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Paper IPM / M / 7907  


Abstract:  
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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