Metric spaces and sparse graphs

TL;DR

This paper explores how different types of connected graphs can represent finite metric spaces derived from Euclidean points, introducing a new sparse graph construction that often simplifies to a tree, with practical implications.

Contribution

It introduces a novel sparse graph, CS, that approximates finite metric spaces and demonstrates that in general, CS reduces to a tree structure, enhancing understanding of graph-metric space relationships.

Findings

The CS graph often simplifies to a tree structure.

Different graph types can recover or approximate original metric spaces.

The new CS graph is sparse and practically useful.

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.