  In 2011, Haemers asked the following question: If $S$ is the Seidel matrix of
a graph of order $n$ and $S$ is singular, does there exist an eigenvector of
$S$ corresponding to $0$ which has only $\pm 1$ elements?
  In this paper, we construct infinite families of graphs which give a negative
answer to this question. One of our constructions implies that for every
natural number $N$, there exists a graph whose Seidel matrix $S$ is singular
such that for any integer vector in the nullspace of $S$, the absolute value of
any entry in this vector is more than $N$. We also derive some characteristics
of vectors in the nullspace of Seidel matrices, which lead to some necessary
conditions for the singularity of Seidel matrices. Finally, we obtain some
properties of the graphs which affirm the above question.
