Isometric embeddings of graphs into Riemannian manifolds

TL;DR

This paper investigates conditions under which graphs can be embedded into Riemannian manifolds while preserving distances, providing classifications for such embeddings in specific cases.

Contribution

It offers new classifications of graphs that can be isometrically embedded into certain Riemannian manifolds, advancing understanding of graph embeddings in geometric spaces.

Findings

Classified graphs embeddable into specific Riemannian manifolds

Identified conditions for isometric embeddings

Extended known results on graph embeddings

Abstract

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.