Is there any information on the dimension of the null space of the (signless, i.e., $0-1$) incidence matrix of a simple graph?
Some simple(st) examples and numerical experiment with maple have led me to conjecture that the null space is non-trivial if and only if the graph contains a cycle of even length as an induced subgraph, because in that case it is possible to consider a 2-coloring of the cycle, assign 1 to the red edges and -1 to the blue edges, and extend this function by 0 outside the cycle. But what about the converse implication?
If this property is (true and) known, references would be highly appreciated.
EDIT: By Theorem 8.2.1 in the book of Godsil-Royle, the rank of the incidence matrix is $n-c_0$, where $n$ is the number of nodes and $c_0$ the number of bipartite connected components. By the rank-nullity theorem, it follows that the nullity is then $m-n+c_0$, where $m$ is the number of edges. Hence, it seems the null space is non-trivial if and only if $m+c_0>n$: for instance, this is the case if $G$ is the graph obtained connecting two triangles by one edge. Still: is it possible to identify the null space of the matrix?