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

Cyclic structure, vertex degree and number of linear vertices in Minimal Strong Digraphs