论文标题
关于seidel矩阵零空间中矢量的大声虫的问题
On a question of Haemers regarding vectors in the nullspace of Seidel matrices
论文作者
论文摘要
在2011年，Haemers问以下问题：$ s $是否是订单$ n $和$ s $的Seidel矩阵，是否存在$ s $的特征向量为$ s $，对应于$ 0 $，只有$ \ pm pm 1 $元素？ 在本文中，我们构建了无限的图表家庭，为这个问题提供了负面答案。我们的构造之一意味着，对于每个自然数$ n $，存在一个图表，其Seidel矩阵$ s $是单数的，因此对于$ s $的NullSpace中的任何整数矢量，该矢量中任何条目的绝对值都超过$ n $。我们还得出了Seidel矩阵零空间中向量的某些特征，这导致了Seidel矩阵奇异性的一些必要条件。最后，我们获得了图表的一些属性，这些属性肯定了上述问题。
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.
