{\stackrel{\mbox{\scriptsize def}}{=}}\bigoplus_{k=0}^\infty
\Graph_{n,k}$ bears the structure of an associative algebra: one defines a product of the graphs $G_1 = (e_1 {, \dots ,} e_{k_1}) \in \Gamma_{n,k_1}$ and $G_2 = (h_1 {, \dots ,} h_{k_2}) \in \Gamma_{n,k_2}$ as $G_1*G_2 {\stackrel{\mbox{\scriptsize def}}{=}}(e_1 {, \dots ,}
e_{k_1}, h_1 {, \dots ,} h_{k_2}) \in \Gamma_{n,k_1+k_2}$; then $*$ is extended to the whole $\Graph_n$ as a bilinear operation. Note that $G_1*G_2 \ne
G_2*G_1$ (the edges are the same but the edge numbering is different), so the algebra $\Graph_n$ is not commutative.

We call a graph $G \in \Gamma_{n,k}$ [*strongly connected*]{} if every two its vertices can be joined by a directed path. A graph is [*strongly semiconnected*]{} if every its connected component (in the topological sense) is strongly connected; equivalently, if every its edge is a part of a directed cycle. A strongly semiconnected graph may contain isolated vertices (i.e. vertices not incident to any edge); by $\SSC_{n,k}^{\{i_1
{, \dots ,} i_s\}}$ we denote the set of strongly semiconnected graphs $G \in
\Gamma_{n,k}$ such that the vertices $i_1 {, \dots ,} i_s$, and only they, are isolated. By $\SSC_{n,k} {\stackrel{\mbox{\scriptsize def}}{=}}\bigcup_{I \subset \{1 {, \dots ,} n\}}
\SSC_{n,k}^I$ we will denote the set of all 