Minimal tiled orders of finite global dimension

A directed graph is associated to any basic tiled order, and it turns out that the graph is connected for all known examples of tiled orders of finite global dimension. It is proved that the minimal connected tiled orders of finite global dimension in a fixed algebra are of global dimension two, and that up to isomorphism, these minimal orders are characterized by their unoriented graph which is a tree. Their irreducible representations are in one-to-one correspondence with the possible orientations of this tree.