Distance geometry

Definition

Let $G = (V, E)$ be a finite graph, let $\delta \colon E \to [0, \infty)$ be a weight on its edges, and let $k \ge 1$ be an integer. A realization of $(G, \delta)$ in $\R^k$ is a map $p \colon V \to \R^k$ such that $$\norm{p(u) - p(v)} = \delta(uv) \quad \text{for every edge } uv \in E.$$ Here $\norm{p(u) - p(v)}$ is the Euclidean distance of the two points. The distance geometry problem asks, for given $(G, \delta)$ and $k$, whether a realization in $\R^k$ exists, and to find one. Distance geometry is the study of such realizations and of the point sets they describe.