2 claification

I've been working on a talk based on some stuff in Olver's "Classical Invariant Theory" book and have been wondering about a related graph enumeration problem.

Start with a triple $(n,v,e)$ of natural numbers. Take all $\mathbb{Q}$-linear combinations of directed graphs (allowing multiple edges, but no loops) with $v$ vertices, $e$ edges, and each vertex has at most $n$ edges going to it or coming from it. Now, take three relations (images scanned from Olver)

Rule 1:

Rule 2:

Rule 3:

Where the function $v$ with a vertex as subscript next to a graph means that graph multiplied by $n$ minus the number of edges attached to that vertex. (So, for instance, an isolated vertex gets multiplied by $n$)

Denote the space after quotienting by these relations by $V_{n,v,e}$. And so, in final form, my question:

What is $\dim V_{n,v,e}$? Or at least, can we find relatively effective upper bounds?

EDIT: Some clarifications. The colorings on the vertices are just to mark them in the pictures to keep track of where everything goes, the graphs are not marked themselves. Additionally, as Rule 2 is slightly unclear from the scan, the $v$ function is always the vertex not attached to the arrow in the configuration.

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# Enumeration of graphs arising in invariant theory

I've been working on a talk based on some stuff in Olver's "Classical Invariant Theory" book and have been wondering about a related graph enumeration problem.

Start with a triple $(n,v,e)$ of natural numbers. Take all $\mathbb{Q}$-linear combinations of directed graphs (allowing multiple edges, but no loops) with $v$ vertices, $e$ edges, and each vertex has at most $n$ edges going to it or coming from it. Now, take three relations (images scanned from Olver)

Rule 1:

Rule 2:

Rule 3:

Where the function $v$ with a vertex as subscript next to a graph means that graph multiplied by $n$ minus the number of edges attached to that vertex. (So, for instance, an isolated vertex gets multiplied by $n$)

Denote the space after quotienting by these relations by $V_{n,v,e}$. And so, in final form, my question:

What is $\dim V_{n,v,e}$? Or at least, can we find relatively effective upper bounds?