I am interested in graphs $G=(V,E)$ with vertex set $V$ a finite dimensional vector space and edges in $E$ which are not necessarily related to the linear structure of $V$. So my objects are just graphs with a vertex set with additional structure, and I don't require any special morphisms. (The vector space structure is for combinatorial reasons, to count vertices in terms of dimension.) I'd like to use a striking name for those graphs, but neither do I know if

1) there is already a name for them,

2) there would be collision with other terminology if I named them something like "vector graphs", "vector based graphs", etc.

For example "linear graph" would be a bad choice.

If some graph theory terminology specialist out there could help that would be very nice.