On Barycentric Representation Of Graphs

Abstract

We shall prove that the linear system associated to a barycentric representation always has a solution, we shall give a necessary and sufficient condition for this solution to be uniquely defined and then give a formal expression for this solution in terms of weighted arborescences. Then, in the planar case, we shall give some necessary and sufficient conditions for such a representation to be plane, given that the fixed vertices are embedded in convex position with a "compatible" circular order