Euclidean Steiner minimal trees, minimum energy configurations, and the embedding problem of weighted graphs in E3

Abstract

We have found that a triple helix configuration of points in E 3 yields the best value of the Steiner ratio for the Euclidean Steiner Minimal Tree (ESMT) problem. In this paper we explore the properties, configurations, and implications of this topology which yields this best Steiner ratio and its relationship to the Euclidean Graph embedding problem (EGEP) for weighted graphs in E 3. The unique equivalence between these problems is also explored in their application for identification and modelling of minimum energy configurations (MECs) such as the biochemical protein structures of Collagen.