An Optimal Lower Bound For The Size Of Periodic Digraphs

Markov graphs form a special class of digraphs constructed from self-maps on the vertex sets of combinatorial trees. In this paper, the trees that admit cyclic permutations of their vertex sets with non-strongly connected Markov graphs in terms of the existence of a special subset of edges are characterized. Additionally, the structure of self-maps of finite sets, which produce strongly connected Markov graphs for all trees, is described. A similar question, concerning which self-maps produce strongly connected Markov graphs for some trees, is answered for the class of permutations.