Does every primitive digraph have a directed cycle?

Question

A digraph is a directed graph.

A directed cycle or simple directed circuit is a directed circuit in which the only repeated vertices are the first and last vertices.

A digraph is primitive if its adjacency matrix is primitive.

A square non-negative matrix $A$ is said to be primitive if there exists a positive integer $k$ such that $A^k >0$ (all entries of $A^k$ are positive).

I need only the existence of a path with the structure $i_0 i_1...i_k i_0$ (sequence of distinc edges) with $k\geq 1$.

Answer 1

Yes. For all $i,j$, $(A^k)_{ij}>0$, so there is at least one walk of length $k$ from $v_1$ to $v_2$ and there is at least one walk of length $k$ from $v_2$ to $v_1$. This closed directed walk which goes from $v_1$ to $v_2$ and then back to $v_1$ must contain a non-trivial directed cycle (i.e. a cycle of length at least 2).