I am interested in finite directed multi-graphs with one connected component, where each vertex comes with exactly 1 edge pointing out from it, which can point to another vertex, or itself. So self-loops as well as loops between neighbors are allowed, for example 1-->2 and 2-->1. Note that there is no restriction on the number of edges pointing into a vertex. The only rule is that exactly 1 edge must be pointing out of each vertex. For this type of a graph (does it have a name?), I believe that
(# of edges) - (# of vertices) + 1 = (# of loops)
Here loops and cycles are the same, and can be any size. (I believe the tetrahedron is not possible for this family of graphs)
Can anyone prove this or find a counter-example?