Metric spaces and sparse graphs

Abstract

Many concrete problems are formulated in terms of a finite set of points in $R^n$ which, via the ambient Euclidean metric, becomes a finite metric space. To obtain information from such a space, it is often useful to associate a graph to it, and do mathematics on the graph, rather than on the space. Connected graphs become finite metric spaces (rather, their set of vertices) via the path-metric. We consider different types of connected graphs that can be associated to a metric space. In turn, the metric spaces obtained from these graphs recover, in some cases, identically the initial space, while they, more often, only "approximate" it. In this last category, we construct a connected sparse graph, denoted CS, that seems to be new. We show that the "general case" for CS is to be a tree, a result of clear practical importance.