give some classifications of graphs that can be isometrically embedded into certain Riemannian manifolds.