The kernels of the incidence matrices of graphs revisited

In this paper we study the structure of some special bases for the null space of the incidence matrix of a graph. Recently it was shown that if G is a graph with no cut vertex, then G has a {−1, 0, 1}-basis. We generalize this result showing that the statement remains valid for every graph with no cut edge. For the null space of any bipartite graph, we construct a {−1, 0, 1}-basis. For any bipartite graph we obtain the support sizes of all elements in the null space of its incidence matrix. Among other things, we prove that for a graph G, there exists a {−1, 1}-vector for the null space of G if and only if the degree of any vertex of G is even and G has an even number of edges.