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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
$\{-1,0,1\}$-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.
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