Generating a list of tuples $\{G, \eta(G)\}$, where $G$ is connected graph on $n$ vertices and $\eta(G)$ represents its nullity.

Question

Let $G$ be a graph with an adjacency matrix $A(G)$. The nullity of a graph $G$ is defined as the dimension of the null space of $A(G)$.

I want to generate a list of connected simple graphs on $n$ vertices with the respective nullity of its adjacency matrix.

As far as I know, the following command produces all nonisomorphic connected graphs on n=4 vertices using nauty software:

geng -c 4

Can I produce a list of such graphs (for the higher number of vertices, say n=6 or n=7) alongside their nullities using Nauty software only?

OR

Is there any command for producing the nullity of a graph in nauty software?