edges: $G = (e_1 {, \dots ,} e_k) \in \Gamma_{n,k}$ means a graph where the edge $e_\ell$ has number $\ell$, for all $\ell = 1
{, \dots ,} k$. By a slight abuse of notation $e \in G$ will mean that $e$ is an edge of $G$ (regardless of number).

Let $G \in \Gamma_{n,k}$ and $e \in G$. By $G \setminus e$, $G/e$ and $G_e^{\vee}$ we will denote the graph $G$ with $e$ deleted, $e$ contracted and $e$ reversed, respectively. Note for correctness that since $G
\setminus e \in \Gamma_{n,k-1}$, one has to change the edge numbering in $G$ after deleting $e$: namely, if $e$ bears number $s$ in $G$ then the numbers of the edges are preserved if they are less than $s$ and lowered by $1$ otherwise. For $G/e \in \Gamma_{n-1,k-1}$ the same renumbering is applied both to the edges and to the vertices. The contracted edge $e$ should not be a loop.

A graph $H \in \Gamma_{n,m}$ is called a subgraph of $G \in \Gamma_{n,k}$ (notation $H \subseteq G$) if $H$ is obtained from $G$ by deletion of several (possibly zero) edges.

Denote by $\Graph_{n,k}$ a vector space over $\Complex$ spanned by $\Gamma_{n,k}$. The direct sum $\Graph_n 