Sums of distances on graphs and embeddings into Euclidean space

Let $G=(V,E)$ be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices $x_1,\cdots ,x_k$, take $x_{k+1}$ to be any vertex maximizing the sum of distances to the vertices already chosen and iterate, keep adding the “most remote” vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices $m\ll \left|V\right|$. We prove that this suggests that the graph G is, in a suitable sense, “m-dimensional” by exhibiting an explicit 1-Lipschitz embedding $\varphi :V\to {\ell }^1\left(R^m\right)$ with good properties.