I am not familiar with graphs. However, I am curious about a question on graphs. Given a finite set equipped with a metric, is there any studying on the following problem?
Problem: given the metric $d$ for vertexes on $V$, build the undirected graph with the minimum number of edges such that, for each pair of vertices $(v_1, v_2)\in V$, the shortest path from $v_1$ to $v_2$ equals $d(v_1, v_2)$. Is this graph unique?
This kind of graph building would be very important to model depedency of agents on economical dynamic systems.