Factorizations of Polynomials over Noncommutative Algebras and   Sufficient Sets of Edges in Directed Graphs
  To directed graphs with unique sink and source we associate a noncommutative associative alsgebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when coefficients of the polynomial can be rationally expressed via elements of a given set of pseudo-roots (edges). Our results are based on a new theorem for directed graphs also proved in this paper. 
Israel Gelfand, Sergei Gelfand, Vladimir Retakh, and Robert Lee Wilson