Counting paths in digraphs

We begin with some terminology. All digraphs in this paper are finite. For a digraph G, we denote its vertex and edge sets by V (G) and E(G), respectively. Unless otherwise stated, we assume |V (G)| = n. The members of E(G) are ordered pairs of vertices. We use the notation uv to denote an ordered pair of vertices (u, v) (whether or not u and v are adjacent). We only consider digraphs which have no loop edges uu, and at most one directed edge uv for all pairs of vertices u 6= v (are simple). A non-edge in G is an unordered pair of distinct vertices u, v so that uv, vu are both not in E(G). We say a simple digraph G is a tournament if for all pairs of vertices u 6= v, exactly one of uv, vu is an edge.