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.
翻译:2011年,Haemers问了以下问题:如果美元是按序图的Seidel 矩阵的Seidel 矩阵,美元和美元是单数,那么是否存在一个相当于$0的S$(只有$1美元元素)的源代码?在本文中,我们构建了图形的无限系列,对该问题给出了否定的答案。我们的其中一项构造表明,对于每一个自然数字,每个数字的Seidel 矩阵都有一个图,其Sidel 矩阵的美元是单数,因此,对于任何在空格中的整形矢量为$S$(美元),此矢量中的任何条目的绝对值都大于$。我们还得出了Seidel 矩阵空格中的矢量的一些特性,这些特性为Seidel 矩阵的单一性提供了一些必要的条件。最后,我们获得了证实上述问题的图表的一些属性。