Quivers of semi-maximal rings

In this paper, the set of quivers of semi-maximal rings is investigated. It is proved that the elements of this set are formed by the elements of the set of quivers of tiled orders and that the set of quivers of tiled orders with $n$ vertices is determined by the integer points of a convex polyhedral domain that lie in the nonnegative part of the space $\mathbb R^{n^2-n}$. It is also proved that the set of quivers of tiled orders with $n$ vertices contains all simple oriented strongly connected graphs with $n$ vertices and $n$ loops, does not contain any graphs with $n$ vertices and $n-1$ loops, and contains only a part of the graphs with $n$ vertices and $m$ ($m<n-1$) loops.