Quoting from Algebraic graph theory by Biggs:
If $G$ is a directed graph, then the dimension of cut space is rank of incidence matrix of $G$.
Now, my question is: What happens when $G$ is not directed? Note that the cut space for undirected graph is a vector space over $\mathbb Z_2$.
I don't know enough about how this stuff works for directed graphs (can you just stick random orientations on the edges and then apply the result for directed graphs? or maybe double up each edge so that you have one going each way?), so the below (which approximately follows Diestel) might not be the quickest approach.
Let $G$ be a connected graph with $n$ vertices. Consider a spanning tree $T$ and pick some edge $e$ of $T$. The set $D_e$ of edges in $G$ between the components of $T-e$ is called a fundamental cut.
The fundamental cuts form a basis for the cut space, so the cut space has dimension $n-1$.
(If you want to show that the fundamental cuts form a basis, let $D$ be a cut and consider $D+\sum_{e \in T \cap D}D_e$. This is in the cut space but contains no edge of $T$, so must be empty.)
If your graph isn't connected then you could do the same thing with a (maximal) spanning forest and then compare the answer to the rank of the graph.