Embeddings of graphs in euclidean spaces

Optimal distortion embeddings of distance regular graphs into Euclidean spaces

In this paper we give a lower bound for the least distortion embedding of a distance regular graph into Euclidean space. We use the lower bound for finding the least distortion for Hamming graphs, Johnson graphs, and all strongly regular graphs. Our technique involves semidefinite programming and exploiting the algebra structure of the optimization problem so that the question of finding a lowe...

Optimal Embeddings of Distance Transitive Graphs into Euclidean Spaces

In this paper we give an explicit formula for the least distortion embedding of a distance transitive graph into Euclidean space. We use this formula for finding least distortion embeddings for important examples: Hamming graphs, Johnson graphs, and Grassmann graphs. Our technique involves semidefinite programming and exploiting symmetry to simplify the optimization problem so that the question...

On Embeddings of Tori in Euclidean Spaces

Counting Euclidean embeddings of rigid graphs

A graph is called (generically) rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd in a finite number of distinct ways, modulo rigid transformations. Here we deal with the problem of determining the maximum number of planar Euclidean embeddings as a function of the number of the vertices. We obtain polynomial systems which totally capture the structure...

Bilipschitz Embeddings of Metric Spaces into Euclidean Spaces

When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small (“snowflake”) deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, lik...