Split graphs have nullity equal to the nullity of their biadjacency submatrix plus the dimension of their clique-kernel, which is at most one.
Nulldecompositionofbipartite graphs without cycles of length 0 modulo 4.Linear Algebra and its Applications, 614:176–196, 2021
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On the nullspace of split graphs
Split graphs have nullity equal to the nullity of their biadjacency submatrix plus the dimension of their clique-kernel, which is at most one.