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 $k\geq 1$.



Does every primitive digraph have a directed cycle?
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).


