Structural properties of minimal strong digraphs versus trees

In this article, we focus on structural properties of minimal strong digraphs (MSDs). We carry out a comparative study of properties of MSDs versus (undirected) trees. For some of these properties, we give the matrix version, regarding nearly reducible matrices. We give bounds for the coefficients of the characteristic polynomial corresponding to the adjacency matrix of trees, and we conjecture bounds for MSDs. We also propose two different representations of an MSD in terms of trees (the union of a spanning tree and a directed forest; and a double directed tree whose vertices are given by the contraction of connected Hasse diagrams).