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In 2011, Haemers asked the following question: If $S$ is the Seidel matrix ofa 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 negativeanswer to this question. One of our constructions implies that for everynatural number $N$, there exists a graph whose Seidel matrix $S$ is singularsuch that for any integer vector in the nullspace of $S$, the absolute value ofany entry in this vector is more than $N$. We also derive some characteristicsof vectors in the nullspace of Seidel matrices, which lead to some necessaryconditions for the singularity of Seidel matrices. Finally, we obtain someproperties of the graphs which affirm the above question.

On a question of Haemers regarding vectors in the nullspace of Seidel matrices