  Minimal Strong Digraphs (MSDs) can be regarded as a generalization of the
concept of tree to directed graphs. Their cyclic structure and some spectral
properties have been studied in several articles. In this work, we further
study some properties of MSDs that have to do with bounding the length of the
longest cycle (regarding the number of linear vertices, or the maximal in- or
outdegree of vertices); studying whatever consequences from the spectral point
of view; and giving some insight about the circumstances in which an efficient
algorithm to find the longest cycle contained in an MSD can be formulated.
Among other properties, we show that the number of linear vertices contained in
an MSD is greater or equal to the maximal (resp. minimal) in- or outdegree of
any vertex of the MSD and that the maximal length of a cycle contained in an
MSD is lesser or equal to 2n-m, where n,m are the order and the size of the MSD
respectively; we have found a bound for the coefficients of the characteristic
polynomial of an MSD, extending the result in a previous work of Lacalle,
Marijuan and Pozo, and finally, we prove that computing the longest cycle
contained in an MSD is an NP-Hard problem.
